Functional renormalization group approach to correlated fermion systems Walter Metzner Max-Planck-Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany Manfred Salmhofer Institut f¨r Theoretische Physik, Universit¨t Heidelberg, Philosophenweg 19, D-69120 Heidelberg, Germany u a arXiv:1105.5289v3 [cond-mat.str-el] 2 Jun 2012 Carsten Honerkamp Institut f¨r Theoretische Festk¨rperphysik and JARA-Fundamentals of Future Information Technology, u o RWTH Aachen University, D-52056 Aachen, Germany Volker Meden Institut f¨r Theorie der Statistischen Physik and JARA-Fundamentals of Future Information Technology, u RWTH Aachen University, D-52056 Aachen, Germany Kurt Sch¨nhammer o Institut f¨r Theoretische Physik, Universit¨t G¨ttingen, Friedrich-Hund-Platz 1, D-37077 G¨ttingen, Germany u a o o Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems, (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals, (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and non-equilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. We present a self-contained derivation of the exact flow equations and describe frequently used truncation schemes. Reviewing selected applications we then show how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems. Contents I. INTRODUCTION A. Motivation B. RG for interacting Fermi systems C. Functional renormalization group D. Scope of the review II. FUNCTIONAL FLOW EQUATIONS A. Generating functionals B. Exact fermionic flow equations C. Expansion in the fields 1. Hierarchy of flow equations 2. Truncations D. Flow parameters 1. Momentum and frequency cutoffs 2. Temperature and interaction flows E. General properties of the RG equations 1. Inductive structure of the RG hierarchy 2. Truncated hierarchies and their iterative solution 3. Running coupling expansion and power counting 4. Self-energy and Fermi surface shift F. Flow equations for observables and correlation functions G. Flow equations for coupled boson-fermion systems 2 2 2 3 4 4 4 6 8 8 10 12 12 14 14 15 15 16 17 18 19 III. COMPETING INSTABILITIES A. Hubbard model and N -patch RG schemes B. Results for the two-dimensional Hubbard model 1. Antiferromagnetism and Superconductivity 2. Ferromagnetism vs. Superconductivity 3. Charge instabilities 4. Flows with self-energy effects C. Pnictide superconductors D. Other systems 21 21 23 24 25 25 25 26 27 IV. SPONTANEOUS SYMMETRY BREAKING A. Fermionic flows B. Flows with Hubbard-Stratonovich fields 28 29 31 33 34 35 V. QUANTUM CRITICALITY A. Hertz-Millis theory B. Full potential flow C. Coupled flow of fermions and order parameter fluctuations 35 VI. CORRELATION EFFECTS IN QUANTUM WIRES AND QUANTUM DOTS A. Quantum transport B. Functional RG in non-equilibrium C. Impurities in Luttinger liquids 36 36 37 38 2 1. A single local impurity–the local sine-Gordon model 38 2. The functional RG approach to the single impurity problem 39 3. Basic wire model 40 4. Numerical solution of improved flow equations 41 5. Resonant tunneling 44 6. Y-junctions 45 7. Non-equilibrium transport through a contacted wire 46 D. Quantum dots 47 1. Spin fluctuations 47 2. Charge fluctuations in non-equilibrium 48 VII. CONCLUSION A. Summary B. Future directions 49 49 50 Acknowledgments 51 A. Wick-ordered flow equations 51 B. Details of power counting 1. Propagator bounds 2. Power counting 3. Improved power counting a. Effects of curvature on power counting b. Uniform improvement from overlapping loops 51 51 52 53 53 54 References energy [eV] 10 1 0.1 0.01 U (Coulomb repulsion) t (kinetic energy, hopping) J (magnetic interaction) kBTc (transition temperature for SC) FIG. 1 (Color online) Important energy scales in hightemperature superconductors of the cuprate family. Magnetic interactions and superconductivity are generated from the kinetic energy (hopping) and the Coulomb repulsion. scales are plagued by infrared divergences, and are therefore often inapplicable even at weak coupling, especially in low dimensions. It is thus natural to treat degrees of freedom with different energy scales successively, descending step by step from higher to lower scales. This is the main idea behind the renormalization group (RG). 55 B. RG for interacting Fermi systems I. INTRODUCTION A. Motivation The Coulomb interaction between electrons in solids leads to a virtually unlimited variety of phenomena, such as magnetic correlations and magnetic order, hightemperature superconductivity, metal-insulator transitions, phase separation and stripes, and the formation of exotic quantum liquid phases. The latter include Luttinger liquids, quantum critical points, and fractional quantum Hall states. Interacting electron systems usually exhibit very distinct behavior on different energy scales. Composite objects and collective phenomena emerge at scales far below the bare energy scales of the microscopic Hamiltonian. For example, in cuprate high-temperature superconductors one bridges three orders of magnitude from the highest scale, the bare Coulomb interaction, via the intermediate scale of short-range magnetic correlations, down to the lowest scale of d-wave superconductivity and other ordering phenomena (see Fig. 1). This diversity of scales is a major obstacle to a straightforward numerical solution of microscopic models, since the most interesting phenomena emerge only at low temperatures and in systems with a large size. It is also hard to deal with by conventional many-body methods, if one tries to treat all scales at once and within the same approximation, for example by summing a subclass of Feynman diagrams. Perturbative approaches which do not separate different Renormalization group methods have a long tradition in the theory of interacting Fermi systems. Already in the 1970s, various versions of the RG have been used to deal with infrared singularities arising in one-dimensional Fermi systems (S´lyom, 1979). Naturally, the RG was o also applied to (mostly bosonic) effective field theories describing critical phenomena at continuous classical or quantum phase transitions in interacting Fermi systems (Fradkin, 1991; Sachdev, 1999). Renormalization group approaches dealing with interacting fermions in arbitrary dimensions d have been developed much later. Due to the extended (not point-like) geometry of the Fermi surface singularity in dimensions d > 1, the renormalization group flow cannot be reduced to a finite number of running couplings. However, the main reason for the delayed development of a comprehensive RG approach for interacting Fermi systems in higher dimensions was probably not this difficulty, but rather a lack of motivation. The few infrared singularities appearing in three-dimensional Fermi systems could usually be handled by simple resummations of perturbation theory (Abrikosov et al., 1963; Nozi`res, 1964). e Triggered by the issue of non-Fermi liquid behavior in two-dimensional systems, and the related discussion on the validity of perturbation theory, systematic RG approaches to interacting Fermi systems in arbitrary dimensions have been developed by various groups in the early 1990s. Aiming at a mathematical control of interacting 3 Fermi systems, Feldman and Trubowitz (1990, 1991), and independently Benfatto and Gallavotti (1990a,b), have formulated a rigorous fermionic version of Wilson’s momentum-shell RG (Wilson and Kogut, 1974). Important rigorous results have indeed been obtained in onedimensional (Benfatto et al., 1994) and two-dimensional (Benfatto et al., 2006; Disertori and Rivasseau, 2000; Feldman et al., 2003, 2004, 1992, 1996) systems. The essential message from these results is that no hitherto unknown instabilities or non-perturbative effects occur in Fermi systems with sufficiently weak short-range interactions, at least in the absence of special features such as Van Hove singularities at the Fermi level. The Wilsonian RG for interacting Fermi systems was popularized among (non-mathematical) physicists by Shankar (1991, 1994) and Polchinski (1993), who presented some of the main ideas in a pedagogical style. In particular, they provided an intuitive RG perspective of Fermi liquid theory. Subtleties associated with the singularities of the interaction vertex for forward scattering were clarified a bit later (Chitov and S´n´chal, 1995; e e Metzner et al., 1998). A Hamiltonian-based RG interpretation of Fermi liquid theory was presented by Hewson (1994), who discussed not only translation invariant systems but also models for magnetic impurities in metals. As an alternative to the Wilsonian RG one may also use flow equations for Hamiltonians based on infinitesimal unitary transformations, which make the Hamiltonian successively more diagonal (Wegner, 1994). This approach has been used successfully for quantum impurity models and other systems (Kehrein, 2006). A weak coupling truncation of the flow equations has been applied to identify instabilities of the two-dimensional Hubbard model (Grote et al., 2002). There is much current interest in RG methods for correlated fermions in non-equilibrium. The perturbative RG (Mitra et al., 2006; Rosch et al., 2001), Wilson’s numerical RG (Anders and Schiller, 2005), as well as Wegner’s flow equation approach (Kehrein, 2006) were extended to non-equilibrium, and real-time RG methods were developed (Schoeller, 2000, 2009; Schoeller and K¨nig, 2000). o C. Functional renormalization group The Wilsonian RG is not only useful for a deeper and partially even rigorous understanding of interacting fermion systems. A specific version of Wilson’s RG known as exact or functional RG turned out to provide a valuable framework for computational purposes. Approximations derived from exact functional flow equations have played an increasingly important role in the last decade. These developments are the central topic of this review. Exact flow equations describe the evolution of a generating functional for all many-particle Green or vertex functions as a function of a flow parameter Λ, usually an infrared cutoff. They can be derived relatively easily from a functional integral representation of the generating functional. Exact flow equations have been known since the early years of the RG, starting with the work of Wegner and Houghton (1973). Polchinski (1984) employed an exact flow equation to formulate a relatively simple proof of renormalizability of the Φ4 -theory in four dimensions. Renormalizability proofs can be further simplified by using a Wick-ordered variant of Polchinski’s equation (Wieczerkowski, 1988). For computational purposes the exact flow equation for the effective action, first derived in the context of bosonic field theories by Wetterich (1993) turned out to be most convenient. The effective action ΓΛ [φ] is the generating functional for one-particle irreducible vertex functions. The latter are obtained by taking derivatives with respect to the source field φ. The flow parameter Λ describes a regularization of the underlying bare action, which regularizes infrared divergencies in perturbation theory. The regularization is removed at the end of the flow, say for Λ → 0. The initial regulator (for Λ = Λ0 ) can be chosen such that ΓΛ0 [φ] is given by the bare action. The flow of ΓΛ [φ] then provides a smooth interpolation between the bare action of the system and the final effective action Γ[φ], from which any desired information can be extracted. This flow is determined by an exact functional differential equation (Ellwanger and Wetterich, 1994; Morris, 1994; Wetterich, 1993). Expanding in the fields one obtains a hierarchy of flow equations for the one-particle irreducible vertex functions. The advantage of that hierarchy compared to others, obtained, for example, from Polchinski’s equation, is that self-energy feedback is included automatically and no one-particle reducible terms appear. The expression functional RG stems from the feature that the exact flow equations describe the flow of a functional or (equivalently) of a hierarchy of functions. An important difference compared to Wilson’s original formulation is that a complete set of source fields is kept in the flowing generating functionals, not only those corresponding to scales below Λ. Hence, the full information on the properties of the system remains accessible, not only the low energy or long wavelength behavior. Exact flow equations can be solved exactly only in special cases, where the underlying model can also be solved exactly (and more easily) by other means.1 However, the functional RG is a valuable source for devising powerful new approximation schemes, which can be obtained by truncating the hierarchy and/or by a simplified parametrization of the Green or vertex functions. These approximations have several distinctive advantages: i) 1 An instructive example is provided by the exact solution of the Tomonaga-Luttinger model via functional RG flow equations (Sch¨tz et al., 2005). u 4 they have a renormalization group structure built in, that is, scales are handled successively and infrared singularities are thus treated properly; ii) they can be applied directly to microscopic models, not only to effective field theories which capture only some asymptotic behavior; iii) they are physically transparent, for example one can see directly how and why new correlations form upon lowering the scale; iv) one can use different approximations at different scales. Small steps from a scale Λ to a slightly smaller scale Λ′ are much easier to control than an integration over all degrees of freedom in one shot, and one can take advantage of the flexibility provided by the choice of a suitable flow parameter. Approximations derived from exact flow equations have been applied in many areas of quantum field theory and statistical physics (Berges et al., 2002). In the context of interacting Fermi systems, functional RG methods were first used for an unbiased stability analysis of the two-dimensional Hubbard model (Halboth and Metzner, 2000a; Honerkamp et al., 2001; Zanchi and Schulz, 1998, 2000). Since then, approximations derived within the functional RG framework have been applied to numerous interacting fermion systems. D. Scope of the review This review provides a thorough introduction to the functional RG in the context of interacting Fermi systems. It should serve as a manual and reference for manybody theorists who wish to apply approximations based on the functional RG to their own problem of interest. We will first describe the functional RG framework and derive in particular the exact flow equations, which are the starting point for approximations. We will discuss general aspects related to the flow equations such as the choice of cutoffs, power counting and truncations. Links to the use of flow equations in the mathematical literature will be pointed out along the way. We will then review some of the most interesting applications of truncated functional RG equations (see table of content). Our aim is not to deliver an exhaustive overview of all applications, but rather to show via selected applications how the functional RG method can be fruitfully used. The functional RG was recently extended to Fermi systems out of equilibrium (Gezzi et al., 2007; Jakobs, 2003; Jakobs et al., 2007a, 2010a,b; Karrasch et al., 2010a,c). In the derivation of the flow equations in Sec. II we restrict ourselves to the equilibrium formalism. Functional RG flow equations for non-equilibrium Keldysh Green and vertex functions can be derived in close analogy (Gezzi et al., 2007; Jakobs, 2003, 2010; Karrasch, 2010). The necessary extensions are briefly mentioned when discussing the application of this method to finite bias steady state transport through correlated quantum wires and quantum dots in Sec. VI. A number of reviews with a focus on the functional RG are already available. Mathematically rigorous de- velopments until the end of the last millenium were summarized in a book by Salmhofer (1999), a large portion of which is dedicated to interacting Fermi systems. Examples of approximations derived from the exact flow equation for the effective action with many applications in quantum field theory and statistical physics were presented in the review article by Berges et al. (2002). A detailed introduction to the functional RG in a textbook style supplemented by selected applications (including Fermi systems) can be found in the recent book by Kopietz et al. (2010). II. FUNCTIONAL FLOW EQUATIONS In this section we present the general functional RG framework. The reader should be familiar with the functional integral formalism for quantum manybody systems, as described in classic textbooks such as Negele and Orland (1987). After introducing the generating functionals for Green and vertex functions in Sec. II.A, we derive the exact functional flow equations in Sec. II.B. The flow equation (35) for the effective action ΓΛ is the central equation of this review. Expanding in the fields we derive the hierarchy of flow equations for vertex functions in Sec. II.C, which is the starting point for approximations. Possible choices of flow parameters are reviewed in Sec. II.D. The general structure of the RG hierarchy and power counting are discussed in Sec. II.E, with various references to the closely related mathematical literature. Sec. II.F is dedicated to flow equations for observables and correlation functions. Coupled flow equations for fermions and bosons, which are useful for studies of spontaneous symmetry breaking and quantum criticality, are derived in Sec. II.G. A. Generating functionals We consider a system of interacting fermions which can ¯ be described by Grassmann fields ψ, ψ, and an action of the form ¯ ¯ ¯ S[ψ, ψ] = −(ψ, G−1 ψ) + V [ψ, ψ] , 0 (1) ¯ where V [ψ, ψ] is an arbitrary many-body interaction, and G0 is the propagator of the non-interacting system. The bracket (., .) is a shorthand notation −1 −1 ¯ for the sum x ψ(x) (G0 ψ)(x), where (G0 ψ)(x) = −1 ′ ′ x′ G0 (x, x ) ψ(x ). The Grassmann field index x collects the quantum numbers of a suitable single-particle basis set and imaginary time or frequency. In case of continuous variables, the sum over x includes the appropriate integrals. Prefactors such as temperature or volume factors depend on the representation (e.g. real or momentum space) and are therefore not written in this general part. A two-particle interaction has the general 5 form 1 ¯ V [ψ, ψ] = 4 ¯ ¯ V (x′ , x′ ; x1 , x2 ) ψ(x′ )ψ(x′ )ψ(x2 )ψ(x1 ) . 1 2 1 2 x1 ,x2 x′ ,x′ 1 2 (2) In particular, for spin- 1 fermions with a single-particle 2 basis labeled by momentum k and spin orientation σ, one has x = (k0 , k, σ), where k0 is the fermionic Matsubara frequency. If the bare part of the action is translation and spin-rotation invariant, the bare propagator has the diagonal and spin-independent form G0 (x, x′ ) = ′ δk0 k0 δkk′ δσσ′ G0 (k0 , k) with G0 (k0 , k) = 1 , ik0 − ξk (3) where ξk = ǫk − µ is the single-particle energy relative to the chemical potential. Connected Green functions can be obtained from the generating functional (Negele and Orland, 1987) ¯ ¯ η ¯ DψDψ e−S[ψ,ψ] e(¯,ψ)+(ψ,η) , G[η, η ] = − ln ¯ (4) ¯ ¯ where DψDψ . . . = x dψ(x)dψ(x) . . . . Completing squares yields the identity ¯ −1 ¯ DψDψ e(ψ,G0 ¯ ψ) (¯,ψ)+(ψ,η) η e η = Z0 e(−¯,G0 η) , G[η, η ] = − ln Z + (¯, Gη) + ¯ η 1 (2!)2 (5) ¯ ¯ −1 where Z0 = DψDψ e(ψ, G0 ψ) is the partition function of the non-interacting system. Hence G[η, η ] = − ln Z0 + ¯ ¯ (¯, G0 η) in the non-interacting case V [ψ, ψ] = 0. For η vanishing source fields, G[0, 0] = − ln Z, where Z= ¯ ¯ DψDψ e−S[ψ,ψ] (6) is the partition function of the interacting system. The connected m-particle Green functions are given by G(2m) (x1 , . . . , xm ; x′ , . . . , x′ ) = 1 m ¯ ¯ − ψ(x1 ) . . . ψ(xm )ψ(x′ ) . . . ψ(x′ ) c = m 1 ∂ 2m G[η, η ] ¯ m (−1) ∂ η (x1 ) . . . ∂ η (xm )∂η(x′ ) . . . ∂η(x′ ) ¯ ¯ m 1 , (7) η,¯=0 η where . . . c is the connected average of the product of Grassmann variables between the brackets. The oneparticle Green function G(2) is the propagator of the interacting system, which we will usually denote without the superscript by G. Expanding G[η, η ] in the fields ¯ yields a formal power series with the connected Green functions as coefficients, G(4) (x1 , x2 ; x′ , x′ ) η (x1 )¯(x2 )η(x′ )η(x′ ) + . . . . η 1 2 ¯ 2 1 (8) x1 ,x2 ,x′ ,x′ 1 2 Renormalization group equations are most conveniently formulated for the Legendre transform of G[η, η ], ¯ the socalled effective action ¯ ¯ Γ[ψ, ψ] = (¯, ψ) + (ψ, η) + G[η, η ] , η ¯ (9) ¯ with ψ = −∂G/∂ η and ψ = ∂G/∂η , which generates one¯ particle irreducible vertex functions (Negele and Orland, 1987) Γ(2m) (x′ , . . . , x′ ; x1 , . . . , xm ) = 1 m ¯ ∂ 2m Γ[ψ, ψ] ¯ ¯ ∂ ψ(x′ ) . . . ∂ ψ(x′ )∂ψ(xm ) . . . ∂ψ(x1 ) ψ,ψ=0 ¯ m 1 . (10) ¯ In the non-interacting case one obtains Γ[ψ, ψ] = ¯ − ln Z0 − (ψ, G−1 ψ). The Legendre correspondence be0 tween the functionals G and Γ yields relations between the connected Green functions G(2m) and the vertex functions Γ(2m) . In particular, Γ(2) = G−1 = G−1 − Σ , (11) 0 where Σ is the self-energy. The connected two-particle Green function is related to the two-particle vertex by ′ ′ G(x1 , y1 ) G(x2 , y2 ) G(4) (x1 , x2 ; x′ , x′ ) = 1 2 ′ ′ y1 ,y2 ,y1 ,y2 ′ ′ × Γ(4) (y1 , y2 ; y1 , y2 ) G(y1 , x′ ) G(y2 , x′ ) , 1 2 (12) while the three-particle Green function G(6) = G3 Γ(6) G3 + G3 Γ(4) GΓ(4) G3 involves Γ(4) and Γ(6) . More generally, the connected m-particle Green functions are obtained by adding all possible trees that can be formed with vertex functions of equal or lower order and G-lines (Negele and Orland, 1987). The effective action obeys the reciprocity relations ∂Γ = −¯ , η ∂ψ ∂Γ ¯ =η. ∂ψ (13) The second functional derivatives of G and Γ with respect to the fields are also reciprocal (Negele and Orland, 1987). We define the matrices of second derivatives at finite fields 6  G(2) [η, η ] = −  ¯ and  ¯ Γ(2) [ψ, ψ] =  ∂2G ∂ η (x)∂η(x′ ) ¯ 2 ∂ G − ∂η(x)∂η(x′ ) 2 ∂ − ∂ η(x)∂G (x′ ) ¯ η ¯ 2 ∂ G ∂η(x)∂ η (x′ ) ¯ ∂2 Γ ¯ ∂ ψ(x′ )∂ψ(x) ∂2 Γ ¯ ¯ ∂ ψ(x′ )∂ ψ(x) 2 2 ∂ Γ ∂ψ(x′ )∂ψ(x) ∂ Γ ¯ ∂ψ(x′ )∂ ψ(x)  = where the matrix elements in the second matrix of the last equation are just a more conventient notation for those in the first matrix. The reciprocity relation for the second derivatives reads ¯ Γ(2) [ψ, ψ] = G(2) [η, η ] ¯ −1 . (16) Note that anomalous components are involved as long as the source fields are finite. Only at η = η = 0 and ¯ ¯ ψ = ψ = 0, and in the absence of U (1) charge symmetry −1 breaking one has the simple relation Γ(2) = G(2) . Another useful generating functional is the effective interaction (Salmhofer, 1999) V[χ, χ] = − ln ¯ 1 Z0 ¯ −1 ¯ DψDψ e(ψ,G0 ¯ ¯ ψ) −V [ψ+χ,ψ+χ] e . (17) A simple substitution of variables yields the relation V[χ, χ] = G[η, η ] + ln Z0 − (¯, G0 η) , ¯ ¯ η (18) where χ = G0 η and χ = Gt η . Here Gt is the transposed ¯ 0¯ 0 bare propagator, that is, Gt (x, x′ ) = G0 (x′ , x). Hence, 0 functional derivatives of V[χ, χ] with respect to χ and χ ¯ ¯ generate connected Green functions with bare propagators amputated from external legs in the corresponding Feynman diagrams. The term ln Z0 −(¯, G0 η) cancels the η non-interacting part of G[¯, η] such that V[χ, χ] = 0 for η ¯ ¯ V [ψ, ψ] = 0. The effective interaction V can also be expressed via functional derivatives, instead of a functional integral: 1 ¯ ¯ ¯ ¯ −1 DψDψ e(ψ,G0 ψ) e−V [ψ+χ,ψ+χ] Z0 1 −V [∂η ,∂η ] ¯ ¯ ¯ −1 = e DψDψ e(ψ,G0 ψ) Z0 = e −V [∂η ,∂η ] ¯ = e ∆G0 e e η,¯=0 η η,¯=0 η (∂χ ,G0 ∂χ ) (¯,χ)+(η,χ) η ¯ ¯ −V [χ,χ] ¯ e , ¯ ψ(x)ψ(x′ ) ¯ ¯ ψ(x)ψ(x′ ) ψ(x)ψ(x′ ) ¯ ψ(x)ψ(x′ ) , ¯ ¯ ¯¯ ¯ ∂∂Γ[ψ, ψ](x′ , x) ∂ ∂Γ[ψ, ψ](x′ , x) ¯ ′ , x) ∂ ∂Γ[ψ, ψ](x′ , x) ¯ ¯ ∂∂Γ[ψ, ψ](x (14) , (15) with the functional Laplacian ∆G0 = ∂χ , G0 ∂χ = ¯ x,x′ ∂ ∂ G0 (x, x′ ) . (20) ∂χ(x) ∂ χ(x′ ) ¯ It is sometimes convenient (see Sec. II.G) to combine ¯ the fields ψ and ψ in a Nambu-type field ψ(x) ¯ ψ(x) Ψ(x) = , (21) and similarly for the source fields η and η , ¯ H(x) = η(x) −¯(x) η . (22) The minus sign in the definition of H makes sure that the ¯ source term (¯, ψ) + (ψ, η) appearing in the definition of η G, and also in the Legendre transform relating G and Γ, ¯ can be written concisely as (H, Ψ). In Nambu notation, the matrices of second derivatives of G and Γ have the compact form ∂ 2G G(2) [H] = − ¯ ∂ H(x)∂H(x′ ) (23) and Γ(2) [Ψ] = ∂2Γ , ¯ ∂ Ψ(x′ )∂Ψ(x) (24) respectively. In this section we derive exact flow equations describing the evolution of the generating functionals defined above, as a function of a flow parameter Λ which parametrizes a modification of the bare propagator G0 . Usually Λ is an infrared cutoff or another scale dependence. For example, in a translation invariant system one may impose a momentum cutoff, modifying G0 to ¯ η η ¯ ¯ = e−V [∂η ,∂η ] e(¯,G0 η) e(¯,χ)+(η,χ) =− B. Exact fermionic flow equations ¯ e−V[χ,χ] = η ¯ × e(¯,ψ+χ)+(η,ψ+χ)  η,¯=0 η (19) GΛ (k0 , k) = 0 θΛ (k) , ik0 − ξk (25) 7 where θΛ (k) is a function that vanishes for |ξk | ≪ Λ and tends to one for |ξk | ≫ Λ. In this way the infrared singularity of the propagator at k0 = 0 and ξk = 0 (corresponding to the non-interacting Fermi surface in k-space) is cut off at the scale Λ. A simple choice for θΛ (k), which was often used in numerical solutions of truncated flow equations, is θΛ (k) = Θ(|ξk | − Λ) , (26) where Θ is the step function. With this choice momenta close to the Fermi surface are strictly excluded, as illustrated in Fig. 2 for a two-dimensional lattice fermion system. Alternatively, one may also use a smooth cutoff π ky 0 −π −π 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 0 kx ∆GΛ 0 ¯ ¯ V Λ0 [χ, χ] = V [χ, χ] , e−V . (27) ¯ GΛ = G0 − GΛ , 0 0 (28) which has support on scales below Λ, we can write ∆GΛ +∆GΛ ¯ 0 0 e−V = e 2 Λ ˙0 ∂ V − tr GΛ ∂ χ∂χ ¯ , (30) d ˙0 where GΛ = dΛ GΛ and tr denotes the trace trA = 0 x A(x, x). Its derivation is simple: π At the highest energy scale Λ0 one has GΛ0 = 0, and 0 thus V Λ0 = V . Hence, V Λ interpolates smoothly between the bare interaction V and the generating functional V. Introducing the soft mode propagator e−V = e∆G0 e−V = e ∂V Λ ˙ Λ ∂V Λ , G0 ∂χ ∂χ ¯ In the second step we have used Eq. (27). With the initial condition function. In the absence of translation invariance it is more convenient to use a frequency cutoff instead of a momentum cutoff. The cutoff excludes ”soft modes” below the scale Λ from the functional integral. Instead of a cutoff one may also choose other flow parameters such as temperature. The various possibilities will be discussed more extensively in Sec. II.D. For the derivation of the flow equations it does not matter how GΛ depends on Λ. 0 The bare action constructed with GΛ (instead of G0 ) 0 ¯ will be denoted by S Λ [ψ, ψ], and the generating functionals introduced in Sec. II.A by G Λ [η, η ], V Λ [χ, χ], and ¯ ¯ ¯ ΓΛ [ψ, ψ], respectively. The original functionals G, V and Γ are recovered in the limit Λ → 0. In the presence of a cutoff, Eq. (19) becomes Λ d Λ V [χ, χ] = − ¯ dΛ Λ d Λ d Λ V = −eV e−V dΛ dΛ Λ Λ Λ d ∆ Λ e G0 e−V = −eV ∆GΛ e−V = −eV ˙ 0 dΛ = right-hand side of Eq. (30) . FIG. 2 (Color online) Momentum space region around the Fermi surface excluded by a sharp momentum cutoff for fermions with a tight-binding dispersion on a two-dimensional square lattice (lattice constant = one). e−V = e V Λ obviously plays a dual role: It is the generating functional for (amputated) Green functions of a system with a cutoff Λ, and at the same time the interaction for the remaining low energy degrees of freedom (Morris, 1994; Salmhofer, 1999). The effective interaction satisfies the following exact renormalization group equation (Brydges and Wright, 1988; Salmhofer, 1999) ∆GΛ ¯ 0 Λ e−V . (29) (31) the RG equation determines the flow of V Λ uniquely for all Λ < Λ0 . The initial value Λ0 must be chosen such that GΛ0 vanishes. For a sharp momentum cutoff, Λ0 can be 0 chosen as the maximal value of |ξk |; for a frequency cutoff Λ0 = ∞. An expansion of the functional V Λ [χ, χ] in the renor¯ malization group equation (30) in powers of χ and χ leads ¯ to the fermionic analog of Polchinski’s (Polchinski, 1984) flow equations for amputated connected m-particle Green functions V (2m)Λ . From the flow equation for V Λ , Eq. (30), and the relation (18) applied to V Λ and G Λ , one obtains an exact flow equation for G Λ : d Λ G [η, η ] = ¯ dΛ 2 Λ ∂G Λ ˙ Λ ∂G Λ ˙ ∂ G +tr QΛ , Q0 0 ∂η ∂η ¯ ∂ η ∂η ¯ , (32) where QΛ = (GΛ )−1 and the dot denotes a Λ-derivative. 0 0 This flow equation can also be derived more directly, by applying a Λ-derivative to the functional integral representation of G Λ . The flow equations for G(2m)Λ and V (2m)Λ generate, among others, also one-particle reducible terms, which require some special care. In this respect the flow equations for one-particle irreducible vertex functions Γ(2m)Λ , obtained from the scale-dependent effective action, ¯ ¯ ΓΛ [ψ, ψ] = (¯Λ , ψ) + (ψ, η Λ ) + G Λ [η Λ , η Λ ] , η ¯ (33) are easier to handle. Note that η Λ and η Λ are Λ¯ ¯ dependent functions of ψ and ψ, as they are determined by the Λ-dependent equations ψ = −∂G Λ /∂ η and ¯ ¯ ψ = ∂G Λ /∂η. Since the Λ-dependence does not change 8 the structure of the action as a function of the fields, all standard relations between the connected Green functions G(2m) and the vertex functions Γ(2m) carry over to the ones for G(2m)Λ and Γ(2m)Λ . d ¯ The Λ-derivative of ΓΛ can be written as dΛ ΓΛ [ψ, ψ] = d Λ ¯ d η Λ ) + d G Λ [η Λ , η Λ ], where the deriva( dΛ η , ψ) + (ψ, dΛ ¯ ¯ dΛ tive in front of G Λ acts also on the Λ-dependence of ¯ η Λ and η Λ . Due to the relations ∂G Λ /∂η = ψ and ¯ Λ ∂G /∂ η = −ψ, most terms cancel and one obtains ¯ d Λ Λ Λ d Λ ¯ Γ [ψ, ψ] = G [η , η ] ¯ dΛ dΛ η Λ ,¯Λ η fixed . (34) Inserting the flow equation (32) for G Λ and using the reciprocity relations (13) and (16), one obtains the exact functional flow equation for the effective action d Λ 1 ¯ ¯ ¯ ˙ ˙ Γ [ψ, ψ] = − ψ, QΛ ψ − tr QΛ Γ(2)Λ [ψ, ψ] 0 0 dΛ 2 −1 . (35) ¯ Here Γ(2)Λ [ψ, ψ] is the matrix of second functional derivatives defined in Eq. (15), and QΛ = 0 QΛ 0 0 0 −QΛt 0 = diag(QΛ , −QΛt ) , 0 0 (36) where QΛt (x, x′ ) = QΛ (x′ , x). 0 0 Alternative definitions of the effective action ΓΛ , differing by interaction-independent terms, have also been used. One variant is to normalize the functional integral Λ defining G Λ at V = 0, dividing by Z0 . This yields an Λ Λ additional contribution ln Z0 to G and to its Legendre transform ΓΛ . In the flow equation for ΓΛ this leads to ˙0 0 an additional term tr(QΛ GΛ ), which is field independent and therefore does not couple to the other contributions (Salmhofer and Honerkamp, 2001). Another variant is (Berges et al., 2002; Ellwanger and Wetterich, 1994) ¯ ¯ ¯ ΓΛ [ψ, ψ] = ΓΛ [ψ, ψ] + (ψ, RΛ ψ) , R (37) where RΛ = QΛ − Q0 . The additional quadratic term 0 cancels the first (trivial) term in the flow equation (35) for ΓΛ , and one obtains the equivalent flow equation d Λ 1 (2)Λ ¯ ˙ ¯ Γ [ψ, ψ] = − tr RΛ ΓR [ψ, ψ] + RΛ dΛ R 2 −1 , (38) where RΛ = diag RΛ , −RΛt . The functional ΓΛ and R its analog for bosonic fields is known as effective average action in the literature (Berges et al., 2002). Both ΓΛ R and ΓΛ tend to the same effective action Γ in the limit Λ → 0, where RΛ vanishes. At the initial scale Λ0 , one ¯ ¯ has ΓΛ0 [ψ, ψ] = S[ψ, ψ], while R ¯ ¯ ¯ ¯ Γ [ψ, ψ] = −(ψ, QΛ0 ψ) + V [ψ, ψ] = S Λ0 [ψ, ψ] 0 ¯ ¯ (39) = S[ψ, ψ] − (ψ, RΛ0 ψ) . the final effective action Γ, while ΓΛ interpolates between the regularized bare action S Λ0 and Γ. On the other hand, the functional ΓΛ has the advantage that its second functional derivative directly yields the inverse propagator (GΛ )−1 without the need to add RΛ . In Appendix A we present yet another version of exact flow equations, based on a Wick ordered effective interaction. That version also contains one-particle reducible contributions, but it has the distinct advantage that the vertices are connected by propagators with an energy scale at or below Λ. This facilitates a systematic power counting (Salmhofer, 1999), and also a numerical evaluation of flow equations, since the integration regions shrink upon lowering Λ. It is instructive to compare the functional RG flow equations with the traditional Wilsonian momentum shell RG (Wilson and Kogut, 1974), which was applied to Fermi systems by Shankar (1991, 1994) and Polchinski (1993). In the commonly used version of Wilson’s RG, the flow of the effective action is computed only for soft fields, that is, for fields with energy or momentum variables below the scale Λ, while in the functional RG the effective action with unrestricted source fields is computed. This allows for a direct calculation of correlations functions with arbitrary external variables such as momenta or Matsubara frequencies. Furthermore, in the traditional implementations of Wilson’s RG the integration of degrees of freedom is combined with a rescaling of momenta and fields, which is chosen such that the momentum space and certain terms in the quadratic part of the action remain invariant during the flow. This facilitates the classification of interactions as relevant, marginal or irrelevant, and helps to identify fixed points of the flow. The functional RG flow equations derived above do not involve any rescaling. Rescaling momentum space in a shell around the Fermi surface requires a non-linear transformation in dimensions d > 1, which spoils the simple linear form of momentum conservation (Kopietz and Busche, 2001; Metzner et al., 1998; Shankar, 1994), and is therefore of questionable value. Power-counting can be done also without rescaling, as shown in Sec. II.E. Rescaling of the fields can be implemented easily by a simple substitution of variables (Kopietz and Busche, 2001; Shankar, 1994). However, in many applications of the functional RG, quantitative results including power-laws with anomalous scaling dimensions are obtained simply by direct calculation of the (unscaled) physical quantities. C. Expansion in the fields Λ0 Hence, ΓΛ has the attractive feature that it interpolates R smoothly between the (unregularized) bare action S and 1. Hierarchy of flow equations The functional flow equation for the effective action can be expanded in powers of the fields. To this end we 9 ¯ where A(2m)Λ [ψ, ψ] is homogeneous of degree 2m in the fields, expand the effective action as ¯ ΓΛ [ψ, ψ] = ∞ m=0 ¯ A(2m)Λ [ψ, ψ] , (−1)m ¯ A(2m)Λ [ψ, ψ] = (m!)2 (40) ¯ ¯ Γ(2m)Λ (x′ , . . . , x′ ; x1 , . . . , xm ) ψ(x′ ) . . . ψ(x′ )ψ(xm ) . . . ψ(x1 ) , 1 m 1 m for m ≥ 1. The field-independent constant A(0)Λ yields the grand canonical potential: A(0)Λ = T −1 ΩΛ . (42) Here we have restored the explicit temperature factor, since it is independent of the representation of the fields. To expand the inverse of Γ(2)Λ on the right hand side of the flow equation, we isolate the field-independent part of Γ(2)Λ as ¯ ¯ ˜ Γ(2)Λ [ψ, ψ] = (GΛ )−1 − ΣΛ [ψ, ψ] , (43) equations for ΣΛ , the two-particle vertex Γ(4)Λ , and the higher-order vertices Γ(6)Λ , Γ(8)Λ , etc. The first three equations in this hierarchy are shown diagrammatically in Fig. 3. Note that only one-particle irreducible oneloop diagrams contribute, and internal lines are dressed by self-energy corrections. The hierarchy does not close at any finite order, since the flow of each vertex Γ(2m)Λ receives a contribution from a tadpole diagram involving Γ(2m+2)Λ , and m-particle vertices with arbitrary m are generated by the flow, irrespective of their presence in the bare action. SΛ where Λ G = (2)Λ Γ ¯ [ψ, ψ] −1 ¯ ψ,ψ=0 Λ Λt = diag(G , −G ) (44) ¯ ¯ ¯¯ ¯ ∂∂ΓΛ [ψ, ψ] ∂ ∂ΓΛ [ψ, ψ] Λ ¯ ∂ ∂ΓΛ [ψ, ψ] ¯ ¯ ∂∂Γ [ψ, ψ] + GΛ −1 d Λ ¯ ˙ ¯ ˙0 Γ [ψ, ψ] = −tr QΛ GΛ − ψ, QΛ ψ + 0 dΛ 1 ¯ ¯ ¯ ˜ ˜ ˜ tr SΛ ΣΛ [ψ, ψ] + ΣΛ [ψ, ψ] GΛ ΣΛ [ψ, ψ] + . . . 2 , (46) where ˙ SΛ = diag(S Λ , −S Λt ) = −GΛ QΛ GΛ . 0 (47) Using the Dyson equation (GΛ )−1 = QΛ − ΣΛ , the so0 called single-scale propagator S Λ can also be written as Λ-derivative of the propagator at fixed self-energy, ΣΛ fixed . Γ (4)Λ = (48) ¯ The expansion of the flow equation in powers of ψ, ψ is now straightforward and leads to a hierarchy of flow SΛ SΛ + (4)Λ Γ (4)Λ GΛ Γ Γ (6)Λ Γ (4)Λ . (45) ¯ ˜ Note that ΣΛ [ψ, ψ] contains all contributions to ¯ Γ(2)Λ [ψ, ψ] which are at least quadratic in the fields. We −1 ˜ −1 GΛ as a can now expand Γ(2)Λ = 1 − GΛ ΣΛ geometric series. Inserted in (35), this yields d SΛ = GΛ dΛ d Λ Σ dΛ d (4)Λ = Γ dΛ is the full propagator, and (cf. Eq. (15)) ¯ ˜ ΣΛ [ψ, ψ] = − (41) x1 ,...,xm x′ ,...,x′ m 1 SΛ d (6)Λ Γ = dΛ Γ (4)Λ GΛ Λ Λ Γ (4)Λ S + Γ (4)Λ SΛ Γ (6)Λ + GΛ Γ (8)Λ G FIG. 3 (Color online) Diagrammatic representation of the flow equations for the self-energy ΣΛ , the two-particle vertex Γ(4)Λ , and the three-particle vertex Γ(6)Λ in the one-particle irreducible version of the functional RG. Lines with a dash correspond to the single scale propagator S Λ , the other lines to the full propagator GΛ . Let us derive explicitly the first two flow equations from the hierarchy. Comparing coefficients of quadratic ¯ contributions (proportional to ψψ) to the exact flow equation yields d (2)Λ ¯ ¯ ˙ A = −(ψ, QΛ ψ) − tr S Λ ∂∂A(4)Λ 0 dΛ . (49) Inserting Eq. (41), and using Γ(2)Λ = QΛ − ΣΛ , one ob0 tains the flow equation for the self-energy, d Λ ′ Σ (x , x) = dΛ S Λ (y, y ′ ) Γ(4)Λ (x′ , y ′ ; x, y) . y,y ′ (50) 10 Comparing coefficients of quartic contributions (propor¯ tional to (ψψ)2 ) yields 1 d (4)Λ ¯ ¯ ¯ ¯ A = tr S Λ ∂∂A(4)Λ GΛ ∂∂A(4)Λ + S Λt ∂ ∂A(4)Λ GΛt ∂ ∂A(4)Λ dΛ 2 1 ¯ ¯¯ ¯¯ − tr S Λ ∂ ∂A(4)Λ GΛt ∂∂A(4)Λ + S Λt ∂∂A(4)Λ GΛ ∂ ∂A(4)Λ − tr S Λ ∂∂A(6)Λ . 2 (51) Inserting Eq. (41), one obtains the flow equation for the two-particle vertex, d (4)Λ ′ ′ Γ (x1 , x2 ; x1 , x2 ) = dΛ ′ ′ GΛ (y1 , y1 ) S Λ (y2 , y2 ) ′ y1 ,y1 ′ y2 ,y2 × ′ ′ Γ(4)Λ (x′ , x′ ; y1 , y2 )Γ(4)Λ (y1 , y2 ; x1 , x2 ) 1 2 − ′ ′ ′ ′ Γ(4)Λ (x′ , y2 ; x1 , y1 )Γ(4)Λ (y1 , x′ ; y2 , x2 ) + (y1 ↔ y2 , y1 ↔ y2 ) 1 2 + ′ ′ ′ ′ Γ(4)Λ (x′ , y2 ; x1 , y1 )Γ(4)Λ (y1 , x′ ; y2 , x2 ) + (y1 ↔ y2 , y1 ↔ y2 ) 2 1 − S Λ (y, y ′ ) Γ(6)Λ (x′ , x′ , y ′ ; x1 , x2 , y) . 1 2 Note that there are several distinct contributions involving two two-particle vertices, corresponding to the familiar particle-particle, direct particle-hole, and crossed particle-hole channel, respectively, as shown diagrammatically in Fig. 4. Similarly, one can obtain the flow 1 1’ pp 2 2’ 1’ 1 1’ ph 2 2’ 2 2’ 1 2’ 1 2’ ph’ 2 The flow equation (35) and the ensuing equations for the vertex functions can be easily generalized to cases with U (1)-symmetry breaking by allowing for offdiagonal elements in the matrices QΛ , GΛ and SΛ . 0 2. Truncations d (4)Λ = Γ dΛ 1 (52) y,y ′ 1’ 2 1’ FIG. 4 Contributions to the flow of the two-particle vertex with particle-particle and particle-hole channels written explicitly, without the contribution from Γ(6)Λ . equation for Γ(6) and all higher vertices. ¯ ¯ Since Γ[ψ, ψ] at ψ = ψ = 0 is essentially (up to a factor T ) the grand canonical potential Ω, the flow equation (35), evaluated at vanishing fields, yields also a flow equation for the grand canonical potential: d Λ ˙ Ω = −T tr QΛ GΛ 0 dΛ . (53) The exact hierarchy of flow equations for the vertex functions can be solved only for systems which can also be solved more directly, that is, without using flow equations. Usually truncations are unavoidable. A natural truncation is to neglect the flow of all vertices Γ(2m)Λ beyond a certain order m0 . We call this the level-m0 truncation. The structure of the resulting equations and general properties of their solution will be discussed in Sec. II.E. Note that the level-m0 truncation contains all perturbative contributions to order m0 in the bare twoparticle interaction. In practice, in applications to physically interesting systems, vertices Γ(2m)Λ with m > 3 have so far been neglected, and the contributions from Γ(6)Λ to the flow of Γ(4)Λ are usually restricted to self-energy corrections (see below) or discarded completely. In particular, the analysis of competing instabilities (see Sec. III) is based entirely on a level-2 truncation given by the flow equation (52) for the two-particle vertex, with Γ(6)Λ replaced by zero, where in addition the self-energy feedback is neglected. This seemingly simple approximation captures the complex interplay of fluctuations in the particleparticle and particle-hole channel, which leads to interesting effects such as the generation of d-wave super- 11 conductivity from antiferromagnetic fluctuations. In the quantum transport phenomena reviewed in Sec. VI, the self-energy as given by the flow equation (50) plays a crucial role. Some of the phenomena described there are already obtained by a level-1 approximation where the flowing two-particle vertex in Eq. (50) is approximated by the bare one. That truncation might look like a HartreeFock approximation, but it is in fact very different, and it works well in cases where Hartree-Fock fails completely. The truncated flow equations are still rather complicated. They involve the flow of functions, not just a limited number of running couplings. For example, the effective two-particle interaction in a translation invariant system is a function of three independent momentum and energy variables. Hence, a simplified parametrization of effective interactions is necessary even for a numerical solution. A useful strategy is to neglect dependences which become irrelevant in the low-energy limit, that is, whose contributions to the flow scale to zero. Contributions to the effective action are called “relevant”, “marginal”, and “irrelevant”, if their importance increases, stays fixed, or decreases, respectively, upon lowering the scale Λ. This classification can be obtained from power counting. To this end, one traditionally considers a renormalization group transformation where one rescales momenta and fields after the integration over fields in a momentum shell of width dΛ such that a certain quadratic part of the action (the Gaussian fixed point) remains invariant (Wilson and Kogut, 1974). From the behavior of the other terms of the action under this transformation one can assess directly whether they increase, remain invariant, or decrease compared to the quadratic part. For Fermi systems in dimensions d > 1 the conventional RG transformation is not applicable, since the reduction of momentum space by the mode elimination cannot be compensated by a linear rescaling of momenta (Shankar, 1991, 1994). However, one can perform the power counting more directly by estimating the scale dependences of Feynman diagrams on the right hand side of the flow equations. As described in Sec. II.E.3, this can be done rigorously and to all orders. At the crudest level the power counting is independent of dimensionality and corresponds to what one would get from the above-mentioned RG transformation applied to onedimensional systems (Shankar, 1994), that is: (i) the self-energy has a relevant piece describing a Fermi surface shift, while linear dependences on frequency and momentum perpendicular to the Fermi surface are marginal; (ii) a regular two-particle interaction is marginal; its dependences on frequencies and momenta perpendicular to the Fermi surface are irrelevant, such that one can parametrize it by its static value on the Fermi surface; (iii) regular m-particle interactions with m ≥ 3 are irrelevant. This basic classification does not depend on dimensionality because the bare propagator G0 , Eq. (3), is singular on a (d− 1)-dimensional surface, such that the codimension of the singularity in the (d + 1) dimensional space spanned by momentum and frequency is always two. One should, however, not jump to the conclusion that the m ≥ 3 terms can simply be discarded from the RG hierarchy in general. This is because effective interactions with m ≥ 3 may diverge for small Λ even in case of finite two-particle interactions. For example, the first contribution to the flow of Γ(6)Λ in Fig. 3 generates a three-particle interaction of order Λ−1 if the external momenta add up to zero at each vertex. When inserted into the equation for Γ(4)Λ in Fig. 3, this may give rise to a marginal term of third order in Γ(4)Λ . For d = 1, this term is indeed marginal. However, if d ≥ 2 and the Fermi surface is curved, this and other contributions are suppressed below the basic power counting estimate due to geometrically reduced integration volumes (Feldman and Trubowitz, 1990; Shankar, 1994). This improved power counting is described in App. B.3. It can also be used to give a precise, scale-dependent meaning to nesting of the Fermi surface. A less obvious effect is that this improvement becomes uniform, that is, independent of the external momenta, in graphs with overlapping loops (Feldman et al., 1996; Salmhofer, 1998a), so that their contribution gets further suppressed (see also App. B.3). It is this effect which implies that the derivative of the self-energy is not marginal, but irrelevant for curved Fermi surfaces in dimension d ≥ 2. Moreover, it justifies truncated flows beyond the weak-coupling regime, as follows. Consider again the first contribution to the flow of Γ(6)Λ , shown also in Fig. 5 (a). When this term is inserted in the equation for Γ(4)Λ , the two lines can be joined in two ways, shown in Fig. 5 (b) and (c). The graph in (b) gets no extra small factor, but the graph in (c) has overlapping loops and for positively curved Fermi surfaces in d = 2, its contribution gets suppressed by an additional small factor Λ ∼ ΛI log ΛI at scales below a scale ΛI , which depends Λ only on the geometry of the constant energy surfaces of the initial dispersion ǫk (Feldman et al., 1998b). This suppression holds uniformly for all values of the external momenta. For d = 3, a similar bound holds, without the logarithm. Similar (in general, weaker) estimates are shown in (Feldman et al., 1996) for general non-nested regular Fermi surfaces in d ≥ 2 and for Fermi surfaces with Van Hove singularities in (Feldman and Salmhofer, 2008a,b). The contribution from graph (c) remains small Λ compared to the second-order term if |Γ(4)Λ | ΛI log ΛI is Λ small. Note that this condition does not require |Γ(4)Λ | itself to be small: curvature effects justify dropping these terms beyond the weak-coupling regime, provided that the above condition is satisfied. This will be used in Sec. III. The detailed argument and a discussion of the consequences for the functional RG flow, are given in Sections 1 and 5 of (Salmhofer and Honerkamp, 2001). In the theory of interacting Fermi systems, one is not only interested in low-energy fixed points and scaling, but also in the behavior at intermediate scales, and formally irrelevant terms may play an important role. There are 12 D. Flow parameters a b c FIG. 5 (a) Third order graph contributing to Γ(6)Λ . (b) Tadpole contraction. (c) Contraction to form a graph with overlapping loops. cases where one would like to know the full temperature, momentum, or frequency dependence of physical quantities, because a low-energy expansion contains insufficient information. One of the advantages of the functional RG framework is that such dependences can be computed directly. In many situations, a comparison to standard resummations of the perturbation expansion is desirable, and it is also an interesting question to what extent such resummations can be reproduced by truncations of the functional RG flow equations. A very important observation regarding this was made by Katanin (2004), who showed that a partial inclusion of the six-point vertex in the flow allows to recover approximations of the type Hartree-plus-ladder summations (in cases where these approximations are a good starting point). This also allows to continue fermionic flows into symmetry-broken phases (see Sec. IV). If we drop the eight-point vertex from the equation for Γ(6)Λ , it is determined by a Feynman graph containing three four-point vertices, depicted in Fig. 5a. When backsubstituted into the equation for the four-point vertex, two external legs get contracted in all possible ways. We have just discussed that the contribution from the graph in (c) is suppressed by improved power counting. When two legs of a single-four-point vertex are contracted to form a tadpole (see Fig. 5b), ˙ the value of the thus obtained subgraph is ΣΛ , by the ˙ flow equation for the self-energy. Thus a factor GΛ ΣΛ GΛ appears in the integral for the value of the graph. By Dyson’s equation, ˙Λ ˙ G =G Σ G +S , Λ Λ Λ Λ (54) so this can be combined with the second-order contri˙ bution to replace S Λ by GΛ . If all other effects of the six-point function, corresponding to graphs of the type shown in Fig. 5c are dropped, the equation for the four-point function gets changed to one where the product GΛ (k)S Λ (k ′ ) + S Λ (k)GΛ (k ′ ) is replaced with d ˙ ˙ GΛ (k)GΛ (k ′ )+GΛ (k)GΛ (k ′ ) = dΛ GΛ (k)GΛ (k ′ ) . If one now restricts further to a single channel in the four-point equation, it becomes explicitly solvable by a ladder summation in that channel. Backsubstitution in the equation for the self-energy gives the corresponding Hartreetype term. This is explained in (Katanin, 2004), and, also in its extension to flows with symmetry breaking, in (Salmhofer et al., 2004). In the derivation of the exact functional flow equation, the scale dependence of the bare propagator GΛ was not 0 specified. The derivation holds for any choice of GΛ , 0 provided all functions involved are indeed differentiable with respect to Λ, and provided that the resulting flow equation is well defined. These conditions are not trivial; in fact, badly chosen flow parameters may lead to divergences on the right hand side of the flow equation. On the other hand, one can exploit the flexibility provided by the choice of the Λ-dependence to ones own advantage. Besides regularity issues, the scale dependence of GΛ is only constrained by the initial condition 0 GΛ0 = 0 , 0 (55) GΛ→0 = G0 . 0 (56) and the final condition The functional Γ = ΓΛ→0 reached at the end of the exact flow is independent of the choice of GΛ . However, 0 in most practical calculations, where approximations are unavoidable, a judicious choice of GΛ is mandatory. Im0 portant aspects related to the choice of GΛ are: regular0 ization of infrared singularities, minimization of truncation errors, respecting symmetries, technical convenience. In the following we will review the most frequently used cutoff schemes along with their merits and drawbacks. 1. Momentum and frequency cutoffs For the sake of a concise discussion, let us focus on translation and spin-rotation invariant one-band systems, such that the bare propagator G0 can be written as a simple function of frequency and momentum as in Eq. (3). The scale dependence can then be introduced by multiplying G0 with a suitable cutoff function θΛ , GΛ (k0 , k) = θΛ (k0 , k) G0 (k0 , k) , 0 (57) with θΛ0 = 0 and θΛ→0 = 1. To regularize the infrared divergence of G0 at zero frequency and for momenta on the Fermi surface (ξk = 0), the cutoff function θΛ (k0 , k) has to vanish sufficiently quickly for k0 → 0, ξk → 0 at fixed Λ > 0. The most frequently used cutoff functions are either pure momentum cutoffs of the form θΛ (k) = ϑ(|ξk |/Λ) or frequency cutoffs θΛ (k0 ) = ϑ(|k0 |/Λ), where ϑ(x) is a function that vanishes for x ≪ 1 and tends to one for x ≫ 1. Mixed momentum and frequency cutoffs 2 2 of the form θΛ (k0 , k) = ϑ[(k0 + ξk )/Λ2 ] are preferred in the mathematical literature, as they facilitate power counting and rigorous estimates. A technical advantage of momentum cutoffs compared to frequency cutoffs is that Matsubara sums on the right hand side of the flow equations can often be performed analytically. Furthermore, a momentum cutoff does not spoil the analytic structure of propagators and vertex 13 functions in the complex frequency plane. However, there are also serious drawbacks, which are specific to Fermi systems. Once self-energy effects are taken into account, the Fermi surface is usually deformed in the course of the flow, such that the momentum cutoff has to be continuously adapted to the new Fermi surface, which complicates the flow equations considerably. Second, particle-hole excitations with a small momentum transfer q are suppressed by the momentum cutoff for |ξk+q − ξk | < 2Λ. As a consequence, the limit of vanishing momentum transfer q → 0 in interaction vertices and response functions does not commute with the limit Λ → 0 (Metzner et al., 1998). In other words, forward scattering interactions and the response to homogeneous fields can be obtained only by taking the limit q → 0 at the end of the flow, at Λ = 0 (Honerkamp and Salmhofer, 2001a). This is a serious drawback in stability analyses (see Sec. III), where one compares the increase of the effective interaction in different momentum channels (including forward scattering), or different susceptibilities, upon lowering Λ until a divergence occurs in at least one channel at a finite scale Λc > 0. A frequency cutoff has the advantage that it does not interfere with Fermi surface shifts, and that particle-hole processes with small momentum transfers are captured smoothly by the flow (Husemann and Salmhofer, 2009). It can also be used in systems without translation invariance (Andergassen et al., 2004), where a momentum cutoff is less useful since the propagator is not diagonal in momentum space. However, a frequency cutoff affects the analytic properties of propagators and vertex functions in the complex frequency plane. Depending on the sort of truncation used, this may pose a serious problem if one likes to continue results to real frequency. For a frequency cutoff the initial cutoff is Λ0 = ∞. Since the contributions to the self-energy flow are of order Λ−1 at large Λ, one has to retain the convergence fac+ tor eik0 0 on the right hand side of the flow equation (Andergassen et al., 2004), analogously to the convergence factor in the perturbation expansion of the selfenergy (Negele and Orland, 1987); for a rigorous justification, see Pedra and Salmhofer (2008). For a bare propagator G0 (k0 , k) = (ik0 − ξk )−1 and a multiplicative cutoff as in Eq. (57), the full propagator has the form GΛ (k0 , k) = θΛ (k0 , k) , ik0 − ξk − θΛ (k0 , k)ΣΛ (k0 , k) (58) ˙ and the single-scale propagator S Λ = −GΛ QΛ GΛ , see 0 Eq. (47), reads S Λ (k0 , k) = (ik0 − ξk )∂Λ θΛ (k0 , k) [ik0 − ξk − θΛ (k0 , k)ΣΛ (k0 , k)] 2 . (59) For a sharp cutoff function such as θΛ (k0 ) = Θ(|k0 | − Λ), the single-scale propagator seems ill-defined, since ∂Λ Θ(|k0 | − Λ) = −δ(|k0 | − Λ), so that a delta peak in the numerator of Eq. (59) coincides with a discontinuity (due to the step function) in the denominator. However, this ambiguity can be easily removed by viewing the step function Θ(x) as a limit of increasingly sharp regularized step functions Θǫ (x), where the discontinuity is smeared over a width ǫ (Morris, 1994). With δǫ (x) = ∂x Θǫ (x), a simple substitution of variables yields 1 ǫ→0 δǫ (x) f (x, Θǫ (x)) −→ δ(x) duf (0, u) , (60) 0 for any continuous function f . Note that the right hand side is unique, that is, it does not depend on the shape of the smeared step function Θǫ (x) for ǫ > 0. For a sharp frequency cutoff θΛ (k0 ) = Θ(|k0 | − Λ), for example, the single-scale propagator thus simplifies to S Λ (k0 , k) = − δ(|k0 | − Λ) , ik0 − ξk − ΣΛ (k0 , k) (61) as long as it does not appear in products where other factors are also discontinuous at |k0 | = Λ. Otherwise, for example in products of the form S Λ (k0 , k)[GΛ (k0 , k)]m , one has to apply Eq. (60) to the entire product. A sharp cutoff has the obvious technical advantage that the integration over the cutoff variable (k0 or ξk ) can be carried out analytically, thanks to the delta-function in the numerator of S Λ . On the other hand, a sharp cutoff generates discontinuities in the momentum or frequency dependences of the vertex functions, corresponding to a pronounced non-locality of the effective action (Morris, 1994), which is often not amenable to a simple parametrization. At finite temperature the flow equations are ill-defined for a sharp frequency cutoff, since the Matsubara frequencies are discrete: k0 = (2n + 1)πT with integer n. Continuous cutoff functions at T > 0 are conveniently chosen such that the Λ-derivative is nonzero only in a frequency range of width 2πT , since then only two frequencies contribute to the Matsubara sum on the right hand side of the flow equation (Enss et al., 2005). There are useful cutoff schemes which are formulated more naturally by adding a regulator function RΛ to the inverse propagator (instead of multiplying): QΛ (k0 , k) = GΛ (k0 , k) 0 0 −1 = Q0 (k0 , k) + RΛ (k0 , k) , (62) with RΛ0 = ∞ and RΛ→0 = 0. In particular, regulator functions of the form (Litim, 2001) RΛ (k) = −Z Λ [sgn(ξk )Λ − ξk ]Θ(Λ − |ξk |) , (63) or its frequency dependent analogue, RΛ (k0 ) = iZ Λ [sgn(k0 )Λ − k0 ]Θ(Λ − |k0 |), have some distinct advantages. The prefactor Z Λ is initially one and is then determined by a momentum (or frequency) derivative of the flowing self-energy ΣΛ (k0 , k). The Litim cutoff satisfies a criterion of ”optimal” regularization of the infrared singularity of the propagator (Litim, 2001). For simple truncations it also leads to a very convenient form of the integrands, facilitating the integrations. 14 It is easy to choose the cutoff function in a way that does not affect the global symmetries of the system, such as global charge conservation or global spin rotatation invariance. However, local conservation laws are typically spoiled. The corresponding Ward identities are modified by cutoff dependent additional terms, which vanish only in the limit Λ → 0 (Enss, 2005). It is very hard to devise truncations which satisfy the modified Ward identities at each scale Λ, and hence truncated flows often violate Ward identities also in the limit Λ → 0 (Katanin, 2004). In these cases it is better to compute only independent quantities from the flow, and determine the remaining quantities, which are fixed by local conservation laws, via the Ward identity. 2. Temperature and interaction flows For fermion systems the infrared singularity of the bare propagator can also be regularized by temperature, instead of a cutoff, since the fermionic Matsubara frequencies stay away from zero at a distance πT . A flow equation with temperature as a flow parameter can be obtained from the general flow equation derived in Sec. II.B, if one manages to shift all temperature dependences of the bare action to the quadratic part. This is indeed possible by a simple rescaling of the fields (Honerkamp and Salmhofer, 2001a). Let us con1 sider a translation invariant system of spin- 2 fermions for definiteness, where the fields depend on a momentum k, a spin index σ, and a Matsubara frequency ′ ωn = (2n + 1)πT . Rescaling the fields as ψσ (n, k) = 3/4 ¯′ (n, k) = T 3/4 ψσ (ωn , k) removes all ¯ T ψσ (ωn , k) and ψσ explicit T -factors from the (quartic) interaction in the bare action. The temperature dependence is thereby shifted entirely to the quadratic part of the action, given by the inverse bare propagator for the rescaled fields, QT (n, k) = 0 T 1/2 . iωn − ξk (64) ¯ The effective action ΓT [ψ ′ , ψ ′ ] for the rescaled fields obeys the exact flow equation Eq. (35), with temperature as the flow parameter. The unscaled vertex functions Γ(2m) are recovered from the vertex functions Γ(2m)T by multiplying with T 3m/2 . The temperature flow has several advantageous features. First, it generates directly a temperature scan of the computed quantities. In cutoff schemes one has to run a full flow for each temperature separately. Second, the temperature flow includes particle-hole excitations with small momentum transfers uniformly at each scale. Third, local symmetries and the corresponding Ward identities are respected at each step at least for the exact flow, which makes the still difficult issue of Ward identities in truncated flows at least more transparent. A particularly simple choice of a flow parameter is provided by a uniform factor λ scaling the bare propagator (Honerkamp et al., 2004), Gλ = λG0 , 0 (65) with λ0 = 0, and λ → 1 at the end of the flow. By a simple rescaling of the fields one can see that this is equivalent to multiplying the bare quartic interaction with a factor λ2 , which means that the interaction is scaled up continuously from 0 to its full strength in the course of the flow. Hence the name ”interaction flow” for this scheme. In the absence of self-energy feedback the interaction flow has the technical advantage that the propagator has the same form at each scale, such that certain loop integrals need to be done only once. However, the global scaling of the propagator does not regularize the infrared singularities, such that one easily runs into infrared divergences. Nevertheless, for suitable problems and simple truncations the interaction flow has been shown to yield results similar to flows with a cutoff, and with less computational effort (Honerkamp et al., 2004). E. General properties of the RG equations In this section we discuss the general structure of the RG hierarchy of equations and provide power counting bounds for its solution. These bounds are simple, but mathematically exact, and they provide a strict sense to the notion of relevant and irrelevant terms. We shall also briefly discuss improved power counting bounds, which provide sharper estimates for bulk Fermi systems in d ≥ 2 and exhibit the role of Fermi surface geometry. The generating functionals were introduced to obtain the Green functions and vertex functions of the model by differentiation, cf. (7) and (10). In the framework of the RG as an iterated convolution, they acquire an independent importance. Indeed, in many situations in bosonic field theory, an expansion in the fields is avoided in favor of a gradient expansion (Berges et al., 2002) or other types of parametrization (see also Sec. V), and the flow may lead to a non-analytic function of the fields. Functions of Grassmann variables are, however, defined only by power series expansions in these variables, so in this case the meaning of the RG is strictly that of the infinite hierarchy. This is only a seeming disadvantage because by the anticommutation properties of Grassmann variables, the fully regularized functionals (as they appear in the flow equations) have convergent expansions in the fields (Abdesselam and Rivasseau, 1998; Feldman et al., 1998a, 2002, 1986; Gawedzki and Kupiainen, 1985; Lesniewski, 1987; Salmhofer and Wieczerkowski, 2000). In contrast, the expansion in the fields of bosonic functionals is almost always divergent, even in the regularized theory. Convergent expansions then take the form of cluster expansions that distinguish between regions of small and large fields (see, for example, Balaban et al. (2010) and references therein). 15 1. Inductive structure of the RG hierarchy ¯ ˜ The functional ΣΛ [ψ, ψ] appearing in (46) has an ˜ ¯ expansion similar to (40), namely ΣΛ [ψ, ψ](x′ , x) = ¯ ˜ (2m)Λ [ψ, ψ](x′ , x), where Σ(2m)Λ is homogeneous ˜ m≥1 Σ ˜ Σ(2m)Λ (x′ , x) 11 of degree 2m in the fields, hence has a representation with ˜ coefficient functions Σ(2m)Λ similar to (41). By definition ˜ the Σ(2m)Λ are determined by Γ(2m+2)Λ , for ˜ (45) of Σ, example (x′ , . . . , x′ ; x1 , . . . , xm ) = −Γ(2m+2)Λ (x′ , x′ , . . . , x′ ; x1 , , . . . , xm , x) . 1 m 1 m (66) Here the indices refer to the matrix structure of (45). The other matrix elements are given by similar expressions. ¯ We use this to expand (46) in homogeneous parts in ψ and ψ and compare coefficients. This gives (2m)Λ d ¯ [ψ, ψ] dΛ A = 1 2 + 1 ¯ ¯ ¯ ˜ ˜ ˜ tr SΛ Σ(2m)Λ [ψ, ψ] + 2 tr SΛ Σ(2)Λ [ψ, ψ]GΛ Σ(2m−2)Λ [ψ, ψ] 1 2 p p≥2 m0 ,...,mp ≥1 m0 +...+mp =m In the sum over p, each of m0 , . . . , mp is at least one ˜ because Σ only contains field-dependent terms, and m0 + . . . + mp = m must hold since A(2m)Λ is homogeneous of ¯ degree 2m in the fields [ψ, ψ]. These two conditions imply that p ≤ m and that mq ≤ m − p for all 0 ≤ q ≤ p, so that for every given m, the sum only runs over finitely many terms. Since the coefficient in A(2m)Λ is Γ(2m)Λ and ˜ Σ(2m)Λ ∼ Γ(2m+2)Λ , comparing coefficients of powers of ¯ ψ and ψ in (67) gives a hierarchy of differential equations for the Γ(2m)Λ , labelled by m. We rewrite (67) as d (2m)Λ dΛ Γ = HΛ Γ(2m+2)Λ + KΛ Γ(4)Λ Γ(2m)Λ m + p=2 LΛ Γ(<2m)Λ . p ˜ ¯ GΛ Σ(2mq )Λ [ψ, ψ] ˜ ¯ tr SΛ Σ(2m0 )Λ [ψ, ψ] (68) The three summands on the right hand side are obtained from the three terms in (67), and it is understood that both sides are functions of 2m variables x1 , . . . , x2m . The action of the operator HΛ on Γ(2m+2)Λ is linear, as is that of KΛ (Γ(4)Λ ) on Γ(2m)Λ , while LΛ is nonlinear in the p lower-m vertex functions Γ(<2m)Λ = Γ(4)Λ , . . . , Γ(2m−2)Λ . Specifically, the action of HΛ is given by a tadpole-type contraction and summation, the action of KΛ (Γ(4)Λ ) is given by the evaluation of a one-loop diagram formed from Γ(2m)Λ and the four-point function Γ(4)Λ , and LΛ p is given by a sum over one-loop diagrams involving p + 1 vertex functions, each of which has mq < m. Thus HΛ , KΛ (Γ(4)Λ ) and LΛ also depend on Λ and on the selfp energy ΣΛ via the propagators S Λ and GΛ . The HΛ term couples the higher vertex function Γ(2m+2)Λ into the equation for Γ(2m)Λ . Thus the hierarchy does not close among finitely many m, and therefore truncations need to be employed to obtain solutions. . (67) q=1 2. Truncated hierarchies and their iterative solution If for some m0 ≥ 1, the initial vertex functions Γ(2m)Λ0 vanish for all m > m0 + 1, one may employ the approximation of setting Γ(2m)Λ = Γ(2m)Λ0 for all m ≥ m0 + 1. That is, all vertices with m > m0 + 1 remain zero, and the (m0 + 1)-particle vertex is kept fixed at its initial value. This level-m0 truncation reduces the infinite hierarchy to a system of finitely many differential equations for (Γ(2m)Λ )m≤m0 . The vertex Γ(2m0 +2)Λ0 enters in the equation for Γ(2m0 )Λ . Specifically, in the level-1 truncation, the two-particle vertex Γ(4)Λ is fixed to its bare value Γ(4)Λ0 , and the self-energy is the solution of (50). The level-2 truncation is given by (52), with Γ(6)Λ fixed to its initial value Γ(6)Λ0 (which may vanish), together with (50). The term KΛ (Γ(4)Λ )Γ(4)Λ is quadratic in Γ(4)Λ . In the level-m0 truncation of the hierarchy, with ′ m0 > 2, and at given ΣΛ and (Γ(2m )Λ )m′ r vanish. As we shall now explain, one can solve the RG hierarchy in terms of a similar expansion in the scale-dependent four-point function V Λ = Γ(4)Λ , again by integrating the RG hierarchy downwards in scale, but keeping the 2m-point functions for m > 2 only to a fixed order in V Λ . The equation for V Λ itself then becomes an integrodifferential equation with a power r nonlinearity on the right hand side (the equation for ΣΛ remains unchanged). This leads in a natural way to power counting estimates for the higher 2m-point functions in terms of the maximal value of the four-point vertex that occurs in the flow. We denote the O (V Λ )r contribution to Γ(2m)Λ by (2m)Λ Γr . Its scale derivative equals d (2m)Λ dΛ Γr = (2m+2)Λ H Λ Γr ′ + p≥2 Λ + K (V Λ (2m)Λ ) Γr−1 LΛ Γ(2m0 )Λ , . . . , Γ(2mp )Λ . (69) rp p r0 The primed sum runs over all sequences (m0 , . . . mp ) and all sequences (r0 , . . . , rp ) with mq ≥ 1 and rq ≥ 1 for all 1 ≤ q ≤ p, m0 + . . . + mp = m + p, and r0 + . . . + rp = r. The solution of the RG hierarchy for an initial (2m)Λ quartic interaction has the property that Γr = 0 for all m > r. Therefore, for m = r, the HΛ term drops out of (68), and all remaining terms contain only V Λ or terms of order at most r − 1 in V Λ . Thus, given these (2r)Λ can be obtained by integration. lower-order Γ’s, Γr Then the right hand side of the equation for m = r − 1 (2r−2)Λ can be determined, and so is determined, so Γr on. Successive backsubstitution then leads to a system of d (2m)Λ equations where dΛ Γr gets contributions from a sum ′ of graphs with r vertices of type V Λ , where Λ0 ≥ Λ′ ≥ Λ, ′′ ′′′ and propagators GΛ and S Λ , and all the intermediate scales Λ′ etc. are integrated. Thus the equation becomes nonlocal in the flow parameter Λ but the right hand side is known once V Λ and ΣΛ are known. V Λ is given by a degree r nonlinear equation, with a similar graphical background as discussed above, and ΣΛ by the standard self-energy equation (50). While more restricted than the level-m truncation, the running coupling scheme also captures effects that cannot be seen in any fixed order of bare perturbation theory, such as screening or asymptotic freedom of certain coupling functions. We use this inductive structure to derive basic power counting bounds for the vertex functions in terms of the flowing four-point function, for a d-dimensional bulk fermion system (d ≥ 1). For simplicity, we focus 1 on spin- 2 fermions with translation-invariant action, so that we can use xi = (ki , σi ) = (k0,i , ki , σi ), as discussed at the beginning of Section II.A. We also assume that the symmetries of the action remain unbroken. These specific assumptions are for presentation only; power counting can be done without them. By (2m)Λ ((k1 , σ1 ), . . . (k2m , σ2m )) = translation invariance, Γr (2m)Λ ˆr (k, σ), where the delta function forces δ ( i ki ) Γ conservation of the spatial momentum k (up to reciprocal lattice vectors) and conservation of the frequency variable k0 , and we have introduced the abbreviations σ = (σ1 , . . . σ2m ) and k = (k1 , . . . , k2m−1 ). For Λ = Λ0 , ˆ (2m)Λ is smooth and bounded because the the function Γr initial interaction is short-range, and this stays so during the flow above critical scales. Consider the maximal size of the vertex functions, (2m)Λ ˆ (2m)Λ (k, σ)|. Then, for m ≥ 3, = sup |Γr Γr k,σ (2m)Λ (2m) Γr ≤ γr sΛ r−m+1 fΛ r Λ2−m , (2m) where γr (70) is independent of Λ and β, fΛ = sup Vℓ (71) Λ≤ℓ≤Λ0 is the maximal value of the four-point coupling on all scales between Λ and Λ0 , and dk |Sα,α′ (k)| , ¯ ˆΛ sΛ = max α α′ d (72) d k with dk . . . = T k0 (2π)d . . . (for the general power ¯ counting, we do not need to assume that the propagator 17 ˆ is diagonal in the spin indices α, α′ , so S Λ also carries these indices). The dependence of sΛ on Λ is determined by the shape of the Fermi surface. As shown in Appendix B.1, sΛ is of order one for regular Fermi surfaces. If the Fermi surface contains Van Hove points, sΛ grows logarithmically in Λ for Λ → 0. At first sight, one may worry about the factor Λ2−m , which diverges for m ≥ 3 as Λ → 0. For the maximum value of the vertex functions, it is indeed true – and easily verified in examples – that there are always particular values of the external momenta where these vertex functions become very large in β = 1/T (and diverge at zero temperature). However, this happens only on a “small” set of momenta. For a general m-point function, it is very involved to determine this set, but this is not necessary for power counting. One can use the L1 norm instead, i.e. (2m)Λ (2m)Λ consider Γr dk1 . . . dkm |Γr ¯ ¯ (k, σ)|. 1 = σ Using generalizations of (70), one can then show that if fΛ remains finite (2m)Λ Γr 1 (2m) r ≤ cr fΛ (73) B.3. This growth of fΛ with decreasing scale Λ is usually called the “flow to strong coupling” in RG studies, and is described in more detail in Section III. A singularity of the two-particle vertex in momentum space means that the interaction becomes long-range in position space. This is associated with the formation of critical fluctuations, and in case of spontaneous breaking of a continuous symmetry, with the appearance of Goldstone bosons (see Section IV). 4. Self-energy and Fermi surface shift The self-energy is important for all effective oneparticle properties of the system, and it can cause drastic effects, as compared to the non-interacting fermions. Accordingly, in the RG flow, the self-energy is a relevant term. In absence of symmetry breaking, it modifies the inverse propagator to ik0 − ξk − Σ(k0 , k). The longdistance behaviour of the fermionic propagator is determined by the behaviour of this function around its zero set. A Taylor expansion around k0 = 0 gives (2m) with constants cr that are independent of Λ, β and the system size L (see Salmhofer (1999), Section 4.4.3). This implies that, even in the limit β → ∞, the 2mpoint vertices can become singular only on a set of zero Lebesgue measure in momentum space. In general, this set can be rather complicated, but, loosely speaking, it will have codimension at least one. It is one of the appealing features of the flow-equation RG that exact statements like (70) can be proven in a few lines, see Appendix B. The argument given there also implies an at-most logarithmic growth of the coefficients in the equation for fΛ itself. The self-energy then comes out of order fΛ , provided that renormalization is done correctly, see Section II.E.4. At small fΛ , the size of the vertices Γ(2m)Λ with m ≥ 3 is thus determined by fΛ . The terms with m ≥ 3 are the RG-irrelevant ones, m = 2 is marginal, and m = 1 is relevant. This classification is explained in detail in Appendix B. In a Taylor expansion of the vertex functions in the Matsubara frequency around zero, and in momentum around the Fermi surface, additional small factors arise, which cancel the small denominators of the propagators; at the same time, the vertex function is replaced by a differentiated one. Hence, the flow obtained by projecting the frequencies to zero and the momenta to the Fermi surface gives the dominant contribution for small Λ. This is expected from a simple counting of bare scaling dimensions, and can be established more rigorously by power-counting arguments similar to those used above and in Appendix B. In the case of a curved Fermi surface in d ≥ 2, fΛ indeed stays small in a weakly interacting system above a BCS-like temperature (see, e.g., Salmhofer (1998b)), indicating the absence of symmetry-breaking. At zero temperature, fΛ grows as Λ decreases, and the four-point function has singularities at points corresponding to nesting vectors of the Fermi surface; for details, see Appendix ik0 − ξk − Σ(k0 , k) = ik0 − ek + ρ(k0 , k) Zk (74) −1 −1 with Zk = 1 + i(∂0 Σ)(0, k), Zk ek = ξk + Σ(0, k) and a Taylor remainder ρ. If ρ vanishes faster than linearly in k0 as k0 → 0, we thus obtain an effective description in terms of quasiparticles with dispersion relation ek , hence “interacting” Fermi surface {k : ξk + Σ(0, k) = 0}, Fermi velocity ∇ek , and quasiparticle weight Zk . If Zk is bounded and nonvanishing for all k, the long-distance decay of the fermion propagator in position space is the same as for the free theory. The question whether Σ is smooth enough for the above to hold is nontrivial. In the one-dimensional Luttinger model, ∂0 Σ(0, kF ) diverges, and the small-k0 behaviour of the self-energy, Σ(k0 , kF ) ∼ |k0 |α with α < 1 depending on the interaction strength, implies that the self-energy effects dominate at small k0 and the decay in position space becomes more rapid, so that the occupation number density n(k) becomes a continuous function of k even at zero temperature (Giamarchi, 2004). An even more drastic change is spontaneous symmetry breaking, where the propagator cannot be written any more in the simple form given above (see Section IV). In the RG flow, Σ is replaced by the Λ-dependent self-energy ΣΛ = (GΛ )−1 − (GΛ )−1 . An important phe0 nomenon in this context is the shift in the Fermi surface entailed by ΣΛ . In terms of power counting, this shift is the most relevant term. Cutting off the propagator around the free Fermi surface then fails to regularize the propagator, which leads to spurious singularities in the RG flow. A convenient method to avoid this is to introduce a counterterm. In the context of the bare perturbation expansion, the counterterm method was already described in (Nozi`res, 1964). The main idea is to antice ipate the form that the propagator takes at the end of 18 the flow and to rearrange the flow such that this form, not the bare one is used as the starting point for the RG analysis, hence the Fermi surface is fixed to that of the interacting system in the flow. The difference between the two dispersion functions appears as a (finite) counterterm. To obtain a one-to-one relation between the model given by the Hamiltonian and the one with fixed interacting Fermi surface, one has to solve a selfconsistency equation. Feldman and Trubowitz (1990) used the counterterm method for the radius shift of a circular Fermi surface in a RG flow. Feldman et al. (1996, 1998b, 1999, 2000) generalized this to the case of non-circular curved Fermi surfaces, solved the selfconsistency equation, and showed that Zk remains finite for d ≥ 2 to all orders. The corresponding fixedpoint problem for the Fermi surface was also considered by Ledowski and Kopietz (2003). The role of Van Hove singularities was analyzed by Feldman and Salmhofer (2008a,b). Feldman and Trubowitz (1991) also used the counterterm method to derive the equation for the superconducting gap from an RG flow; for further work in that direction, see also Section IV. In alternative to counterterms, one can try to avoid a momentum space cutoff altogether (Honerkamp and Salmhofer, 2001a,b; Husemann and Salmhofer, 2009), or to use an adaptive scheme (see the appendix in Honerkamp et al. (2001) and Benfatto et al. (2006); Salmhofer (2007)). F. Flow equations for observables and correlation functions All observables of the fermionic system are given by polynomials in the fields, so they can be calculated from the connected Green functions G(2m)Λ , hence by the above-mentioned tree relations also from the irreducible vertex functions Γ(2m)Λ . It is nevertheless convenient, and due to the limitations of approximations often mandatory, to calculate the flow of observables and their correlation (or response) functions by separate flow equations, which we derive and discuss now. For simplicity we restrict the presentation to the particularly important class of observables that are correlations of fermionic bilinears. Charge-invariant bilinears are of the form a charge invariant bilinear is of the form B(k) = dp ψσ (p) bσ,σ′ (p, k) ψσ′ (p + k) . ¯ ¯ (77) The frequency k0 is an integer multiple of 2πT . The case k = 0 and b = σi , where σi denotes the ith Pauli matrix, corresponds to a uniform spin density. The case k0 = 0, k = (π, π, . . . , π) and b = σi corresponds to a staggered spin density. Similarly, the same choices of k but with bσ,σ′ = δσ,σ′ correspond to charge densities. Other choices of k can be used to test tendencies towards noncommensurate magnetic or charge ordering. Cooper pair fields correspond to the non-charge invariant combinations B(k) = ¯ dp ψσ (p) ∆σ,σ′ (p, k) ψσ′ (−p + k) ¯ ¯ + ψσ (p) ∆σ′ ,σ (−p + k, k) ψσ′ (−p + k) . (78) Again, the simplest choice is uniform singlet pairing, where k = 0 and ∆σ,σ′ (p, k) = ∆(p)εσ,σ′ . In this case ∆(p) is the gap function. Triplet pairing, extended Cooper pairs, and spatially nonuniform gaps, are described by suitable generalizations. A convenient way of generating correlation functions of the bilinears B is to couple the B(x) to external source fields J(x), i.e. to add a term (J, B) = x J(x)B(x) to the action. The external field J is not an integration variable, so it can be regarded as a (functional) parameter on which G depends. Writing G = G(J, η, η ), we then ¯ have B(x) B(y) − B(x) B(y) = − ∂ 2 G(J, η, η ) ¯ ∂J(x)∂J(y) J=0 η,η=0 ¯ . (79) ¯ In presence of J, the effective action Γ = Γ(J, ψ, ψ), as well as all other quantities appearing in the fermionic Legendre transform (9), depend on J as well. Since re∂Γ ¯ ¯ lations such as ∂ ψ (J, ψ, ψ) = η(J, ψ, ψ) remain valid for ¯ any J, straightforward differentiation yields ∂ 2 G(J, η, η ) ¯ ∂J(x)∂J(y) J=0 η,η=0 ¯ = ¯ ∂ 2 Γ(J, ψ, ψ) ∂J(x)∂J(y) J=0 ¯ ψ,ψ=0 . (80) Graphically, this relation is intuitive in that the bilinears always couple to the (effective) vertices by two lines, and ¯ B(x) = ψ(y) B(x; y, y ′ )ψ(y ′ ). (75) the fermionic vertices are all even, so that the graphs y,y ′ that contribute are automatically irreducible. Again, because J plays the role of a parameter, the flow Charge-non-invariant bilinears are of the form equation (35) is unchanged. Flow equations for the response functions are then obtained simply by expanding ¯ ¯ ˜ ψ(y) B(x; y, y ′ )ψ(y ′ ) + ψ(y) B(x; y, y ′ )ψ(y ′ ) . Γ in the fields J and comparing coefficients. This again B(x) = y,y ′ leads to a hierarchy of equations for the vertex functions (76) Γ(2m,n) that have 2m fermionic and n external bosonic ˜ The functions B and B determine the spatial and spin lines. The J-independent term corresponds to the stanstructure of these bilinears. For translation-invariant sysdard fermionic hierarchy for the Γ(2m,0) = Γ(2m) , which tems, we can choose a momentum representation where therefore remains unchanged. When counting powers x = (k0 , k) and y = (p0 , p, σ), as explained above Eq. (3). in the fermionic fields, each J corresponds to a bilinWith the notations p = (p0 , p) and dp = T p0 dp, ¯ ear, so that the truncation Γ(2m) = 0 for m ≥ m0 19 for the fermionic vertices corresponds to a truncation Γ(2m,n) = 0 for m + n ≥ m0 . The flow equations remaining after a truncation for m0 = 3 are shown diagrammatically in Figure 6. We first introduce some notation for bosons, and write down the bosonic analogues of some of the most important equations from Sec. II.B. Bosonic particles are described by complex fields φ. It is convenient to combine φ and its complex conjugate φ∗ in a bosonic Nambu field Φ(x) = φ(x) φ∗ (x) . (81) The generating functional for connected Green functions can be written as (Negele and Orland, 1987) G[H] = − ln DΦ e−S[Φ] e(H ∗ ,Φ) , (82) where S[Φ] is the bare action, and FIG. 6 The truncation of the hierarchy for the response function that corresponds to keeping only the irreducible twoparticle vertex in the fermionic hierarchy. Clearly, the two-point correlation B(x) B(y) − B(x) B(y) of any fermionic bilinear B only involves the fermionic two- and four-point functions, hence could simply be calculated from the knowledge of Γ(2) and Γ(4) by the reciprocity relation and (12). Eq. (80) shows that the route via external fields in the one-particle irreducible equations is strictly equivalent to this if the hierarchy is treated exactly. When making truncations to the hierarchy and other approximations, the two are no longer the same. Anomalous scaling dimensions of fermionic bilinears (or other composite objects) are captured easily by separate flow equations for these quantities, while they are hard to obtain from Γ(2) and Γ(4) , if the latter are computed from a truncated flow equation. An instructive example is given by the calculation of the density profile near a static impurity in a Luttinger liquid in Andergassen et al. (2004). H(x) = . (83) the source field. Connected Green functions are obtained as functional derivatives G(2m) (x1 , . . . , xm ; x′ , . . . , x′ ) = 1 m φ(x1 ) . . . φ(xm )φ∗ (x′ ) . . . φ∗ (x′ ) m 1 − c = ∂ 2m G[H] ∂h∗ (x1 ) . . . ∂h∗ (xm )∂h(x′ ) . . . ∂h(x′ ) H=0 m 1 . (84) The effective action is defined as Legendre transform Γ[Φ] = (H ∗ , Φ) + G[H] , (85) with Φ = −∂G/∂H ∗ . Functional derivatives of Γ[Φ] yield the bosonic m-particle vertex functions Γ(2m) (x1 , . . . , xm ; x′ , . . . , x′ ) = 1 m ∂ 2m Γ[Φ] ∂φ∗ (x1 ) . . . ∂φ∗ (xm )∂φ(x′ ) . . . ∂φ(x′ ) Φ=0 m 1 . (86) The matrices of second derivatives at finite fields G(2) [H] = − G. Flow equations for coupled boson-fermion systems The focus of this review is on fermion systems. However, even if the bare action involves only fermionic fields, bosonic degrees of freedom are frequently generated as fermion composites and order parameter fields. For example, Cooper pairs and the order parameter in a superconductor are bosons. Technically, bosonic fields are introduced in an originally purely fermionic theory by a Hubbard-Stratonovich decoupling of an interaction between fermions (Popov, 1987). Often the fermionic fields are subsequently intregrated out, such that an effective action involving only bosons remains. Otherwise one has to deal with a coupled theory of fermions and bosons. In this section we generalize the flow equations derived in Sec. II.B to interacting boson-fermion systems. Flow equations for coupled boson-fermions systems have been derived by various groups, with slight differences in the notation (Berges et al., 2002; Kopietz et al., 2010). h(x) h∗ (x) = ∂2G ∂H ∗ (x)∂H(x′ ) φ(x)φ∗ (x′ ) φ∗ (x)φ∗ (x′ ) φ(x)φ(x′ ) φ∗ (x)φ(x′ ) (87) and Γ(2) [Φ] = ∂2Γ ∂Φ∗ (x)∂Φ(x′ ) (88) obey the reciprocity relation Γ(2) [Φ] = (G(2) [H])−1 . Endowing the bare propagator G0 with a cutoff or another scale dependence, one can derive exact flow equations for the generating functionals in complete analogy to the fermionic case. In particular, the flow equation for the effective action ΓΛ [Φ] has the form 1 1 d Λ ˙ ˙ Γ [Φ] = (Φ∗ , QΛ Φ) + tr QΛ Γ(2)Λ [Φ] 0 0 dΛ 2 2 −1 , (89) 20 where QΛ = diag(QΛ , QΛt ) with QΛ = (GΛ )−1 . Note 0 0 0 0 0 that the first term on the right hand side can also be ˙ written as (φ∗ , QΛ φ). The above flow equation is equiva0 lent to the frequently used flow equation for the effective average action (Berges et al., 2002) 1 ΓΛ [Φ] = ΓΛ [Φ] − (Φ∗ , RΛ Φ) , R 2 involves also mixed boson-fermion propagators, which vanish only for H = 0. The matrix of second derivatives of the effective action ∂2Γ ¯ ∂ Ξ(x)Ξ(x′ )  ∂2Γ ∂ 2Γ  ∂Φ∗ (x)∂Φ(x′ ) ∂Φ∗ (x)∂Ψ(x′ ) =   ∂ 2Γ ∂2Γ ¯ ¯ ∂ Ψ(x)∂Φ(x′ ) ∂ Ψ(x)∂Ψ(x′ ) Γ(2) [Ξ] = (90) with RΛ = QΛ − Q0 , which reads (Wetterich, 1993) 0 −1 d Λ 1 (2)Λ ˙ Γ [Φ] = tr RΛ ΓR [Φ] + RΛ dΛ R 2 . (91) Order parameters are often associated with real (not complex) bosonic fields. In that (simpler) case the above equations are still valid if one replaces the complex Nambu fields Φ and H by the real fields φ and h. A generalization to coupled fermion-boson systems is now straightforward. Bosonic and fermionic fields are conventiently collected in a ”super-field” Ξ= Φ Ψ , (92) where Φ and Ψ are the bosonic and fermionic Nambu fields defined above (see Sec. II.A). The conjugate superfield is given by ¯ Ξ= Φ∗ ¯ Ψ . G[Hb , Hf ] = − ln ∗ where ¯ ∗ ¯ Γ[Φ, Ψ] = (Hb , Φ) + (Hf , Ψ) + G(Hb , Hf ) , (95) ∗ ¯ where Φ = −∂G/∂Hb and Ψ = −∂G/∂ Hf . The source fields may also be collected in a super-field H= Hb Hf . (96) The Legendre transform can then be written more con¯ cisely as Γ[Ξ] = (H, Ξ) + G(H). The matrix of second functional derivatives of G at finite fields ∂ 2G G(2) [H] = − ¯ ∂ H(x)∂H(x′ ) = d Λ 1 1 ¯ ˙ ˙ Γ [Φ, Ψ] = (Φ∗ , QΛ Φ) − (Ψ, QΛ0 Ψ) b0 f dΛ 2 2 −1 1 ˙ + Str QΛ Γ(2)Λ [Φ, Ψ] , (99) 0 2 DΦDΨ e−S[Φ,Ψ] e(Hb ,Φ)+(Hf ,Ψ) , (94) where S[Φ, Ψ] is the bare action, and Hb and Hf are the Nambu source fields for bosons and fermions, respectively. Functional derivatives with respect to the source fields generate connected Green functions with an arbitrary number of bosonic and fermionic fields, the only general constraint being that the number of fermion fields is always even. The effective action Γ[Φ, Ψ] is given by the Legendre transform ¯ Φ(x)Φ∗ (x′ ) − Φ(x)Ψ(x′ ) ¯ Ψ(x)Φ∗ (x′ ) − Ψ(x)Ψ(x′ ) QΛ = 0 QΛ 0 b0 0 QΛ0 f . (100) The supertrace Str incorporates a minus sign in the fermionic sector. The flow equation (99) is equivalent to the flow equation (Berges et al., 2002) 1 d Λ (2)Λ ˙ Γ [Φ, Ψ] = Str RΛ ΓR [Φ, Ψ] + RΛ dΛ R 2 −1 , (101) with RΛ = QΛ − Q0 , for the effective average action 0 1 ¯ 1 ΓΛ [Φ, Ψ] = ΓΛ [Φ, Ψ]− (Φ∗ , RΛ Φ)+ (Ψ, RΛ Ψ) . (102) b f R 2 2 The expansion of the exact functional flow equation (99) proceeds in complete analogy to the purely fermionic case. Inserting ˜ Γ(2)Λ [Φ, Ψ] = (GΛ )−1 − ΣΛ [Φ, Ψ] , (103) with GΛ = Γ(2)Λ [Φ, Ψ] (97)   . (98)  is related to G(2) [H] by the reciprocity relation Γ(2) [Ξ] = (G(2) [H])−1 . A flow of the generating functionals is generated by modifing the bare propagators for bosons and fermions, Gb0 and Gf 0 , such that they depend on some scale parameter Λ. We denote the scale dependent bare propagators by GΛ and GΛ0 , and their inverse by QΛ and QΛ0 . b0 f b0 f The generalization of the exact flow equations for the effective action in purely bosonic or fermionic systems to coupled boson-fermion systems reads (93) The generating functional for connected Green functions involving both bosons and fermions reads  −1 Φ=Ψ=0 = diag(GΛ , GΛ ) , b f into the functional flow equation (99), one obtains (104) 21 1 1 1 ¯ ˙ d Λ ˙ b0 ˙0 Γ [Φ, Ψ] = Str QΛ GΛ + (Φ∗ , QΛ Φ) − (Ψ, QΛ0 Ψ) f dΛ 2 2 2 1 ˜ ˜ ˜ − Str SΛ ΣΛ [Φ, Ψ] + ΣΛ [Φ, Ψ] GΛ ΣΛ [Φ, Ψ] + . . . 2 with the single-scale propagator ˙ SΛ = −GΛ QΛ GΛ = diag(SΛ , SΛ ) . 0 b f (106) The expansion in powers of the fields is now straightforward and leads to a hierarchy of flow equations for all vertex functions. The first few terms are shown diagrammatically in Fig. 7. d Λ = Σ dΛ f + d Λ = Σ dΛ b + d dΛ = d dΛ = d dΛ = + + + + + + + ... + ... + + ... FIG. 7 Diagrammatic representation of the flow equations for the (fermionic and bosonic) self-energies and some of the interaction vertices in a coupled boson-fermion theory. Solid lines denote fermionic, dashed lines bosonic propagators. Propagators with a dash are single-scale propagators. The flow equations derived above are also valid in case of U (1)-symmetry breaking, if one allows for offdiagonal elements in the matrices QΛ , QΛ0 , GΛ , GΛ b0 f b f etc. (Berges et al., 2002; Sch¨tz and Kopietz, 2006). u Coupled flow equations for fermions and bosonic Hubbard-Stratonovich fields are particularly convenient to treat fluctuations associated with spontaneous symmetry breaking (see Sec. IV) and quantum criticality (see Sec. V), but they may also be used to study Luttinger liquids and other symmetric states in interacting Fermi systems (Bartosch et al., 2009a; Ledowski and Kopietz, 2007a,b; Sch¨tz et al., 2005). u III. COMPETING INSTABILITIES In this section we describe how one can apply the level2 truncation of the fermionic RG, mainly without selfenergy corrections, to two-dimensional fermion systems, , (105) and study the interplay of ordering tendencies. In resummations of perturbation theory, their manifestation are singularities in the four-point function and in certain susceptibilities. In the RG, the precursor to a singularity is the growth of some parts of the vertex function (often termed “flow to strong coupling”). Since singularities in the vertex function change the power counting drastically, this truncated flow then has to be stopped before a singularity happens, at a scale Λ∗ > 0, where one can read off the dominant interactions and infer a tentative phase diagram (in this, susceptibilities are used to compare the strength of different ordering tendencies and to determine Λ∗ ). As discussed in Sec. II.C.2, curvature effects of the Fermi surface imply that the truncations discussed here can be used also when the interaction is no longer small, provided that the power counting improvement factor times the interaction strength remains small. To obtain a true phase diagram, however, one needs to integrate over all degrees of freedom, also those with scales below Λ∗ . This has been achieved in some cases (see Section IV), but much remains to be done. This first step of monitoring the flow to strong coupling above Λ∗ , as described in this section, is important for the following reasons. (1) It allows to determine the effective interaction just above transition scales from the given microscopic model without any additional a priori assumptions about the nature of symmetry-breaking, and thereby provides an initial condition for the integration at scales below Λ∗ . (2) It exhibits how the interplay of the scale-dependent scattering processes on different parts of the Fermi surface gradually builds up the effective interaction. (3) It has by now become a versatile tool for analyzing models with an elaborate microscopic structure, such as multiple bands. A. Hubbard model and N -patch RG schemes The Hubbard model and its extensions have become standard in correlated fermion systems: on the square lattice as a candidate model for high-temperature superconducting cuprates (Anderson, 1997; Fulde, 1991), in a multiband-generalization, for the newly discovered iron superconductors (Miyake et al., 2010), on triangular lattices for organic crystals (Kino and Fukuyama, 1996; McKenzie, 1997), on the honeycomb lattice for graphene (Herbut, 2006; Lopez-Sancho et al., 2009). The Hamil- 22 tonian for the simplest one-band Hubbard model reads H=− ti−j c† cj,s + U i,s ni,↑ ni,↓ (107) i i,j,s where ti−j = tj−i is the hopping amplitude between sites i and j and U is the Hubbard on-site repulsion. We consider here mainly the case with only nearest-neighbor hopping t and next-to-nearest neighbor hopping t′ on a square lattice. Additional hopping terms can be added if a more detailed description of the band structure is required, and other interaction terms may be added. The chemical potential µ and t and t′ determine the band structure ξk = −2t(cos kx + cos ky ) − 4t′ cos kx cos ky − µ, and hence the shape of the Fermi surface. Resummations of perturbation theory in U suggest singularities in different channels, arising from Fermi surface nesting and Van Hove singularities (Schulz, 1987), hence competing effects, which are best treated by RG methods. After two-patch studies, which provided a very crude approximation to the momentum dependence of the four-point vertex (Dzyaloshinskii, 1987; Furukawa et al., 1998; Gonzalez et al., 1996; Lederer et al., 1987; Schulz, 1987), more careful analyses with momentum-dependent vertices were done using the Polchinski (Zanchi and Schulz, 1997, 1998, 2000), the Wick ordered (Halboth and Metzner, 2000a,b), and the one-particle irreducible flow equations (Honerkamp et al., 2001), all with a momentum space regulator. To include ferromagnetism, the temperature flow was introduced by Honerkamp and Salmhofer (2001a,b) and Honerkamp (2001), and further developed by Katanin and Kampf (2003). The results of these studies at Van Hove filling were confirmed using a refined parametrization of the wavevector dependence (Husemann and Salmhofer, 2009). The decoupling of the various ordering tendencies in the limit of small U very close to the instability and the influence of non-local interactions were discussed by Binz et al. (2002, 2003). In the general RG setup of Section II, the fermion fields now carry a spin index s and a multiindex K consisting of Matsubara frequencies ω, wavevectors k, and possibly a band index b. To avoid bias, the action is required to retain all symmetries of the initial action. This implies (see Honerkamp et al. (2001); Salmhofer and Honerkamp (2001)) that Γ(4)Λs3 s4 (K1 , K2 ; K3 , K4 ) = V Λ (K1 , K2 ; K3 , K4 )δs1 s3 δs2 s4 − V Λ (K2 , K1 ; K3 , K4 )δs1 s4 δs2 s3 s1 s2 (108) for a spin-rotation invariant system. By lattice- and time-translation invariance, K4 is fixed by K1 , K2 and K3 in the one-band model (in multiband models, the fourth band index b4 still remains free). We therefore abbreviate notation to V Λ (K1 , K2 , K3 ). In the truncation Γ(6)Λ = 0, the flow equations for the self-energy and for the coupling function become Λ d dΛ Σ (K) =− dK ′ 2V Λ (K, K ′ , K) − V Λ (K, K ′ , K ′ ) S Λ (K ′ ) , Λ d dΛ V Λ Λ Λ = TP P + TP H,d + TP H,cr (109) Λ Λ Λ with the particle-particle term TP P and the direct and crossed particle-hole terms TP H,d and TP H,cr : Λ TP P (K1 , K2 ; K3 , K4 ) = Λ TP H,d (K1 , K2 ; K3 , K4 ) = dK V Λ (K1 , K2 , K) LΛ (K, −K + K1 + K2 ) V Λ (K, −K + K1 + K2 , K3 ) , (110) dK −2V Λ (K1 , K, K3 ) LΛ (K, K + K1 − K3 ) V Λ (K + K1 − K3 , K2 , K) +V Λ (K1 , K, K + K1 − K3 ) LΛ (K, K + K1 − K3 ) V Λ (K + K1 − K3 , K2 , K) +V Λ (K1 , K, K3 ) LΛ (K, K + K1 − K3 ) V Λ (K2 , K + K1 − K3 , K) , Λ TP H,cr (K1 , K2 ; K3 , K4 ) = dK V Λ (K1 , K + K2 − K3 , K) LΛ (K, K + K2 − K3 ) V Λ (K, K2 , K3 ) . (111) (112) Here LΛ (K, K ′ ) = S Λ (K)GΛ (K ′ ) + GΛ (K)S Λ (K ′ ) is the product of single-scale propagators S Λ and full propagators GΛ with momentum assignments corresponding to the diagrams in Fig. 8. For the Hubbard Hamiltonian (107), the initial condition is V Λ0 (K1 , K2 , K3 ) = U . Other interactions can be dealt with by modifying this initial condition. The truncation Γ(6)Λ = 0 is justified only for a sufficiently small bare coupling, since a contribution to Γ(6)Λ is generated at third order in the two-particle interaction, which leads to third order contributions to the flow of V Λ (see Sec. II). In most studies the self-energy feedback into the flow of 23 V Λ was also neglected, since it also affects the flow only at third order in V Λ . The coupling function V Λ (K1 , K2 , K3 ) depends on three wavevectors and three Matsubara frequencies, so that the RG equation for a two-dimensional system is a differential equation in a 9-dimensional space. As discussed in Section II.E, its most singular part sits at zero Matsubara frequency. Hence one may neglect the frequency dependence. Then V Λ defines an effective Hamiltonian. Similarly, the k-dependence is most important in the angular direction along the Fermi surface. This dependence can then be taken into account by a discretization, i.e. by devising patches in the Brillouin zone in which the coupling function is kept constant. Feldman et al. (1992) showed that using N patches leads to a natural N -vector model in two dimensions. Zanchi and Schulz (1998, 2000) were the first to use it in studies of the Hubbard model. Usually one forms elongated patches that extend roughly perpendicular to the Fermi surface but are rather narrow parallel to the Fermi surface (see Fig. 9). The coupling function is then computed for wavevectors k1 to k3 at the Fermi surface in the center of the patches. We label the patches by κi = 1, . . . N . The function V Λ is thus approximated by O(N 3 ) interpatch couplings V Λ (κ1 , κ2 , κ3 ). Even if k1 , k2 and k3 are on the Fermi surface, k4 can be anywhere. In the calculation of the loop integrals it is however necessary to assign a patch number κ4 to k4 , which amounts to an approximation of projecting k4 on the Fermi surface. Note that this projected N -patch discretized coupling function V Λ (κ1 , κ2 , κ3 ) then has fewer symmetries; for instance V Λ (κ1 , κ2 , κ3 ) = V Λ (κ2 , κ1 , κ4 ) in general, as in the latter object k3 is not necessarily on the Fermi surface. For sufficiently large N , this discretization captures the angular variation of the coupling function along the Fermi surface with good precision. The results obtained within this approximation, described in the following, have been found to be robust when the dependence on frequencies ωi (Honerkamp et al., 2007; Klironomos and Tsai, 2006) and the component of ki transversal to the Fermi surface (Halboth and Metzner, 2000a; Honerkamp, 2001; Honerkamp et al., 2004) are included. Katanin (2009) performed a flow to third order in the scale-dependent four-point-vertex (see Section II.E.3), with the frequency dependence in the same approximation as Honerkamp and Salmhofer (2003). B. Results for the two-dimensional Hubbard model Starting from the initial condition given by the Hubbard model, the flow is run from Λ0 down to a characteristic scale Λ∗ , where the largest coupling reaches some multiple α of the bandwidth. The choice of α varies widely in the literature; the discussion here is based on the comparably cautious choice α = 2 or 3, as well as s, K2 s, K3 s′ , K 1 s′ , K 4 Λ TP H,cr Λ TP P Λ TP H,d FIG. 8 Top row: The coupling function V Λ (K1 , K2 , K3 ) with the spin convention, and the diagrams entering in the flow equation for the self-energy (middle and right diagram). Middle and bottom row: The diagrams for the flow of the coupling function. The internal lines are either full propagators GΛ or single-scale propagators S Λ . FIG. 9 (Color online) N -patch discretization of the Brillouin zone for the one-band Hubbard model on the 2D square lattice. The colored region is a patch in which the coupling function is approximated as a constant. on the consistency check that the results do not change drastically as α is changed. The characteristic scale Λ∗ corresponds to a temperature T∗ . If T is clearly above T∗ , the flow can be integrated to scale zero without any instabilities. T∗ is only an upper bound for the temperature where ordering can set in because of order parameter fluctuations at scales below Λ∗ . In two dimensions they are so strong that long-range order that breaks continuous symmetries does not occur at any T > 0, thus “ordering” is to mean either short-range order with a very 24 a) 1 25 b) 23 27 29 10 19 31 17 15 01 k1 y 30 21 0.5 k a /π large correlation length, or ordering in a related system with a small coupling in the third direction, as is present in most materials. 3 1. Antiferromagnetism and Superconductivity 5 10 13 −0.5 20 5 11 7 −1 −1 9 0 k a /π 10 1 x c) 17 15 13 5 k1 19 31 3 20 0 10 11 −10 7 −1 −1 0 10 21 29 y 30 27 0.5 −0.5 30 25 23 0 1 20 k2 d) 1 k a /π The results discussed here are obtained with a slightly smeared-out step-function as cutoff on k (no cutoff on the frequencies) and by dropping the self-energy. Antiferromagnetism. For t′ = 0 and µ = 0, the band is half-filled and the Fermi surface a perfect square. Every vector connecting parallel sides of the Fermi surface is a nesting vector, and ∇ξk = 0 at (π, 0) and (0, π). This strongly enhances particle-hole terms at wavevector Q = (π, π). A random-phase approximation summation of these bubbles results in a divergent static spin susceptibility at Q for any U > 0 at sufficiently low T , indicating the formation of an antiferromagnetic (AF) spin-density wave (SDW), in accordance with mean-field studies (Fulde, 1991). The basic RG results at low T are shown for U = 2t in Fig. 10. The labelling of the N = 32 patches along the Fermi surface can be read off Fig. 10 a). Fig. 10 b) shows V Λ as a function of the patch indices κ1 and κ2 , at Λ∗ ∼ 0.16t and with κ3 = 1 (i.e. k3 near (−π, 0)). Strongly enhanced repulsive interactions appear as a vertical line at κ2 = 24 (i.e. for k2 − k3 = Q), almost κ1 -independent, and as a horizontal line at κ1 = 24 (corresponding to k1 − k3 = Q) with only a weak dependence on κ2 , roughly half as large as the vertical feature. In an extrapolation where the regular profiles are narrowed down to delta functions with an appropriate prefactor J, V Λ (κ1 , κ2 , κ3 ) = J (2δk2 −k3 ,Q + 4 δk1 −k3 ,Q ), corresponding to a mean-field AF spin iniQ·(Ri −Rj ) teraction Hamiltonian J Si · Sj , with i,j e 1 + Si = 2 ci σci . The effective Hamiltonian consisting of the low-scale hopping term and this interaction exhibits AF long-range order at sufficiently low T . An analysis of the flow of susceptibilities(Halboth and Metzner, 2000a; Honerkamp et al., 2001) as described in Sec. II.F confirms this picture. The extrapolation to a mean-field Hamiltonian is a drastic oversimplification, in which the spin fluctuations are lost, but they are retained in the V Λ obtained by the RG flow. As the leading instability is clearly exposed by this analysis, one can also resort to a bosonized description that treats the collective infrared physics (Baier et al., 2004). d-wave Cooper pairing. For t′ = −0.3t and µ = −1.2t, the Fermi surface still contains the saddle points (π, 0) and (0, π) but is curved away from these points (Fig. 10c)). Now Cooper pair scattering dominates, well visible in Fig. 10d) on the diagonal lines k1 + k2 = 0 ( |κ1 −κ2 | = N/2 in terms of patch indices). It is attractive when the incoming pair k1 , −k1 is near the same saddle point (±π, 0) as the outgoing pair k3 , −k3 , and repulsive when incoming and outgoing pairs are at different saddle points. This is the symmetry of the formfactor 9 0 kxa /π 1 10 k 20 30 2 FIG. 10 (Color online) N -patch functional RG data obtained with the momentum-shell functional RG for the repulsive Hubbard model on the 2D square lattice. Upper plots: µ = 0, t′ = 0 and initial U = 2t, lower plots: µ = 1.2t, t′ = −0.3t, U = 3t. To the left: Fermi surfaces for the two cases and the N = 32 discretization points for the two incoming k1 , k2 and the 1st outgoing wavevector k3 . To the right: the coupling function V Λ∗ (κ1 , κ2 , κ3 ) with κ3 = 1 and κ1 and κ2 moving around the Fermi surface. The colorbars on the right indicate the values of the interactions. d(k) = d0 (cos kx − cos ky ) for dx2 −y2 -Cooper pairing. In an extrapolation as above, V Λ (k1 , k2 , k3 ) gives rise to the mean-field Hamiltonian Λ HdSC = VdSC k,k′ d(k)d(k′ ) c† ′ ,↑ c† ′ ,↓ c−k,↓ ck,↑ . k −k which has a d-wave singlet-paired ground state. This d-wave pairing instability was found in a number of studies using different functional RG schemes (Halboth and Metzner, 2000a,b; Honerkamp, 2001; Honerkamp and Salmhofer, 2001a,b; Honerkamp et al., 2001; Tsai and Marston, 2001; Zanchi and Schulz, 1998, 2000), in a rather large parameter region. This constitutes convincing evidence that the weakly coupled Hubbard model possesses a d-wave superconducting ground state. Interplay of AF and SC. In Fig. 10d), the sign structure of the d-wave term goes together, and fits perfectly with, enhanced repulsive interactions near κ1 = 8 and κ2 = 24, which are the remnants of the SDW feature in Fig. 10b). Their larger width is due to the Fermi surface curvature. As Λ is decreased, these SDW features appear first, due to approximate nesting at high scales, and then create an attractive component in the dx2 −y2 -pairing channel, which then grows as Λ is lowered further, while the SDW is cut off by Fermi surface curvature, as discussed also in Appendix B.3. When the SDW-enhancing terms are removed by hand from the right hand side of the RG equation, the d-wave terms are suppressed as well. Thus 25 the d-wave pairing interaction is induced by AF-spin fluctuations that appear on higher scales. At fixed U , t and t′ , there is a sizable interval of µ for which the Fermi surface remains close to the saddle points. Since both AF-SDW and d-wave SC are driven by repulsive scattering between (π, 0) and (0, π), both grow and reinforce one another. In the saddle point regime, it becomes impossible to single out one over the other in the truncation used here. By analogy with the quasi-one-dimensional ladder systems, it has been argued that in this regime, the Fermi surface gets truncated (Furukawa et al., 1998; Honerkamp et al., 2001; L¨uchli et al., 2004). a 2. Ferromagnetism vs. Superconductivity At the Van Hove filling, ferromagnetic (FM) tendencies are enhanced by the logarithmic divergence of the density of states, and the Stoner criterion for the bare interaction suggests an FM ordered state at arbitrarily small U . However, the Van Hove singularities also make the O(U 2 ) Cooper pair scattering log2 -divergent, hence put the two terms into direct competition. As discussed in Section II.D.1, the momentumshell cutoff artificially suppresses FM. For this reason, the T -flow (see Section II.D.2) was invented (Honerkamp and Salmhofer, 2001a,b), and we discuss results obtained by T -flow here. The main difference to the AF/SC scenario discussed above is that at zero transfer momentum, scattering processes driving FM must have the opposite sign from those driving singlet SC, hence mutually suppress one another. This simple picture is confirmed by the RG with momentum-dependent vertices, in a study where t′ and µ are varied at fixed U and t, such that the Fermi surface always contains the saddle points: near to t′ = −t/3, T∗ gets strongly suppressed, hinting at a quantum critical point between the dSC and FM phases (lower left plot in Fig. 11). These results were later confirmed by a two-particle self-consistent approach (Hankyevych et al., 2003) and in the so-called Ω-scheme, which employs a soft infrared regulator on the Matsubara frequencies (Husemann and Salmhofer, 2009); see the lower right plot in Fig. 11. In the latter study, the N -patch scheme was replaced by a parametrization of the vertex functions in terms of exchange bosons. The much higher value of Λ∗ in the transitional regime near t′ = −t/3 is believed to be due to a form factor that was not fully resolved there. 3. Charge instabilities The effective interaction develops a pronounced momentum dependence also in the charge sector. In the forward scattering channel, this amounts to the formation of non-uniform contributions to the Landau interaction. If strong enough, the latter can lead to a Pomeranchuk instability (Pomeranchuk, 1959), that is, a symmetrybreaking deformation of the Fermi surface. In particular, the antiferromagnetic peak drives the combination of couplings VcΛ (κ1 , κ2 , κ3 ) = 2V Λ (κ1 , κ2 , κ3 ) − V Λ (κ2 , κ1 , κ3 ) at certain Q = k3 − k1 . Near to Q ≈ 0 and Q ≈ (π, π), VcΛ (κ1 , κ2 , κ3 ) ≈ −fd (k1 )fd (k2 )Vd (k3 − k1 ) , (113) where fd (k) has the same symmetries as d(k) = cos kx − cos ky , but is more strongly peaked near the saddle points. For Q = (π, π) the corresponding mean-field state is the d-density wave state, which breaks timereversal invariance (Chakravarty et al., 2001) and gaps the single-particle states, except at nodal points on the Brillouin zone diagonal. For forward scattering, Q = 0, the mean-field state only breaks the lattice rotational symmetry of the electronic dispersion and hence of the Fermi surface. This tendency to form a nematic state (Fradkin et al., 2010) via a d-wave Pomeranchuk instability driven by forward scattering interactions was discovered using functional RG (Halboth and Metzner, 2000b). Although the Pomeranchuk instability is not leading in the flow for the Hubbard model (Honerkamp et al., 2002), a nematic state can coexist with the superconducting state (Neumayr and Metzner, 2003; Yamase and Metzner, 2007), and it may get less suppressed by fluctuations since it breaks no continuous symmetry. The d-wave Pomeranchuk instability has been investigated as a possible source of nematicity of the electronic state in relation with experiments on various correlated electron systems (Honerkamp, 2005; Metlitski and Sachdev, 2010c; Okamoto et al., 2010; Yamase, 2009; Yamase and Metzner, 2006). 4. Flows with self-energy effects We briefly summarize functional RG studies where the self-energy has been included. If a frequency-independent vertex function V Λ is directly inserted in the right hand side of Eq. (50), then ΣΛ is real and independent of the frequency, hence only changes the dispersion. This was taken into account in the appendix of Honerkamp et al. (2001), where the adaptive scale decomposition method later detailed in Salmhofer (2007) was used. To keep the density fixed, µ is adjusted as a function of Λ. Since the interaction grows in the flow, it is a nontrivial check of the validity of the truncation that the feedback from the interaction does not lift the low-kinetic-energy modes to high energies, which would drastically shift the Fermi surface and lead to spurious divergences. The first study by Honerkamp et al. (2001) showed that the Fermi surface tends to become flatter as Λ decreases, but that it indeed shifts very little before the flow is stopped at Λ∗ . Thus including the real part of the self-energy does not lead to any essential changes in the AF/SC scenario described above. However, correlations that only feed on the immediate vicinity of the saddle points, like FM, are 26 0.16 0.12 liquid temperature dependence and to correlate with the strength of the generated d-wave pairing interaction (Ossadnik et al., 2008), similar to what is observed experimentally in overdoped cuprates. More refined studies of the frequency-dependence revealed, however, that a simple parametrization in terms of a quasiparticle weight is insufficient (Katanin and Kampf, 2004; Rohe and Metzner, 2005). It was shown that near Λ∗ , the small-|ω|-behavior of ΣΛ (ω, k) leads to a split-up of the quasiparticle peak. All these findings are consistent with an anisotropic break-up of the Fermi surface that one would like to connect with the phenomenology of the high-Tc cuprates (Honerkamp et al., 2001; Lee et al., 2006), but a quantitative comparison is difficult due to the strongly coupled nature of the cuprates. 0.14 0.1 T/t T/t 0.1 0.06 0.06 d−wave regime 0.04 0.08 AF regime 0.04 0.02 saddle point regime 0.02 0 0.08 0.07 0.06 0.05 0.04 0.03 µ/t 0.02 0.01 0 0 −0.01 0 d−wave regime −0.8 −0.9 −1 µ/t −1.1 −1.2 −1.3 0 10 10 AF dominated d−wave dominated −1 10 Stoner −1 10 −2 10 c AF Ω Tc / t approximate nesting regime 0.12 0.08 −3 10 FM dominated −4 10 −2 10 FM dSC −5 10 0 0.1 0.2 0.3 −t’ / t 0.4 0.5 −3 10 0 0.1 0.2 −t’ 0.3 0.4 0.5 FIG. 11 (Color online) Leading instabilities as found by N patch functional RG in the t-t′ -Hubbard model. Left upper plot: T∗ vs. µ for band filling larger than unity, at t′ = −0.3t and U = 3t. There is a high-energy-scale AF SDW instability with a weaker dx2 −y 2 -wave pairing instability when the AFSDW is cut off. Data from Honerkamp (2001). Right upper plot: Data for the same t′ and U on the ’hole-doped’ side with band fillings smaller than one, from Honerkamp et al. (2001). Now there is a broad crossover ’saddle point regime’ between the nesting-driven AF-SDW instability and the dx2 −y 2 -wave pairing regime. Lower left plot: T∗ vs. t′ at the Van Hove filling where the Fermi surface contains the points (π, 0) and (0, π). For large t′ one finds a ferromagnetic instability. Data from Honerkamp and Salmhofer (2001a,b) obtained with the T -flow. Right lower plot: Ω∗ vs. t′ at Van Hove filling, now obtained with the simplified vertex parametrization of Husemann and Salmhofer (2009) and with a soft frequency regulator Ω. affected more strongly, and a full analysis of the coupled flow of self-energy and vertex directly at the saddle points remains an open problem, in spite of partial results (Feldman and Salmhofer, 2008b). The imaginary part and the frequency-dependence of the self-energy can be approximated by inserting the integrated flow of the interaction vertex in the self-energy equation (Honerkamp, 2001). This effectively includes two-loop frequency-dependence effects, and captures the T 2 -dependence of the quasiparticle scattering rate in a Fermi-liquid situation and the exponent of the vanishing quasiparticle weight in the Luttinger liquid up to second order in the bare couplings (Honerkamp and Salmhofer, 2003). For the 2D Hubbard model, the quasiparticle lifetime and renormalization factor was calculated in Honerkamp (2001); Honerkamp and Salmhofer (2003), exhibiting a strongly k-dependent quasiparticle degradation as Λ∗ is approached. This trend was also found by Zanchi (2001) in a slightly different approximation for the self-energy, and is also robust in a more elaborate treatment (Katanin, 2009), where the six-point vertex was included partially. The anisotropy of the quasiparticle lifetime was found to have a non-Fermi- C. Pnictide superconductors The functional RG has been very useful in the study of the newly discovered iron pnictide superconductors (Hirschfeld and Scalapino, 2010; Ishida et al., 2009; Norman, 2008). Here the functional RG may work even better, as the pnictides are less strongly correlated than the high-Tc cuprates. This can already be inferred from the experimental phase diagram, where one only finds metallic antiferromagnetic phases (if at all), but never Mott insulating antiferromagnetism. Theoretical works that try to assess the iron d-orbital onsiteinteraction strengths find values that put the materials into the range of weak to moderate correlations (Anisimov et al., 2009; Miyake et al., 2010). Regarding the electronic structure, the pnictides are more complex than the cuprates. At least three of the five iron dorbitals have non-negligible weight near the Fermi level (Daghofer et al., 2010; Mazin et al., 2008). Therefore, even if one is only interested in the vicinity of the Fermi surface, the multi-band character has to be kept. The Fermi surface (see Fig. 12 b)) is divided into two hole pockets, centered around the origin of the Brillouin zone at k = 0, and two electron pockets around k = (π, 0) and k = (0, π) in the unfolded zone corresponding to the small unit cell with one iron atom (or k = (π, π) in the folded zone corresponding to the large unit cell with two iron atoms). As pointed out early (Kuroki et al., 2008; Mazin et al., 2008), there is approximate nesting of electron- and hole pockets which enhances particlehole susceptibilities with the wavevector connecting these pockets. In addition, depending on the parameters and approximations (Ikeda et al., 2010), there can be a third hole pocket at (π, π) in the unfolded zone. The first N -patch studies of the pnictides were performed by Wang et al. (2009a, 2010, 2009b) for a fiveband model. These authors obtained a sign-changing s-wave pairing instability driven by AF fluctuations as the dominant pairing instability. Further they found strongly anisotropic gaps around the electron pockets, with possibility of node formation. The basic struc- 27 ture of the phase diagram with the sign-changing pairing gap between electron- and hole-pockets can be understood already from simplified few-patch RG approaches (Chubukov et al., 2008). This would however predict isotropic gaps around these pockets (Platt et al., 2009). To understand the gap anisotropy one has to take into account the multi-orbital nature of the electronic spectrum in the iron pnictides, as was done already in the initial studies (Wang et al., 2009a, 2010, 2009b). In order to understand this point, let us start with a single-particle Hamiltonian in wavevector-Fe-d-orbital space h(k)oo′ c† ck,o′ ,s k,o,s H= (114) where the matrices h(k)oo′ take into account intra- and inter-orbital terms for orbital index o = o′ or o = o′ respectively. s is the spin quantum number. The energy bands are obtained by a unitary transformation from orbital to band operators (index b), ck,b,s = o ubo (k)ck,o,s . The standard choice for the interaction between the electrons is to introduce orbital-dependent intra- and inter-orbital onsite repulsions, plus Hund’s rule and pair hopping terms. While these local terms lead to k-independent interactions in the orbital basis, parametrized by a tensor Vo1,o2,o3,o4 , after the transformation to bands one arrives at a k-dependent interaction function k,s,o Vo1,o2,o3,o4 ub1,o1 (k1 )ub2,o2 (k2 )u∗ (k3 )u∗ (k4 ) . b3,o3 b4,o4 Vb1,b2,b3,b4 (k1 , k2 , k3 , k4 ) = (115) o1,o2,o3,o4 Summarizing this brief section, the iron superconductors pose an interesting problem where the functional RG has been instrumental in obtaining the main ordering tendencies in good agreement with current experiments. For future research, one goal should be to make the functional RG a useful bridge between ab-initio descriptions providing the effective model at intermediate energy scales and the many-body effects seen in the experiments at low scales. In particular it will be interesting to relate experimentally observed materials trends in, e.g., the gap structure or the energy scales of the different systems, to changes in the microscopic Hamiltonian taken from ab-initio descriptions. Furthermore, the functional RG studies may have to be extended to include the dispersion orthogonal to the iron-pnictide planes, as this would yield additional possibilities for nodes in the gap function (Hirschfeld and Scalapino, 2010; Ishida et al., 2009; Norman, 2008; Platt et al., 2011b). a) b) 0.2 π 0.1 0.0 78 79 80 65 66 67 77 76 68 69 75 74 73 72 71 70 46 45 44 c) 42 41 33 34 40 39 35 36 37 30 −0.1 29 28 26 13 32 0 17 16 32 48 64 63 25 −100 38 5 49 50 23 20 21 0 62 61 60 59 64 24 9 1 18 19 cdw − 50 PI sdw d−wave sc −150 27 31 −0.2 0 43 47 48 ky The combination of ubo s behind the interaction tensor is sometimes called the ’orbital make-up’ (Graser et al., 2009; Maier et al., 2009). These prefactors cause a marked k-structure already in the initial interaction which is then renormalized during the functional RG flow. It turns out that this orbital make-up has an essential influence on the competition between different channels in the flow and is responsible for the gap anisotropies found in the multi-band functional RG studies by Wang et al. (2009a, 2010, 2009b) and in subsequent functional RG studies (Platt et al., 2011a; Thomale et al., 2011a). A typical result for the predicted pairing gaps is shown in Fig. 12 a). Note that according to the functional RG analysis, the pairing state should be strongly doping-dependent (Thomale et al., 2011a,b, 2009). 22 57 −200 s+ −wave sc − 56 51 52 53 54 π kx 58 55 −250 10−3 10−2 10−1 Λ [eV ] 100 FIG. 12 (Color online) Functional renormalization group results for the iron pnictide compound LaFeAsO at moderate hole doping. a) Superconducting form factor as the outcome of functional RG, plotted versus the position on the hole pockets at Γ and M and electron pockets at X, numbered as depicted in b). The competing fluctuations manifest themselves in diverging ordering susceptibilities at low RG scales as shown in c), including in particular spin density wave (SDW), superconductivity (SC), Pomeranchuk (PI) and charge density wave (CDW) instabilities (Thomale et al., 2011a). D. Other systems Besides the above-described two larger fields of application, the functional RG truncations described in this Section have also been employed in a number of other models in strongly correlated electron physics. Here we briefly list some of these activities. In relation to possible unconventional superconductivity in organic crystals and layered cobaltates, Hubbardtype models on the triangular lattice have been studied (Honerkamp, 2003; Tsai and Marston, 2001). At large U , the spin exchange interaction between the sites of the triangular lattice is geometrically frustrated, leading to a much weaker appearance of antiferromagnetism and a possible non-magnetic insulating phase (Morita et al., 2002; Sahebsara and S´n´chal, e e 2008; Yoshioka et al., 2009). At weak coupling and for nearest-neighbor hopping, Fermi surface nesting is ab- 28 sent, so that near to or at half band filling, only low-scale Kohn-Luttinger-like superconducting instabilities occur out of an innocuous Fermi liquid. However, there appears to be a strong dependence on details of the microscopic modelling. To study interaction effects in graphene, the N -patch functional RG has been applied to the extended Hubbard model on the honeycomb lattice. In nominally undoped graphene, the Fermi surface becomes a set of Dirac points where the density of states vanishes, and no instabilities are found for sufficiently small interactions. If the interaction strength exceeds a certain value, various instabilities driven by particle-hole fluctuations between the two Dirac points (Honerkamp, 2008) are found. Interestingly, for larger second-nearest neighbor interactions, there is the possibility of an instability towards a quantum spin Hall phase (Raghu et al., 2008). However, a spin-liquid phase for intermediate strength of the Hubbard onsite repulsion that was recently found in quantum Monte Carlo calculations (Meng et al., 2010) is not reflected in the functional RG results on this level of approximation. When the Fermi level is moved away from the Dirac points, the functional RG again detects pairing instabilities. In the case of dominant nearest neighbor repulsion, the leading pairing tendency is in the f -wave triplet channel (Honerkamp, 2008). The unbiasedness of the functional RG, and the access it gives to k- and ω-dependence of vertex functions, is also of great use in (quasi-)one-dimensional models. The half-filled extended Hubbard model in one dimension has been studied in the search for bond-order-wave phases, which could indeed be found with a refined patching of the k-dependence of the interaction away from the Fermi points (Tam et al., 2006). For quasi-1D models with a small transverse hopping in a second direction the change from a gapless Luttinger liquid in a strictly one-dimensional situation to Fermi liquid instabilities toward ordering can be monitored as a function of the transverse hopping (Honerkamp and Salmhofer, 2003). The Fermi surface in coupled metallic chains was studied by Ledowski and Kopietz (2007a,b); Ledowski et al. (2005). The possibility of triplet pairing driven by density wave fluctuations has been explored in such situations (Nickel et al., 2005, 2006). In these quasi-onedimensional systems, including the frequency dependence of the interaction vertex is numerically more feasible than in two dimensions. This has been used to study the interplay of phonon-mediated and direct electronelectron interactions for chains (Tam et al., 2007a), ladders (Tam et al., 2007b) and systems with small transverse hopping (Bakrim and Bourbonnais, 2010). Many-fermion lattice Hamiltonians can also be realized with ultracold atoms in optical lattices, opening up new directions. For example, mixtures of more than two hyperfine states (Honerkamp and Hofstetter, 2004) and boson-mediated pairing on two-dimensional lattices (Klironomos and Tsai, 2007; Mathey et al., 2006, 2007) have been investigated using fermionic N -patch methods. Another promising development is the application of the functional RG to quantum spin systems (Reuther and W¨lfle, 2010). Here, an auxiliary-fermion o representation is used for the spins in generalized Heisenberg models, and the functional RG can be formulated in terms of these fermions. As important difference to systems of itinerant electrons, in the quantum spin system the kinetic energy for the pseudo-fermions is zero and the interactions only depend on one spatial or wavevector variable. This allows one to keep the full frequency dependence of the self-energy and interaction vertex on the imaginary axis, in the usual truncation where the six-point vertex is neglected. The Katanin modification (Katanin (2004), see also Section II.C.2) of the flow hierarchy turns out to be crucial here. If it is employed, the auxiliary-fermion functional RG describes the transitions from N´el order to collinear order through an intermedie ate paramagnetic phase in the J1 -J2 spin-1/2 model on the square lattice as function of J1 /J2 in good agreement with numerical approaches. Furthermore, similar systems on the triangular lattice (Reuther and Thomale, 2011) and with longer-ranged couplings (Reuther et al., 2011) were studied. The success of a relatively simple truncation in such an intrinsically strongly coupled system is explained by these authors in that the diagrams summed in this flow contain the leading contributions in both 1/N - and 1/S-expansions plus particle-particle diagrams, hence those contributions that are believed to be most important. IV. SPONTANEOUS SYMMETRY BREAKING In many interacting Fermi systems a symmetry of the bare action is spontaneously broken at sufficiently low temperatures and, in particular, in the ground state. In the fermionic flow equations, the common types of spontaneous symmetry breaking such as magnetic order or superconductivity are associated with a divergence of the effective two-particle interaction at a finite scale Λc > 0, in a specific momentum channel. In Sec. III we discussed several examples for such divergences. The truncation for the effective two-particle vertex leading to the N -patch scheme described and used in Sec. III is insufficient to describe the symmetry-broken phase. To continue the flow below the scale Λc , an appropriate order parameter has to be introduced. There are two distinct ways of implementing spontaneous symmetry breaking in the functional RG. In one approach the fermionic flow is computed in presence of a small (ideally infinitesimal) symmetry breaking term added to the bare action, which is promoted to a finite order parameter below the scale Λc (Salmhofer et al., 2004). A relatively simple truncation of the exact flow equation captures spontaneous symmetry breaking in mean-field models such as the reduced BCS model exactly, although the effective two-particle interactions diverge. Another possibility is to decouple the interac- 29 tion by a bosonic order parameter field, via a HubbardStratonovich transformation, and to study the coupled flow of the fermionic and order parameter fields (Baier et al., 2004). In case of competing instabilities a reliable calculation based on either of the above-mentioned routes to symmetry breaking is quite involved. For a rough estimate of order parameters and phase diagrams, one may also neglect low energy fluctuations and combine flow equations at high scales with a mean-field treatment at low scales. In this functional RG + mean-field approach, one stops the flow of the effective two-particle interaction at a scale ΛMF > Λc , that is, before it diverges. The remaining low energy degrees of freedom are treated in mean-field approximation, with a reduced effective interaction extracted from the effective two-particle vertex Γ(4)ΛMF . In a first application of this “poor man’s” approach to symmetry breaking the interplay and possible coexistence of antiferromagnetism and d-wave superconductivity in the (repulsive) two-dimensional Hubbard model were studied (Reiss et al., 2007). As in any hybrid method, the results depend quantitatively on the choice of the intermediate scale ΛMF (except for mean-field models), and there is no unique criterion for this choice. We now review the purely fermionic and the HubbardStratonovich approaches to spontaneous symmetry breaking in the functional RG. The methods will be explained for the case of a superconductor as a prototype for continuous symmetry breaking, and the reader is referred to the literature on applications involving other order parameters. A. Fermionic flows The effective action ΓΛ obtained from the exact flow equation or from symmetry-conserving truncations thereof exhibits the same symmetries as the bare action S. To analyze spontaneous symmetry breaking, one therefore has to add a symmetry-breaking term δS to the bare action and compute the flow of ΓΛ in presence of this term. In case of spontaneous symmetry breaking an arbitrarily small symmetry breaking term is promoted to a finite order parameter at a scale Λc , which survives until the end of the flow. A crucial issue is then to find a managable truncation of the exact flow equation which captures the essential features of the flow into the symmetry broken phase. This is non-trivial since the effective two-particle interactions driving the symmetry breaking become large. Indeed, truncations based on neglecting vertices Γ(2m)Λ with m > 2 in the hierarchy of flow equations fail miserably. A benchmark for truncations is the requirement ¯ ΓΛ [Ψ, Ψ] = Γ(0)Λ − that they should at least provide a decent solution for mean-field models. This requirement is met by an approximation introduced by Katanin (2004) to implement Ward identities in truncated flow equations. Katanin’s truncation, which was described already in Sec. .II.C, conS d dΛ = Σ Γ d dΛ = Γ Γ G Γ FIG. 13 Coupled flow equations for the self-energy and the two-particle vertex determining the fermionic flow with symmetry breaking. sists of two coupled flow equations for the self-energy ΣΛ and the two-particle vertex Γ(4)Λ , see Fig. 13. They are almost identical to the first two equations in the hierarchy described in Sec. II.C, with Γ(6)Λ = 0, but in the flow equation for Γ(4)Λ the single-scale propagator S Λ is replaced by ∂Λ GΛ = S Λ + GΛ ∂Λ ΣΛ GΛ . This modification takes tadpole contributions obtained from contractions of the three-particle vertex Γ(6)Λ into account. It is easy to see that the Katanin truncation solves mean-field models for symmetry breaking such as the Stoner model for ferromagnetism or the reduced BCS model exactly (Salmhofer et al., 2004). The exact selfenergy in such models is given by the Hartree-Fock term Σ = V G (schematically), where V is the bare interaction, and the two-particle vertex by a ladder sum of the form Γ(4) = V (1 − GGV )−1 . These equations hold also in presence of a cutoff Λ. Applying Λ-derivatives one finds immediately that ΣΛ and Γ(4)Λ obey flow equations of the (schematic) form ∂Λ ΣΛ = Γ(4)Λ S Λ and ∂Λ Γ(4)Λ = Γ(4)Λ ∂Λ (GΛ GΛ )Γ(4)Λ , which corresponds exactly to Katanin’s truncation. To be more specific, we now consider the case of singlet superconductivity, where the continuous U (1) symmetry associated with charge conservation is spontaneously broken, while spin-rotation invariance remains conserved. Superconductivity can be induced by adding a term of the form δS = k ¯ ¯ ∆0 (k)ψ↑ (k)ψ↓ (−k) + ∆∗ (k)ψ↓ (−k)ψ↑ (k) , 0 (116) with a (generally complex) external pairing field ∆0 (k), to the bare action. It is convenient to use Nambu spinors ¯ ¯ ¯ Ψα (k) and Ψα (k) with Ψ+ (k) = ψ↑ (k), Ψ+ (k) = ψ↑ (k), ¯ − (k) = ψ↓ (−k), Ψ− (k) = ψ↓ (−k). The effective action ¯ Ψ as a functional of the Nambu fields, truncated beyond two-particle terms, has the form ¯ Γ(2)Λ2 (k) Ψα1 (k)Ψα2 (k) α1 α k α1 ,α2 G 30 + 1 4 ¯ ¯ Γ(4)Λ2 α3 α4 (k1 , k2 , k3 , k4 ) Ψα1 (k1 )Ψα2 (k2 )Ψα3 (k3 )Ψα4 (k4 ) . α1 α Due to spin-rotation invariance only terms with an equal ¯ number of Ψ and Ψ fields contribute. The Nambu propagator GΛ = (Γ(2)Λ )−1 can be written as a 2 × 2 matrix of the form GΛ (k) GΛ (k) ++ +− GΛ (k) GΛ (k) −+ −− (117) k1 ,...,k4 α1 ,...,α4 propagator, and d (V Λ ± W Λ ) = −(V Λ ± W Λ )2 dΛ d |GΛ (k)|2 ∓ |F Λ (k)|2 . (123) × dΛ ¯ (−ik0 + ξk ) ψσ (k)ψσ (k) 0.06 k,σ ∆Λ ¯ S[ψ, ψ] = = ¯ ¯ V (k, k ) ψ↑ (k)ψ↓ (−k)ψ↓ (−k ′ )ψ↑ (k ′ ).(119) ′ + k,k′ Note that the interaction is restricted to particles with strictly opposite momenta and spins. It is well-known that mean-field theory solves this model exactly in the thermodynamical limit (Haag, 1962; M¨hlschlegel, 1962). u The restricted momentum dependence of the bare interaction carries over to similar restrictions for the effective two-particle vertex Γ(4)Λ in Eq. (117). Only two independent components appear, namely (4)Λ V Λ (k, k ′ ) = Γ+−+− (k, k ′ , k ′ , k) , Λ ′ W (k, k ) = (4)Λ Γ++−− (k, k ′ , k ′ , k) (120) . where d Λ dΛ F |∆Λ fixed k d FΛ dΛ ∆Λ fixed 0.02 0 0 0.02 0.04 0.06 0.08 Λ −50 −100 0 0.02 0.04 0.06 0.08 Λ FIG. 14 (Color online) Flow for a reduced BCS model with a constant density of states at zero temperature; the band width is one and the bare interaction V = −0.3. Left: Flow of the gap ∆Λ ; the thick line is for a bare gap ∆0 = 2.4·10−4 and the thin line for ∆0 = 6 · 10−8 , in units of the band width. Right: Flow of the linear combinations V Λ + W Λ (solid lines) and V Λ − W Λ (dashed lines) of normal and anomalous vertices. Thick lines are again for ∆0 = 2.4 · 10−4 and thin lines for ∆0 = 6 · 10−8 . (121) The first component is a normal interaction between two particles, and its initial value V Λ0 (k, k ′ ) is the bare interaction. The second component is an anomalous term describing the creation of four particles. It is initially zero, but is generated by charge symmetry breaking terms in the course of the flow. Another anomalous term describing the destruction of four particles is given by the complex conjugate of W Λ (k, k ′ ). The diagonal element of the Nambu self-energy vanishes for the reduced BCS model, while the off-diagonal element is given by the gap function ∆Λ (k). For the special case of a momentum-independent swave interaction V , the flow equations obtained from the procedure described above are particularly simple. Choosing a momentum-independent and real bare gap ∆0 > 0, the flowing quantities ∆Λ , V Λ and W Λ are real and momentum-independent, too. Their (exact) flow is given by d Λ ∆ = −(V Λ + W Λ ) dΛ 0.04 (VΛ ± WΛ) GΛ (k) F Λ (k) . k ∗Λ F (k) −GΛ (−k) (118) A typical flow for an attractive bare interaction V < 0 It is instructive to discuss the flow of the superconis shown in Fig. 14, for two different choices of the bare ducting gap and the two-particle vertex for the reduced gap ∆0 . The gap increases monotonically from the iniBCS model (Salmhofer et al., 2004), which is defined by 0.08 0 an action of the form GΛ (k) = , (122) is the anomalous Nambu single-scale tial value ∆0 upon lowering Λ, and reaches a finite value ∆ ≫ ∆0 for Λ → 0. A finite ∆0 regularizes the squareroot singularity in the gap flow at Λ = Λc . The normal vertex V Λ reaches a large negative value at the critical scale Λc , while the anomalous vertex W Λ becomes large and positive. The linear combination V Λ + W Λ , which drives the gap flow, is also strongly negative at Λc , but it saturates at a moderately negative value for Λ below Λc . By contrast, V Λ − W Λ decreases monotonically and reaches a final value of order 1/∆0 for Λ → 0, which diverges for ∆0 → 0. This divergence is the mean-field remnant of the Goldstone mode associated with the broken continuous symmetry. In the case of a discrete broken symmetry, the effective interaction becomes large only at the critical scale, while no large components remain for Λ → 0. This has been exemplified in a study of the RG flow of a mean-field model for a commensurate charge density wave (Gersch et al., 2005). The performance of the Katanin truncation for models with full (not reduced) interactions has not yet been fully explored, since an accurate parametrization of the flowing vertex is quite demanding. However, the results obtained so far are encouraging. Staying with su- 31 perconductivity as a an example, the (Nambu) vertex contains 16 components, most of them corresponding to anomalous interactions. In addition to the anomalous terms appearing already in the reduced BCS model, there are anomalous interactions corresponding to the creation of three particles and destruction of one particle, and vice versa (Gersch et al., 2008; Salmhofer et al., 2004). Making full use of spin-rotation invariance, all the Nambu components can be actually expressed by only three independent functions of momenta and frequencies (Eberlein and Metzner, 2010). The main challenge is an adequate parametrization of the (three-fold) momentum and frequency dependence, since singularities associated with symmetry-breaking and the Goldstone mode appear in the course of the flow. Surprisingly, in a test case study for the weakly attractive Hubbard model, a rather crude parametrization using the N patch discretization described in Sec. III turned out to yield a reasonable flow into the superconducting phase, with results for the gap in good agreement with results obtained earlier by other means (Gersch et al., 2008). This is encouraging, but the low energy fluctuations are clearly not well described in such a parametrization. To deal with the singular momentum and frequency dependence in the Cooper channel (and possibly also in the forward scattering channel), the channel decomposition devised by Husemann and Salmhofer (2009) seems very useful, since it allows one to isolate singular dependences in functions of only one momentum and frequency variable, similar to a description of singular interactions by exchange bosons. The channel decomposition has been formally extended already to the superconducting state (Eberlein and Metzner, 2010), but a concrete calculation beyond mean-field models has not yet been performed. In systems with a first order phase transition one may miss the symmetry broken phase if one tests only for local stability of the symmetric phase by offering a small symmetry breaking field, since the latter may be metastable. However, one can escape from the metastable state by adding a scale dependent symmetry breaking counterterm RΛ to the effective action, which has to be choosen sufficiently large at the beginning of the flow and fades out for Λ → 0, such that the system is ultimately not modified. Formally this is just another choice of regularization within the general framework described in Sec. II.B. The counterterm method has been implemented for the exactly soluble test case of a chargedensity wave mean-field model by Gersch et al. (2006). Popular approximations also beyond mean-field theory can be retrieved from the functional RG by keeping a suitable subset of contributions. In particular, the Eliashberg theory for frequency-dependent (usually phonon-induced) pairing interactions can be obtained as an approximation to the exact flow equations both in the symmetric (Tsai et al., 2005) and in the symmetrybroken state (Honerkamp and Salmhofer, 2005). This is achieved by keeping the Cooper channel for zero total momentum and frequency and the crossed particle-hole channel for zero transfer in the flow of the interaction, and the Fock term for the self-energy. B. Flows with Hubbard-Stratonovich fields Collective order parameter fluctuations associated with spontaneous symmetry breaking in interacting many-body systems are often treated by introducing an auxiliary order parameter field via a HubbardStratonovich transformation (Popov, 1987). A combination of the functional RG with the Hubbard-Stratonovich route to spontaneous symmetry breaking in an interacting Fermi system was first used by Baier et al. (2004). They studied the formation of an antiferromagnetic state in the repulsive two-dimensional Hubbard model at halffilling and managed to recover the low-energy collective behavior (described by a non-linear sigma model) from a truncated set of coupled flow equations for the fermions and the order parameter field. In the following we describe the method for the case of a superfluid phase, summarizing the work of several groups. We consider an interacting continuum or lattice Fermi system with a local attraction V < 0. For continuum systems a suitable ultraviolet regularization is necessary. A local attraction can act only between particles with opposite spin and leads to singlet pairing. It is thus natural to decouple this interaction by a Hubbard-Stratonovich transformation with a complex bosonic field φ(q) corresponding to the bilinear composite of fermionic fields V k ψ↓ (−k)ψ↑ (k + q). This leads to an action of the form ¯ S[φ, ψ, ψ] = − k,σ mb + 2 + k,q ¯ ψσ (k) (ik0 − ξk ) ψσ (k) φ∗ (q)φ(q) q ¯ ¯ ψ↑ (k + q)ψ↓ (−k)φ(q) + h.c. , (124) where φ∗ is the complex conjugate of φ and mb = −1/V > 0. Spontaneous symmetry breaking can now be studied by using the flow equation for the effective action ¯ ΓΛ [φ, ψ, ψ] for coupled bosonic and fermionic fields derived in Sec. II.G. Relatively simple truncations capture several non-trivial fluctuation effects. Effective interactions beyond quartic order in the fields are generally neglected. Also boson-fermion vertices beyond the order appearing already in the bare action are discarded. The truncations are usually formulated as an ansatz for the effective average action ¯ ¯ ΓΛ [φ, ψ, ψ] = ΓΛ [φ, ψ, ψ] − regulator term R ¯ ¯ = ΓΛ [φ] + ΓΛ [ψ, ψ] + ΓΛ [φ, ψ, ψ] , (125) b f bf which obeys the initial condition ΓΛ0 = S, see Sec. II.G. R 32 The ansatz used for the bosonic part is guided by the usual strategy of a double expansion in φ and gradients (see, for example, Tetradis and Wetterich (1994)): ΓΛ [φ] = b Λ Uloc (φ(x)) + gradient terms , (126) x where x = (x0 , x1 , . . . , xd ) collects imaginary time and real space coordinates. Note that we use the same letter φ for the real space and momentum space representations of the bosonic field. The shape of the local potential Λ Uloc (φ) depends on the scale. For Λ above a critical scale Λc it has the convex form Λ Uloc (φ) = mΛ |φ|2 + uΛ |φ|4 , b (127) with a minimum at φ = 0. For Λ < Λc the potential assumes a mexican hat shape Λ Uloc (φ) = uΛ |φ|2 − |αΛ |2 2 , (128) with a circle of minima at |φ| = |αΛ |, where αΛ is the (flowing) bosonic order parameter. The regime Λ > Λc is called the symmetric regime. At Λ = Λc the bosonic mass mb vanishes. In the symmetry-broken regime, for Λ < Λc , the order parameter αΛ rises continuously from zero to a finite value. Its flow can be computed by tracing the Λ minimum of the flowing potential Uloc or, equivalently, (1)Λ by the condition that the bosonic one-point vertex Γb vanishes. For the gradient terms in ΓΛ [φ] various choices have b been made. The simplest one (Birse et al., 2005; Diehl et al., 2007; Krippa, 2007) compatible with the U (1) symmetry has the form of an inverse bare propagator for free bosons, x q Λ Zb φ∗ (x)∂x0 φ(x) − AΛ φ∗ (x)∇2 φ(x) = b Λ φ∗ (q) −iZb q0 + AΛ q2 φ(q) , b (129) where ∇ = (∂x1 , . . . , ∂xd ). For lattice fermions one may replace q2 by a periodic dispersion ωq which is proportional to q2 only at small q (Strack et al., 2008). The term linear in q0 is absent in particle-hole symmetric systems (Strack et al., 2008), such that contributions of 2 order q0 become important. Additional gradient terms have to be taken into account to fully capture the effects of the Goldstone mode, as discussed below. The normal fermionic part of the effective action is usually kept in its bare form, sometimes adjusted by renormalization factors for the frequency and momentum dependences. In the symmetry broken regime, an anomalous term is generated, such that ΓΛ becomes f ¯ ΓΛ [ψ, ψ] = − f + k k,σ Λ ¯ ψσ (k)(iZf k0 − AΛ ξk )ψσ (k) f ¯ ¯ ∆Λ (k)ψ↑ (k)ψ↓ (−k) + h.c. . (130) For a local interaction the k-dependence of the gap function ∆Λ (k) is very weak and in simple truncations fully absent. Quartic terms corresponding to effective twofermion interactions are absent in the bare action by virtue of the Hubbard-Stratonovich decoupling, but are generated again in the course of the flow. These generated terms are neglected in lowest order truncations, and sometimes they are treated by a dynamical decoupling procedure called dynamical bosonization, see below. For the effective boson-fermion interaction one also maintains the bare form of a local 3-point function, ¯ ΓΛ [φ, ψ, ψ] = bf k,q ¯ ¯ g Λ ψ↑ (k + q)ψ↓ (−k)φ(q) + h.c. + anomalous terms , (131) ¯¯ with anomalous terms of the form ψψφ and ψ ψφ∗ contributing only in the symmetry-broken regime. The coupling g Λ is frequently referred to as “Yukawa coupling”. The anomalous terms in the boson-fermion interaction are usually neglected. If taken into account, they remain indeed rather small (Strack et al., 2008). Instead of using a U (1)-symmetric ansatz for the effective action, one may also start from the hierarchy of flow equations for the vertex functions and implement the U (1)-symmetry by Ward identities (Bartosch et al., 2009b). Even with the simple ansatz (129) for the bosonic gradient terms, the effective action described above yields sensible results not only at weak coupling, but actually in the entire regime from BCS superfluidity to Bose Einstein condensation of tightly bound pairs (Diehl et al., 2007). In particular, the transition temperature Tc increases exponentially with the interaction in the weak coupling regime, reaches a maximum, and finally saturates in the strong coupling limit, as it should. The bosonic interaction uΛ and also the bosonic renorΛ malization factors AΛ and Zb vanish in the limit Λ → 0 b (Birse et al., 2005; Krippa, 2007). This fluctuation effect reflects the drastic renormalization of longitudinal order parameter correlations, which are well-known from the interacting Bose gas in dimensions d ≤ 3 (see, for example, Pistolesi et al. (2004)). Note that for a nodeless gap function the low-energy behavior of a fermionic superfluid is equivalent to that of an interacting Bose gas, since fermionic excitations are fully gapped. However, with the simple ansatz (129) the transverse order parameter fluctuations corresponding to the Goldstone mode are also strongly renormalized, which is not correct. To distinguish between longitudinal and transverse fluctuations, one may fix the phase of the order parameter αΛ such that αΛ is real, decompose the complex order parameter field in real and imaginary parts φ(q) = σ(q) + iπ(q) with σ(−q) = σ ∗ (q) and π(−q) = π ∗ (q), and introduce different renormalization factors for σ and π fields (Pistolesi et al., 2004). Using this decomposition, the correct infrared behavior was obtained by Strack et al. (2008) where, however, the cancellation of singular contributions to the renormalization 33 factors for the transverse π fields was implemented by hand. To capture this cancellation intrinsically, one has to include an additional U (1) symmetric gradient term of the form [σ(∂x0 , ∇)σ + π(∂x0 , ∇)π]2 (Strack, 2009; Tetradis and Wetterich, 1994). The fermionic flow based on the Katanin truncation described in Sec. IV.A reproduces the exact solution of the reduced BCS model (and other mean-field models). Within the truncation described above, the bosonized flow yields a reasonable solution without artificial features, but the gap comes out a bit too small. The reason Λ for this is the truncation of Uloc (φ) at quartic order. To recover the exact solution, one has to keep all orders in φ (Strack et al., 2008). ¯ The ansatz (130) for ΓΛ [ψ, ψ] neglects the generation of f fermionic interactions by the flow. In particular, quartic (two-fermion) interactions are generated by box diagrams with four boson-fermion vertices. These terms contain contributions from particle-hole fluctuations which, among other effects, lead to a significant reduction of the transition temperature. The (re-)generated twofermion interaction can be decoupled at each step in the flow by a procedure called dynamical bosonization (Floerchinger and Wetterich, 2009; Gies and Wetterich, 2002, 2004). A general two-fermion interaction cannot be decoupled (exactly) by a single Hubbard-Stratonovich field, such that several fields may be needed to obtain accurate results. Dynamical bosonization was used to include effects from particle-hole fluctuations in attractively interacting Fermi systems by Floerchinger et al. (2008). Following the work of Baier et al. (2004) on the repulsive Hubbard model at half-filling, functional RG flow equations with Hubbard-Stratonovich fields were also applied to the Hubbard model away from half-filling. Commensurate and incommensurate antiferromagnetic fluctuations were investigated (Krahl et al., 2009a). More importantly, it was clarified how the generation of d-wave pairing from antiferromagnetic fluctuations can be captured by a bosonized flow (Krahl et al., 2009b), and the flow was continued into the symmetry broken phase, with coupled order parameter fields describing antiferromagnetism and d-wave superconductivity (Friederich et al., 2010, 2011). Compared to the purely fermionic RG described in Sec. IV.A, the treatment of order parameter fluctuations is facilitated considerably by the Hubbard-Stratonovich field. On the other hand, fluctuation effects associated with other channels (the particle-hole channel in case of superfluidity) look more complicated. For systems with competing instabilities the choice of an adequate Hubbard-Stratonovich field becomes problematic, since the fermionic interaction can be decoupled in different ways, which, in combination with truncations, may lead to ambiguities in the results. In general, several Hubbard-Stratonovich fields must be used, and the analysis done in Section III indicates which ones are the most important. The decomposition of the interaction in (Husemann and Salmhofer, 2009) allows to switch to Hubbard-Stratonovich fields after the fermionic flow has been performed down to a certain scale, and may thus be used to combine the two flow representations. V. QUANTUM CRITICALITY Instabilities of the normal metallic state lead to a rich variety of quantum phase transitions (Sachdev, 1999) in the ground state of interacting electron systems, which can be tuned by a control parameter such as pressure, doping, or a magnetic field. Most interesting are continuous transitions which lead to quantum critical fluctuations (Belitz et al., 2005). Near a quantum critical point (QCP) electronic excitations are strongly scattered by order parameter fluctuations such that Fermi liquid theory breaks down (von Loehneysen et al., 2007; Vojta, 2003). Quantum critical fluctuations are therefore frequently invoked as a mechanism for non-Fermi liquid behavior observed in strongly correlated electron compounds. Quantum phase transitions in interacting Fermi systems are traditionally described by an effective order parameter theory pioneered by Hertz (1976) and Millis (1993). An order parameter field φ is introduced by a Hubbard-Stratonovich decoupling of the fermionic interaction, and the fermionic fields are subsequently integrated out. The resulting effective action for the order parameter is truncated at quartic order and analyzed by standard scaling and RG techniques. However, more recent studies revealed that the Hertz-Millis approach is not always applicable, especially in low-dimensional systems (Belitz et al., 2005; von Loehneysen et al., 2007). Since electronic excitations in a metal are gapless, integrating out the electrons can lead to singular interactions in the effective order parameter action which cannot be approximated by a local quartic term. The nature of the problem was identified and essential aspects of its solution were presented first for disordered ferromagnets by Kirkpatrick and Belitz (1996), and later elaborated on by Belitz et al. (2001a,b). For clean ferromagnets, Belitz et al. (1997) showed that Hertz-Millis theory breaks down, and no continuous quantum phase transition can exist, in any dimension d ≤ 3; the transition is generically of first order (Belitz et al., 1999). The Hertz-Millis approach was also shown to be invalid for the quantum antiferromagnetic transition in two dimensions (Abanov and Chubukov, 2004; Abanov et al., 2003; Metlitski and Sachdev, 2010a). In that case a continuous transition survives, but the QCP becomes non-Gaussian. Applications of the functional RG to quantum phase transitions and quantum criticality in interacting Fermi systems have appeared only recently. In Sec. V.A we explain how the Hertz-Millis theory fits into the functional RG framework and we review some extensions relying on an effective order parameter action truncated at quartic or hexatic order. An application of a non-perturbative truncation, where all orders in φ are (and must be) kept, 34 is discussed in Sec. V.B. Finally, in Sec. V.C we briefly discuss the possibility to study coupled flow equations for fermions and their critical order parameter fluctuations in the functional RG framework, and we refer to first steps in this direction. A. Hertz-Millis theory In his seminal work on quantum phase transitions in metallic electron systems, Hertz (1976) proposed to decouple the electron-electron interaction by introducing an order parameter field φ via a Hubbard-Stratonovich transformation. The resulting action is quadratic in the ¯ fermionic variables ψ and ψ, which can therefore be integrated out. One thus obtains an effective action which depends only on φ. Truncating at quartic order in φ, and discarding irrelevant momentum and frequency dependences (in the sense of standard power counting), leads to the Hertz action. S[φ] = S (0) + + φ(−q) Aq2 + Z q |q0 | φ(q) |q|z−2 Uloc (φ(x)) , (132) x where S (0) is a field-independent term, and Uloc (φ) = rφ2 + uφ4 . (133) We write our equations for the case of a real scalar order parameter for simplicity, using (again) the same letter φ for the real and momentum space representations of the field. Except for the frequency dependence, the action has the form of a φ4 theory for thermal phase transitions. The frequency dependent term stems from low-energy particle-hole excitations. Here the dynamical exponent z is an integer number ≥ 2 depending on the type of transition. Tuning the parameter r one can approach the phase transition, in particular the quantum phase transition at T = 0. The action (132) has been analyzed by standard scaling and RG techniques. Due to the frequency dependence, the scaling behavior at the QCP corresponds to a system with an effective dimensionality deff = d + z, where d is the spatial dimension (Hertz, 1976). As a consequence, the QCP appears to be Gaussian in three- and even in two-dimensional systems. An important insight by Millis (1993) was that the φ4 term in the action is nevertheless crucial to obtain the correct temperature dependences near the QCP. He derived the temperature dependence of the correlation length ξ and other quantities by using a perturbative RG with a mixed momentum and frequency cutoff. From a functional RG perspective, Millis’ scaling theory can be viewed as a simple truncation of the effective average action ΓΛ [φ] evolving from S[φ], namely R ΓΛ [φ] = Γ(0)Λ + R φ(−q) AΛ q2 + Z Λ q |q0 | φ(q) |q|z−2 Λ Uloc (φ(x)) , + (134) x where Λ Uloc (φ) = rΛ φ2 + uΛ φ4 , (135) and Λ parametrizes a mixed momentum and frequency cutoff. The flow equations for the parameters in Eq. (134) are obtained by inserting ΓΛ [φ] in the exact functional R flow equation (91) and comparing coefficients. Due to the local form of the φ4 interaction, the self-energy is momentum and frequency independent such that the parameters AΛ and Z Λ remain invariant. The flow of uΛ , which is driven by a contribution of order (uΛ )2 , is important only in the marginal case d + z = 4 and near the thermal phase transition at Tc > 0. Hence, most of Millis’ results on the region around the QCP in the phase diagram are based on an analysis of the flow of rΛ and the thermodynamic potential ΩΛ = T Γ(0)Λ (for a review, see von Loehneysen et al. (2007)). Various extensions of Millis’ analysis were derived within the functional RG framework. In particular, an extension to the symmetry-broken phase was formulated, for cases where the broken symmetry is discrete and does not gap out the fermionic excitations (Jakubczyk et al., 2008). One such case is a nematic transition driven by a Pomeranchuk instability of interacting electrons on a lattice, where the discrete point-group symmetry of the lattice is spontaneously broken (Fradkin et al., 2010). The symmetry-broken regime was described by the ansatz (134) for ΓΛ [φ], with a quartic local potential which has R a minimum away from zero: Λ Uloc (φ) = uΛ φ2 − (φΛ )2 0 2 . (136) The resulting flow equations were used to compute Tc < and the Ginzburg temperature TG below Tc as a function of the control parameter r. To access the non-Gaussian thermal critical regime near Tc , it is crucial to take the flow of the quartic coupling uΛ into account. The parameters AΛ and Z Λ are now scale dependent, too. While Z Λ remains almost invariant, the flow of AΛ is important near Tc and gives rise to an anomalous scaling dimension. A main result of the calculation was that the leading rdependence of Tc is the same as that of the Ginzburg temperatures below and above Tc (the latter was calculated by Millis (1993)), but a fairly large Ginzburg region opens in two dimensions (Bauer et al., 2011; Jakubczyk et al., 2008). In another extension a φ6 -interaction was included in Λ Uloc to study a possible change of the order of the transition by fluctuations (Jakubczyk, 2009), as well as quantum tricritical points in metals (Jakubczyk et al., 2010). Note that the extensions mentioned above are based on perturbative truncations resulting in flow equations with few running couplings, which could have been obtained also by more conventional RG methods. 35 B. Full potential flow We now review an application to a problem where the effective action cannot be truncated at any finite order in φ, such that the possibility to make non-perturbative truncations becomes crucial (Jakubczyk et al., 2009). The problem arises when asking how a nematic transition caused by a d-wave Pomeranchuk instability in two dimensions is affected by fluctuations. Such a transition can be modelled by tight-binding electrons on a square lattice with an attractive d-wave forward scattering interaction (Metzner et al., 2003): ǫ k nk + k 1 2L fkk′ (q) nk (q) nk′ (−q) , (137) k,k′ ,q where nk (q) = σ c† k−q/2,σ ck+q/2,σ and L is the number of lattice sites. The interaction has the form fkk′ (q) = −g(q)dk dk′ , (138) where dk = cos kx − cos ky is a form factor with dx2 −y2 symmetry. The coupling function g(q) ≥ 0 has a maximum at q = 0 and is restricted to small momentum transfers by a cutoff Λ0 . For sufficiently large g = g(0) the interaction drives a d-wave Pomeranchuk instability leading to a nematic state with broken orientation symmetry, which can be described by the order parameter φ= g L dk nk . (139) k φ2 2T − 2g L ln 1 + e−(ǫk −φdk −µ)/T Λ=Λ0=0.37 Λ=0.22 Λ=0.11 Λ=0.05 Λ=0.00 -0.8936 -0.894 -0.8944 -0.8948 0 0.05 0.1 0.15 0.2 φ In the plane spanned by the chemical potential and temperature a nematic phase is formed below a dome-shaped transition line Tc (µ) with a maximal transition temperature near Van Hove filling. In mean-field theory, the phase transition is usually first order near the edges of the transition line, that is, where Tc is relatively low, and second order at the roof of the dome (Kee et al., 2003; Khavkine et al., 2004; Yamase et al., 2005). Introducing an order parameter field via a HubbardStratonovich transformation, integrating out the fermions, and keeping only the leading momentum and frequency dependences for small q and small q0 /|q| leads to a Hertz-type action S[φ] of the form (132), with z = 3 and a local potential given by the mean-field potential Uloc (φ) = -0.8932 U(φ) H= ansatz (134) for ΓΛ [φ] without expanding the local poR Λ Λ tential Uloc (φ). The flow of Uloc (φ) is then determined by a partial differential equation which contains a second derivative of the potential with respect to φ. In Fig. 15 an exemplary plot of the evolution of the Λ flowing effective potential Uloc (φ) is shown for Λ ranging from the ultraviolet cutoff Λ0 = e−1 ≈ 0.37 to the final value Λ = 0. The flow has been computed for electrons on a square lattice with nearest neighbor hop1 ping t = 1, next-to-nearest neighbor hopping t′ = − 6 , and a coupling strength g = 0.8. The initial (meanfield) potential has a minimum at φ0 = 0.112. The final potential exhibits spontaneous symmetry breaking with an order parameter φ0 = 0.102. Fluctations shift φ0 toward a slightly smaller value compared to the meanΛ field solutions. The flat shape of Uloc (φ) for φ ≤ φ0 at Λ = 0 is imposed by the convexity property of the grand canonical potential. The transition line between normal Λ FIG. 15 (Color online) Flowing effective potential Uloc (φ) for −1 various values of Λ between Λ0 = e and 0, at µ = −0.78 and T = 0.05 (Jakubczyk et al., 2009). and symmetry-broken phases is shown in Fig. 16 for two choices of Λ0 . Compared to the corresponding mean-field result, the transition temperature is suppressed, with a larger reduction for larger Λ0 (corresponding to a larger phase space for fluctuations). For Λ0 = 1 the transition is continuous down to T = 0, leading to quantum critical points at the edges of the nematic dome. Increasing Λ0 further (or reducing g), one may even eliminate the nematic phase completely from the phase diagram (Yamase et al., 2011). . (140) k At low temperatures, the coefficients of a Landau expansion of Uloc (φ) in powers of the field, U (φ) = a0 + a2 φ2 + a4 φ4 + . . ., are typically negative for all exponents 2m ≥ 4. Hence, S[φ] and consequently also the effective action ΓΛ [φ] cannot be truncated at any finite order R in φ. Fortunately, for bosonic fields the functional RG allows also for non-perturbative approximations, where one expands only in gradients, and not in powers of φ (Berges et al., 2002). In particular, one can use the C. Coupled flow of fermions and order parameter fluctuations There are various systems where integrating out the electrons leads to an effective order parameter action with singular interactions which cannot be approximated by a local coupling (Belitz et al., 2005; von Loehneysen et al., 2007). In such cases it can be advantageous to keep the electrons in the action, 36 0.16 VI. CORRELATION EFFECTS IN QUANTUM WIRES AND QUANTUM DOTS 1st order transition 2nd order transition 1st order (MFT) 2nd order (MFT) 0.14 0.12 T 0.1 0.08 0.06 0.04 0.02 0 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 µ FIG. 16 (Color online) Critical temperatures versus chemical potential for Λ0 = e−1 (larger dome with dots and crosses) and Λ0 = 1 (smaller dome with dots). The mean-field transition line is also shown for comparison (Jakubczyk et al., 2009). treating the coupled system consisting of electrons and their order parameter fluctuations. Several coupled boson-fermion systems exhibiting quantum criticality have already been analyzed by various methods; see, for example, Abanov et al. (2003); Belitz et al. (2001a,b); Huh and Sachdev (2008); Kaul and Sachdev (2008); Metlitski and Sachdev (2010a,b); Rech et al. (2006); Vojta et al. (2000). The interplay of bosonic and fermionic infrared singularities at the quantum critical point poses an interesting problem. The functional RG for coupled bosons and fermions described in Sec. II.G provides a suitable framework to study such problems. So far, it has not been applied to quantum phase transitions in metallic electron systems. However, encouraging works on relativistic field-theoretic models with gapless bosons and fermions have already appeared. For example, functional RG flow equations have been used to study the Gross-Neveu model (Rosa et al., 2001), quantum electrodynamics (Gies and Jaeckel, 2004), and supersymmetric Wess-Zumino models (Gies et al., 2009). In the context of condensed matter physics, a toy model for a semimetalto-superfluid quantum phase transition has been studied by coupled flow equations for the electrons and the superfluid order parameter (Obert et al., 2011; Strack et al., 2010). In dimensions d < 3 the fermions and the order parameter fluctuations acquire anomalous scaling dimensions at the QCP of that model, leading to non-Fermi liquid behavior and non-Gaussian criticality. It will be interesting to devise suitable truncations of the coupled boson-fermion flow equations for magnetic and nematic quantum phase transitions in lowdimensional metallic systems, where many open questions need to be clarified. As our last application of the functional RG to correlated fermion systems we discuss many-body effects in quantum wires and dots. The focus is on transport through such systems which are coupled to two or more semi-infinite leads. While in many of the above applications it was crucial to devise an approximation scheme in which the flow of the two-particle vertex (the effective two-particle interaction) was properly described, in the ones reviewed in this section the physics is dominated by the flow of the self-energy. We start out with a brief discussion of quantum transport through a region containing correlations in Sec. VI.A. To study transport beyond the linear response regime the functional RG was recently extended to Fermi systems out of equilibrium. In Sec. VI.B we review the main steps of this generalization. After discussing the most elementary example of linear transport through an inhomogeneous correlated quantum wire – a chain with a single local impurity – in Secs. VI.C.1-VI.C.4, we show how the functional RG can be used (i) to describe transport on all energy scales for more complex setups (Sec. VI.C.5), (ii) to identify unconventional low-energy fixed points of such systems (Sec. VI.C.6), and (iii) to study finite bias nonequilibrium transport (Sec. VI.C.7). As an example of the application of the functional RG to quantum dots, in Sec. VI.D.2 we consider an interacting chain of only three lattice sites corresponding to a dot. A. Quantum transport Experimental progress has made it possible to measure transport through mesoscopic regions like onedimensional (1D) quantum wires of up to micrometer length and “zero-dimensional” quantum dots. The experiments provide evidence for correlation effects (Deshpande et al., 2010; Hanson et al., 2007).2 It is a great theoretical challenge to describe transport when correlations in the mesoscopic system are important. Usually the leads to which this correlated region is connected are modeled as non-interacting. A general formal expression in terms of Keldysh Green functions for the current I through an interacting system coupled to two leads (indices L, R) in the stationary state was presented by Meir and Wingreen (1992). Either for specific models or applying certain approximations to the two-particle interaction for each channel ζ it can be brought into a Landauer-B¨ttiker type form (B¨ ttiker, 1986; Landauer, u u 2 Correlation effects in effectively 1D electronic systems are also studied using photoemission. For a recent review, see Grioni et al. (2009). 37 1957) Iζ = 1 2π Tζ (ǫ, T, Vb ) [fL (ǫ) − fR (ǫ)] dǫ , (141) where we set e = 1 = such that the conductance quantum per channel is given by 1/(2π). Here Tζ is an effective transmission probability, Vb = µL − µR the bias voltage, and fL/R are Fermi functions with the chemical potentials of the left and right leads µL/R . For the transport through a non-interacting system Tζ is the singleparticle transmission probability. The goal is to compute Tζ in the presence of correlations. Here truncations of the functional RG flow equations which lead to frequency independent self-energies are considered. In this case using Eq. (141), with Tζ being proportional to the “contact-tocontact” matrix element of the (retarded) one-particle Green function (see below) does not present an additional approximation as current vertex corrections vanish (Enss, 2005; Oguri, 2001). In this approximation the two-particle interaction affects the transport only via the renormalized self-energy which acts as an additional, T and Vb dependent scattering potential on non-interacting electrons (Langer and Ambegaokar, 1961; Oguri, 2001). For the linear conductance gζ (T ) the transmission probability enters only at zero bias Vb = 0, for fermions the system is assumed to be in the steady state. For interacting bosons functional RG was also used to study dynamics (Gasenzer and Pawlowski, 2008; Kloss and Kopietz, 2011). On a technical level and compared to the equilibrium Matsubara functional RG in the steady state the Keldysh structure only leads to an additional index (the so-called Keldysh index ± referring to the upper and lower branch of the Keldysh contour) to be added to the set of quantum numbers. B. Functional RG in non-equilibrium One of the main challenges of the non-equilibrium functional RG is to devise cutoff schemes which do not violate causality and general Kubo-Martin-Schwinger (KMS) relations (Jakobs, 2010; Jakobs et al., 2010b) after truncation of the infinite hierarchy of flow equations. For a general cutoff fulfilling the requirements discussed in Sect. II.D it is only guaranteed that causality and KMS relations hold up to the truncation order, e.g. first order for the level-1 truncation. In fact, an infrared (real) frequency cutoff similar to Eq. (143) violates causality in second order (Gezzi et al., 2007; Jakobs, 2003). Its breaking constitutes a severe problem as relations connecting the Keldysh contour matrix elements of the Green function cannot be used. In particular, one cannot rotate from the Ga,b Green function, with Keldysh indices a, b = ±, to retarded, advanced, and Keldysh Green functions as it is usually done (Rammer and Smith, 1986). The momentum cutoff scheme used in other sections of this review avoids this problem but is not suitable for models with broken translational invariance. We will return to this issue and discuss appropriate cutoff schemes when reviewing applications of non-equilibrium functional RG to transport through wires in Sec. VI.C.7 and dots in Sec. VI.D.2. Recently, the functional RG approach was extended to study steady state non-equilibrium transport through quantum wires and dots in the presence of a finite bias voltage given by the difference of the chemical potentials of the left and right leads Vb = µL − µR (Gezzi et al., 2007; Jakobs, 2003, 2010; Jakobs et al., 2007a, 2010a; Karrasch, 2010; Karrasch et al., 2010c). The basic idea behind this extension is the use of real time or real frequency Green functions on the Keldysh contour (Rammer and Smith, 1986) instead of Matsubara Green functions. As usual in diagrammatic approaches based on Keldysh Green functions one assumes that the initial statistical operator does not contain any correlations (Rammer and Smith, 1986). One can then either use a functional integral formulation (Kamenev, 2004) of the non-equilibrium many-body problem (Gezzi et al., 2007) or a purely diagrammatic approach (Jakobs, 2003; Jakobs et al., 2007a) to derive the (same) flow equations for the self-energy and higher order vertex functions. Although the method allows to work with two-time Green functions and to study transient dynamics, in the current implementation of the functional RG in non-equilibrium Besides its ability to be applied to non-equilibrium problems the real time (or real frequency) Keldysh functional RG has a distinct advantage even in equilibrium: the analytic continuation from Matsubara to real frequencies can be avoided. In fact, computing the real frequency dependence of observables obtained by a numerical solution of the Matsubara RG flow equations truncated on a level which includes a flowing two-particle vertex and self-energy of initially unknown frequency structure presents a formidable task (Karrasch et al., 2008, 2010b) as it is known from other imaginary time quantum many-body methods such as quantum Monte-Carlo. This advantage was utilized in the real time functional RG study of the single-impurity Anderson model (Jakobs, 2010; Jakobs et al., 2010a) in which a frequency dependent two-particle vertex and self-energy (complete level-2 truncation) were kept (for the imaginary time analog of this study, see Sec. VI.D.1, Hedden et al. (2004), and Karrasch et al. (2008)), and can be expected to be useful also in functional RG studies of other models. gζ (T ) = 1 2π Tζ (ǫ, T, 0) − ∂f ∂ǫ dǫ , (142) i.e. Tζ (ǫ, T, 0) is an equilibrium property. At zero temperature Eq. (142) simplifies further to gζ (0) = Tζ (µ, 0, 0)/(2π). 38 C. Impurities in Luttinger liquids The metallic state of correlated fermions in one dimension is a non-Fermi liquid. It falls into the Luttinger liquid (LL) class (Haldane, 1980). This state of matter is characterized by a powerlaw decay of space-time correlation functions with interaction dependent exponents (Giamarchi, 2004; Luther and Peschel, 1974; Luttinger, 1963; Mattis, 1974; Mattis and Lieb, 1965; Sch¨nhammer, 2005) and o spin-charge separation (Dzyaloshinskii and Larkin, 1974; Meden and Sch¨nhammer, 1992; Voit, 1993). In paro ticular, after Fourier transforming, the 2kF -component of the density-density response function shows a powerlaw divergence (Apel and Rice, 1982; Mattis, 1974) for repulsive interactions, instead of a logarithmic one for non-interacting electrons (Lindhard function in 1D). This indicates that any local inhomogeneity with a nonvanishing 2kF component strongly affects the low-energy physics and thus the transport characteristics of a LL at low temperatures. Perturbation theory is insufficient to describe the interaction effects as it fails to capture the RG flow of the inhomogeneity appearing even to first order in the interaction and higher order diagrams diverge. As will become clear below the functional RG approach captures the impurity flow and does not require a simplified modeling of the inhomogeneity. 1. A single local impurity–the local sine-Gordon model To understand the effect of a single impurity on the low-energy physics of a spinless, infinite LL (absence of non-interacting leads), bosonization was used first (for a review of this method see e.g. von Delft and Schoeller (1998), Giamarchi (2004), and Sch¨nhammer (2005)). o Within this approach the Fourier components of fermionic density operators are split into their chiral parts which obey Bose commutation relations in the low energy subspace (Tomonaga, 1950). In this limit the kinetic energy and the two-particle interaction can be written as bilinears in these operators (Tomonaga, 1950), while a single-particle scattering term (impurity) generally takes a complicated form in the bosonic degrees of freedom. It simplifies if only the pure forward scatter˜ ˜ ing V (0) and backward scattering V (2kF ) contributions are kept. Using bosonization to obtain results beyond this approximate modelling of the impurity is rather involved. The forward scattering term is linear in the bosons and can easily be treated leading to a phase shift. This is irrelevant for the physics described in the following. The backward scattering term consists of the cosine of a local bosonic field and the resulting Hamiltonian is known from field theory as the local sine-Gordon (LSG) model. One can either use the exact Bethe ansatz solution (Fendley et al., 1995) of this model or a bosonic RG ˜ which is perturbative in V (2kF ) (Furusaki and Nagaosa, 1993a; Kane and Fisher, 1992) to obtain analytical results. Alternatively numerical methods can be applied to the LSG model (Egger and Grabert, 1995; Moon et al., 1993). This led to a complete picture of the RG scaling ˜ of V (2kF ) which has direct consequences for the linear conductance. The RG flow connects two fixed points, the perfect chain fixed point with conductance g = K/(2π) and the open chain fixed point with g = 0. In 1D spinless fermion systems only a single transport channel exists and we thus suppress the channel index ζ from now on. Here K > 0 denotes the so-called LL parameter which depends on the underlying model of the quantum wire and its parameters such as the strength and range of the two-particle interaction as well as the band filling. Independently of the model considered K < 1 for repulsive interactions and K > 1 for attractive ones, while the non-interacting case corresponds to K = 1. The corrections to the fixed point conductances are power laws sγp/o with the infrared energy scale s (e.g. the temperature T or the energy cutoff Λ in a RG procedure). The exponents are independent of the bare impurity strength and given by γp = 2(K − 1) and γo = 2(1/K − 1), respectively. The sign of the scaling exponents implies that the open chain fixed point is stable for repulsive interactions and unstable for attractive ones. The opposite holds for the perfect chain fixed point. These insights confirmed earlier indications that impurities with a backscattering component strongly alter the low-energy physics of LLs with repulsive interactions (Apel and Rice, 1982; Mattis, 1974). The behavior can be summarized by saying that even a weak single impurity grows and eventually cuts the chain into two parts with open boundary conditions at the end points. The exponent γo characterizing (for repulsive twoparticle interactions) the suppression of g on small scales is twice the scaling exponent of the local single-particle spectral function of a LL close to an open boundary. This can be understood by viewing transport across the impurity as an end-to-end tunneling between two semi-infinite LLs (see e.g. Kane and Fisher (1992)). Bosonization was not only used for an infinite LL wire but also for the experimentally more relevant case in which an interacting wire containing a single impurity is contacted to two semi-infinite non-interacting leads. Contacts generically lead to single-particle backscattering and thus have an effect similar to the impurity (see below). To disentangle the effect of the contacts and the impurity one often models the contacts such that they do not lead to any backscattering. In this case and in the absence of the impurity the conductance takes the maximal value 1/(2π) (instead of K/(2π) for an infinite LL). Using bosonization this can either be achieved within the so-called local Luttinger liquid picture (Janzen et al., 2006; Maslov and Stone, 1995; Ponomarenko, 1995; Safi and Schulz, 1995) or by appropriate boundary conditions for the bosonic fields (Egger and Grabert, 1997; Egger et al., 2000). The fixed 39 points and scaling exponents turned out to be the same as in the LSG model. For weak two-particle interactions the problem of a single impurity in a LL was also studied using a fermionic RG (Matveev et al., 1993; Yue et al., 1994). In this approach a flow equation for the transmission coefficient at kF is derived using poor man’s RG. An extension of this method to interactions of arbitrary strength was presented by Aristov and W¨lfle (2008) and o Aristov and W¨lfle (2009). o Remarkably, no intermediate fixed points appear within the LSG model and the crossover between the perfect and open chain is characterized by a one-parameter scaling function (Egger et al., 2000; Fendley et al., 1995; Kane and Fisher, 1992; Moon et al., 1993). In one spatial dimension a general impurity can be described by ˜ matrix elements Vk,k′ (only VkF ,−kF = V (2kF ) is kept in the LSG model). The RG analysis for such an impurity involves the coupling of all matrix elements Vk,k′ in the flow and one might wonder if this leads to an intermediate fixed point absent in the LSG model. To lowest order in the impurity strength the flow of VkF ,−kF upon lowering the cutoff Λ is driven by VkF ,−kF itself (see the next section). For repulsive two-particle interactions VkF ,−kF increases and the perturbative RG breaks down. Now the other couplings, absent in the LSG model, might become relevant and eventually cut off the flow of VkF ,−kF to large values, that is towards the open chain fixed point. That this does not happen can nicely be revealed within a functional RG approach. Before reviewing how this question was approached (see Sec. VI.C.4) we first present the most elementary functional RG flow equation to tackle inhomogeneous LLs and analytically show that it leads to the scaling behavior known from bosonization in the limit of weak impurities. obtain the fixed point value of the effective interaction to leading order one can therefore replace the flowing twoparticle vertex by the antisymmetrized bare interaction corresponding to the level-1 truncation introduced in Sec. II.C.2. An improvement which includes the flow of the static two-particle vertex is reviewed in Sec. VI.C.4. After presenting the most elementary functional RG flow equation for the self-energy we show that it leads to the correct scaling properties for a weak impurity. As one deals with systems in which translational invariance is broken it is advantageous to introduce the infrared cutoff Λ in frequency space. To set up the functional RG flow equations for the self-energy the propagator G0 of the non-interacting Hamiltonian H0 containing only the kinetic energy is replaced by GΛ (iωn ) = ΘΛ (ωn )G0 (iωn ) 0 with a function ΘΛ which is unity for |ωn | ≫ Λ and vanishes for |ωn | ≪ Λ. More specifically  |ωn | ≤ Λ − πT  0, 1 n |−Λ ΘΛ (ωn ) = + |ω2πT , Λ − πT ≤ |ωn | ≤ Λ + πT  2 1, |ωn | ≥ Λ + πT (143) was used (Enss et al., 2005), where Λ starts at ∞ and goes down to 0.3 For T → 0, ΘΛ becomes a sharp Θfunction and Matsubara frequencies with |ω| < Λ are suppressed. In this subsection a general continuum or lattice model of spinless fermions is considered with one-particle states |α , which in the following will be either local states |x , where x = j is the site index for the lattice model, with lattice constant a = 1, or momentum states |kn with kn = 2πn/L. A general two-particle interaction Eq. (2) is † assumed and an impurity term Vimp = α,β Vα,β ψα ψβ . The flow for the self-energy Eq. (50) in the level-1 truncation reads (Meden et al., 2002) 2. The functional RG approach to the single impurity problem As in the previous applications because of the necessary truncations the functional RG can (presently) only be used for small to intermediate two-particle interactions. For the application to inhomogeneous LLs it is crucial that it is non-perturbative in the single-particle inhomogeneity. Here the focus is on a description in which the RG flow and the interaction dependent exponents characterizing the physics close to fixed points are kept at least to leading order in the interaction. Following the discussion in the last paragraph of the preceeding subsection the feedback of the impurity into the flow of the self-energy dominates the physics to be studied. It is thus advantageous to use the one-particle irreducible functional RG scheme in which the full propagator including the self-energy appears on the right hand side of the flow equations. For spinless fermions in a homogeneous wire the electron-electron interaction is renormalized only by a finite amount of order interaction squared (S´lyom, 1979). A single impurity does not alter this. To o d Λ Σ =T dΛ α,β × + eiωl 0 ωl dGΛ (iωl ) 0 dΛ γ,δ 1 − GΛ (iωl )ΣΛ 0 1 − ΣΛ GΛ (iωl ) 0 −1 −1 Uα,γ,β,δ , (144) δ,γ where GΛ and ΣΛ are matrices. The initial condition is 0 given by ΣΛ=∞ = Vα,β and Uα,β,γ,δ denotes the antisymα,β metrized bare two-particle vertex. At temperature T = 0 and applying Eq. (60) to products of Θ- and δ-functions, Eq. (144) simplifies to d Λ 1 Σ =− dΛ α,β 2π 3 + ˜ δ,γ Uα,γ,β,δ GΛ (iω) eiω0 ,(145) ω=±Λ γ,δ A significant speed-up of the numerical solution of the differential flow equations can be achieved using the alternative cutoff scheme discussed in the Appendix of Andergassen et al. (2006b). 40 where ˜ GΛ (iω) = Q0 (iω) − ΣΛ −1 (146) is the full propagator for the cutoff dependent self-energy + and Q0 = (G0 )−1 . The convergence factor eiω0 in Eq. (145) is relevant only for determining the flow from Λ = ∞ down to some arbitrarily large Λ0 . For Λ0 much larger than the band width this high energy part of the flow can be computed analytically leading to the initial condition ΣΛ0 = Vα,β + γ Uα,γ,β,γ /2. α,β Within the present scheme ΣΛ is frequency indepenα,β dent and can be considered as a flowing effective impurity potential. To obtain an approximation for the Green function of the original cutoff free problem for arbitrary impurity parameters one has to determine the self-energy ΣΛ at Λ = 0 by numerically solving the set of differenα,β tial equations (144) or (145). To compute the right hand side of the flow equations one has to invert the matrix Eq. (146), i.e., to solve the problem of a single particle moving in the effective scattering potential ΣΛ . α,β Because of the matrix inversion involved in calculating the right hand side of Eq. (145) the flow equations can be solved analytically only in limiting cases, one being the situation of a weak impurity. One then works in momentum space and considers ΣΛ ′ for k = k ′ . The k,k term linear in ΣΛ presents the leading approximation in ˜ the expansion of GΛ on the right hand side of Eq. (145) and one obtains (Meden et al., 2002) d Λ 1 Σk,k′ = − dΛ 2π × Uk,k1 ,k′ ,k2 k1 ,k2 1 1 ΣΛ2 ,k1 + (Λ → −Λ) , (147) k iΛ − ξk2 iΛ − ξk1 where ξk = ǫk − µ with the one-particle dispersion ǫk . The antisymmetrized two-body matrix element Uk,k1 ,k′ ,k2 contains a momentum conserving Kronecker delta, where k + k1 = k ′ + k2 modulo the reciprocal lattice vector 2πn for the lattice model with n = 0, ±1, when the four momenta are in the first Brillouin zone. The umklapp processes n = ±1 involve low energy excitations only for special fillings–half filling for the nearest neighbor hopping model discussed later. First models are considered for which umklapp processes are absent. To determine the backscattering properties of the self-energy one can put k = kF and k ′ = −kF − q with |q| ≪ kF . In order to read off the dominant behavior for small Λ the remaining summation variable ˜ is shifted k1 = −kF + k1 d Λ 1 Σ =− dΛ kF ,−kF −q 2π ˜ k1 UkF ,−kF +k1 ,−kF −q,kF +q+k1 ˜ ˜ 1 ΣΛ × ˜ ˜ iΛ − ξkF +q+k1 kF +q+k1 ,−kF +k1 iΛ − ξ−kF +k1 ˜ ˜ ˜ For |k1 | ≪ kF one can linearize the dispersion ξ−kF +k1 ≈ ˜ ˜1 + q), where vF denotes the ˜1 and ξ −vF k ˜ kF +q+k1 ≈ vF (k Fermi velocity. In the thermodynamic limit the two G0 ˜ factors for q = 0 and Λ → 0 are proportional to δ(k1 ). If only the singular contributions are kept the differential equation for ΣΛF ,−kF reads k ˆ Uk ,−k ,k ,−k 1 d Λ ΣkF ,−kF = − F F F F ΣΛF ,−kF dΛ 2πvF Λ k (149) ˆ where UkF ,−kF ,kF ,−kF = LUkF ,−kF ,kF ,−kF is independent of the system size. For the continuum model ˜ ˜ ˜ ˆ UkF ,−kF ,kF ,−kF = U (0) − U (2kF ), where U (k) is the Fourier transform of the two-particle potential U (x − x′ ). This leads to the scaling relation ΣΛF ,−kF ∼ k 1 Λ ˆ UkF ,−kF ,kF ,−kF /(2πvF ) . (150) As Eq. (149) was derived by expanding the Green function GΛ in powers of the self-energy, the scaling behavior Eq. (150) can be trusted only as long as ΣΛ stays small. Thus a single-particle backscattering term is a relevant perturbation for repulsive interactions and an irrelevant one for attractive ones consistent with the bosonization result. Equation (149) holds even in the half-filled band case for the nearest neighbor hopping lattice model. The additional singular contribution due to umklapp scattering ˆ is proportional to UkF ,kF ,−kF ,−kF which vanishes because of the antisymmetry of the matrix element (Meden et al., 2002). Neglecting all interaction effects beyond the renormalization of the impurity potential and using the Born approximation the correction to the perfect chain conductance is given by the self-energy squared. The correˆ sponding exponent UkF ,−kF ,kF ,−kF /(πvF ) agrees to leading order in the interaction (Sch¨nhammer, 2005) with o γp = 2(K − 1) obtained within bosonization. The opposite limit of a weak link can be treated analytically as well leading to results consistent to those of the bosonization approach with an exponent characterizing the deviation from the open chain fixed point conductance g = 0 which agrees to leading order in the interaction with γo = 2(1/K − 1) (Meden et al., 2002). Next the numerical solution of the RG flow for a specific lattice model with arbitrary impurity strength is discussed. This allows us to address the question of additional fixed points. 3. Basic wire model 1 + (Λ → −Λ) . (148) In the following the tight-binding model of spinless fermions with nearest neighbor interaction supplemented by an impurity is considered. The Hamiltonian is given 41 Vr Vl Vg lead lead 1 ~ jr jr jl N { ~ jl ND FIG. 17 Schematic plot of the quantum dot situation, where the barriers are modeled by two site impurities (Enss et al., 2005). by H = Hkin + Hint + Himp with kinetic energy Hkin = − ∞ c† cj + c† cj+1 , j+1 j (151) j=−∞ where c† and cj are the creation and annihilation opj erators on site j, respectively. The corresponding noninteracting dispersion is ǫk = −2 cos k. The interaction is restricted to electrons on N neighboring sites (Enss et al., 2005) Hint = N −1 j=1 where Vj,j ′ is a static potential. Site impurities are given by Vj,j ′ = Vj δj,j ′ . For a single site impurity Vj = V δj,j0 j0 is chosen to be far away from both leads. Impurities close to the contacts were discussed by Furusaki and Nagaosa (1996) and Enss et al. (2005). Resonant tunneling can be studied considering two site impurities of strengths Vl and Vr on the sites ˜l = jl − 1 j and ˜r = jr + 1. The ND sites between jl and jr define j a quantum dot. The effect of a gate voltage restricted to the dot region is described by a constant Vg on sites jl to jr . This situation is sketched in Fig. 17. Hopping impurities are achieved setting Vj,j ′ = Vj ′ ,j = −tj,j+1 δj ′ ,j+1 . For the special case of a single hopping impurity, tj,j+1 = (t′ − 1) δj,j0 , the unit hopping amplitude is replaced by t′ on the bond linking the sites j0 and j0 + 1. In the doublebarrier problem a hopping tl across the bond (˜l , jl ) and j tr across (jr , ˜r ) is considered. j The homogeneous model H = Hkin + Hint with a constant interaction U across all bonds–not only the ones within [1, N ]–can be solved exactly by the Bethe ansatz and K is determined by a system of coupled integral equations (Haldane, 1980). In the half-filled case they can be solved analytically leading to K= Uj,j+1 [nj − ν(n, U )] [nj+1 − ν(n, U )] , (152) with the local density operator nj = c† cj . The two rej gions of the lattice with j < 1 and j > N constitute the semi-infinite non-interacting leads. To model contacts which do not lead to single-particle backscattering (see above) the interaction Uj,j+1 between electrons on sites j and j + 1 is allowed to depend on the position. A conductance g = 1/(2π) in the absence of single-particle impurities is only achieved if Uj,j+1 is taken as a smoothly increasing function of j starting form zero at the bond (1, 2) and approaching a constant bulk value U over a sufficiently large number of bonds. Equally, the Uj,j+1 are switched off close to the bond (N − 1, N ). The results are independent of the shape of the envelope function as long as it is sufficiently smooth. An abrupt twoparticle inhomogeneity acts similarly to a single-particle impurity. A detailed disussion on the effect of the spatial variation of the two-particle interaction was presented by Meden and Schollw¨ck (2003a) and Janzen et al. (2006). o In Eq. (152) the density nj is shifted by a parameter ν(n, U ), which depends on the filling factor n and the bulk interaction U . This is equivalent to introducing an additional one-particle potential which can compensate the Hartree potential in the bulk of the interacting wire. In the half-filled band case ν(1/2, U ) = 1/2 (Enss et al., 2005). The general form of the impurity part of the Hamiltonian is Vj,j ′ c† cj ′ , j Himp = j,j ′ (153) 2 U arccos − π 2 −1 (154) for |U | ≤ 2. At other fillings the integral equations can be solved numerically. The model shows LL behavior for all particle densities n and any interaction strength except at half filling for |U | > 2 were either phase separation sets in (for U < −2) or the system orders into a chargedensity-wave state (for U > 2). Due to the presence of the semi-infinite leads the direct calculation of the non-interacting propagator related to Hkin + Himp requires the inversion of an infinite matrix. Using a standard projection technique (Taylor, 2000) it can be reduced to the inversion of an N × N matrix. The leads then provide an additional ωn -dependent diagonal one-particle potential on sites 1 and N (Enss et al., 2005) lead Vj,j ′ (iωn ) = iωn + µ 2 1− 1− 4 (iωn + µ)2 ×δj,j ′ (δ1,j + δN,j ) . (155) Since the interaction is only non-vanishing on the bonds between the sites 1 to N the problem including the semiinfinite leads is this way reduced to the problem of an N -site chain. In the functional RG it is then advantageous to replace the projected non-interacting propagator G0 including the impurity by a cutoff dependent one (Enss et al., 2005). 4. Numerical solution of improved flow equations With a minor increase in the numerical effort one can go beyond Eq. (144) for the flow of the selfenergy and include a Λ-dependent static interaction U Λ 42 2 , -0.2 0.1 Σj,j -25 where the function h(Λ) depends only on the cutoff Λ and the Fermi momentum kF . The solution of the flow equation is lengthy for arbitrary fillings (Andergassen et al., 2004) but has a simple form for half-filling U UΛ = 1+ Λ− 2+Λ2 √ 4+Λ2 . Λ Uj,j+r |ωn |≈Λ r=±1 ×Q0 (iωn ) ∂ Λ 1 Σ = ∂Λ j,j±1 2π |ωn |≈Λ ×Q0 (iωn ) 50 75 (157) by this scheme. Thus only the leading order is completely kept in the flow of Σ. The coupled flow equations can be solved numerically by an algorithm which approximately scales as N (Andergassen et al., 2004). Typically systems of 104 lattice sites were considered, roughly corresponding to the length of quantum wires accessible to transport experiments. For the interacting wire of finite length the energy scale δN = vF /N forms a cutoff for any RG flow. The flowing self-energy Eq. (158) depends on the three scales T , δN , and Λ. Saturation of ΣΛ for Λ T or Λ δN sets in “automatically” in contrast to more intuitive RG schemes in which the flowing couplings depend on Λ only and the flow is stopped “by hand” by replacing Λ → T or Λ → δN , respectively (Kane and Fisher, 1992; Yue et al., 1994). 1 Q0 (iωn ) − ΘΛ (ωn )ΣΛ , j+r,j+r 1 Q0 (iωn ) − ΘΛ (ωn )ΣΛ -1 (j-j0) -2 1 Q0 (iωn ) − ΘΛ (ωn )ΣΛ 10 , -4 10 j,j±1 (158) where the matrix Q0 = (G0 )−1 is the inverse of the projected non-interacting propagator with impurity. The symbol |ωn | ≈ Λ stands for taking the positive as well as negative frequency with absolute value closest to Λ.4 The initial conditions for Σ at Λ = Λ0 → ∞ are independent of the precise realization of the inhomogeneity and read ΣΛ0 = [1/2 − ν(n, U )] (Uj−1,j + Uj,j+1 ) and ΣΛ0 = 0. j,j±1 j,j The frequency dependence of the self-energy which appears in the exact solution in order U 2 is not captured 4 25 j-j0 U/(2π) 1 Q0 (iωn ) − ΘΛ (ωn )ΣΛ Λ Uj,j±1 0 FIG. 18 Self-energy near a site impurity of strength V = 1.5 filling n = 1/4, and interaction U = 1; the impurity is located at j0 = 513 with N = 1025 sites. (Data taken from Andergassen et al. (2004).) These approximations are sufficient to correctly describe the RG flow of the two-particle vertex on the Fermi surface of the homogeneous system (Andergassen et al., 2004) to second order, as it is usually done in the so-called g-ology method (S´lyom, 1979). For the inhomogeneous o LLs the flow of the effective interaction leads to improved results for the scaling exponents (see below). Within these approximations and using the projection of the leads, the self-energy at T ≥ 0 is a frequencyindependent tridiagonal matrix in real space determined by the flow equation (j, j ± 1 ∈ [1, N ]) 1 ∂ Λ Σj,j = − ∂Λ 2π 0 -0.1 (156) |∆Σj,j+1| d Λ U = h(Λ) U Λ dΛ -0.1 Σj,j+1 (Andergassen et al., 2004). Its flow equation is derived from the general one for the two-particle vertex Eq. (52) applying the following approximations: (i) The threeparticle vertex is set to zero. (ii) All frequencies are set to zero. (iii) The feedback of the inhomogeneity on the flow of the interaction is neglected. (iv) The interaction is assumed to remain of nearest-neighbor form. Then U Λ obeys a simple differential equation (Andergassen et al., 2004), To achieve this result the cutoff scheme discussed in the Appendix of Andergassen et al. (2006b) was used. -6 10 0 10 1 10 2 10 j-j0 3 10 0 1000 2000 3000 j-j0 FIG. 19 Decay of the oscillatory part of the off-diagonal matrix element of the self-energy away from a single hopping impurity at bond j0 , j0 + 1. Results for t′ = 0.1, j0 = 5000, N = 104 , U = 1, n = 1/2, and different temperatures T = 10−1 (solid line), T = 10−2 (dotted line), T = 10−3 (dashed line), and T = 10−4 (dashed-dotted line) are presented. The left panel shows the data on a log-log scale, the right panel on a linear-log scale. For comparison the left panel contains a power-law (j − j0 )−1 (thin solid line) (Enss et al., 2005). Figure 18 shows the self-energy Σ at the end of the 43 Using scattering theory (Enss et al., 2005) one can show that the effective transmission T (ǫ, T ) is given by the (1, N ) matrix element of the single-particle Green function T (ǫk , T ) = 4 sin2 k | N |G(ǫk + i0)|1 |2 . Via the T -dependent self-energy (see Fig. 19) G and thus T carries a temperature dependence. A typical example for the T -dependence of the linear conductance g(T ) for a strong local impurity is shown as the solid line in Fig. 21. It clearly follows the expected power-law behavior for δN T ≪ B with the band width B = 4. The T −1 scaling at larger T is a band effect. For −0.5 ≤ U ≤ 1.5 and fillings n = 1/2 as well as 1/4 the exponent extracted (see lower panel of Fig. 21) agrees well with the one of the LSG model γo = 2(1/K − 1), with K taken from the Bethe ansatz. Even for U = 1.5 the relative error is less than 5 percent (see Fig. 5 of Enss et al. (2005)). Higher order corrections in U present in the numerical solution of the flow equations (157) and (158) clearly improve the result over the one of the perturbative (in the impurity strength) analytical solution of Sec. VI.C.2 which yields a purely linear exponent. We emphasize that this improvement is not systematic as second and higher order terms are only partly kept in the RG. A similar agreement can be found for γp (Enss et al., 2005). 10 10 0 -2/K x -1 2 -1 (1+x ) 2πg RG flow, for T = 0 in the vicinity of a site impurity of intermediate strength. Both the onsite energy Σj,j as well as the hopping Σj,j+1 become oscillatory functions with wave number 2kF and a decaying amplitude. The asymptotic value of Σj,j+1 away from the impurity leads to a broadening of the band due to the interaction. A more detailed analysis of the oscillatory ¯ part |∆Σj,j+1 | = Σj,j+1 − Σoff , with the spatial av¯ erage Σoff , for different T > 0 is presented in Fig. 19. The left panel shows that for |j − j0 | 10 it decays as 1/|j − j0 | up to a thermal length scale ∼ 1/T (provided T > δN ) beyond which the decay becomes exponential; see the right panel. For U > 0 this is the generic behavior for large bare impurities or on asymptotical large length scales. It is the scattering off such a long-ranged oscillatory potential–so-called Wigner-von Neumann potential (Reed and Simon, 1975)–which leads to the power-law suppression of the conductance and the local spectral weight. One can analytically show that the amplitude of the asymptotic 1/|j − j0 | decay determines the exponent (Barnab´-Th´riault et al., 2005). By virtue of the RG e e flow this amplitude–and thus the exponent–becomes independent of the impurity strength. For this reason first order perturbation theory fails. It also leads to an oscillatory self-energy which decays as 1/|j − j0 | but with an amplitude which depends on the bare impurity strength (Meden et al., 2002) incorrectly leading to a power-law with an impurity dependent exponent. The idea of an oscillatory decaying potential is similarly inherent to a poor man’s fermionic RG approach (Yue et al., 1994). Often these oscillations of the effective renormalized potential are referred to as Friedel oscillations. This is misleading as this term is reserved to the spatial oscillations of the electron density. In particular, in an inhomogeneous LL the effective potential decays as |j − j0 |−1 , while the density oscillations asymptotically decay as |j − j0 |−K (Egger and Grabert, 1995). The latter can also be shown within the functional RG formalism presented here (Andergassen et al., 2004). The application of the self-consistent Hartree-Fock approximation leads to an oscillatory self-energy with a constant amplitude and thus to a charge density wave state (Meden et al., 2002). This is an unphysical artifact of the approximation. 10 10 -2 -3 10 -2 -1 0 10 10 10 K-1 x=[T/T0(U,n,V)] 1 FIG. 20 One-parameter scaling plot of the conductance. Open symbols represent results obtained for U = 0.5, n = 1/2, and different T and V , while filled symbols were calculated for U = 0.851, n = 1/4. Both pairs of U and n lead to the same K = 0.85 (within the present approximation). The solid line indicates the non-interacting scaling function (1+x2 )−1 and the dashed one the LSG model power-law decay with exponent −2/K. (Data taken from Enss et al. (2005).) Within the LSG model no intermediate fixed points appear which is reflected by one-parameter scaling K−1 g = gK (x)/2π with x = [T /T0 ] ˜ and a nonuniversal scale T0 (Egger et al., 2000; Fendley et al., 1995; Kane and Fisher, 1992; Moon et al., 1993). For ap˜ propriately chosen T0 data for different T and V (2kF ) but fixed K can be collapsed onto the K-dependent scaling function gK (x). It has the limiting behavior ˜ gK (x) ∝ 1−x2 for x → 0 and gK (x) ∝ x−2/K for x → ∞. ˜ ˜ One can perform a similar scaling with data from the numerical solution of the flow equations for the microscopic lattice model considering different V and T as well as two sets of (U, n) leading to the same LL parameter (Enss et al., 2005). The perfect collaps of the data of Fig. 20 shows that the improved description of the impurity flow beyond the single amplitude approximation inherent to the LSG model does not lead to additional fixed points. The functional RG scaling function shows a sensible K dependence. The exponent of the large x power-law decay is smaller than the non-interacting one −2 (solid line at large x) and very close to the LSG model exponent −2/K shown as the dashed line in Fig. 20. This has to be contrasted to the K independent (non-interacting) scaling function g = (1 + x2 )−1 resulting from the poor ˜ 44 * T 100 2πgp(T) 10 10 10 10 exponent man’s fermionic RG (Yue et al., 1994); solid line in Fig. 20. The functional RG results for a single local impurity in a LL show that the LSG model describes the physics (two fixed points, exponents, one-parameter scaling) of a broader class of models. The same approach was also used to study the persistent current through a LL ring with a local impurity prierced by a magnetic flux (Gendiar et al., 2009; Meden and Schollw¨ck, 2003a,b) o Aspects resulting from the spin degree of freedom of electrons were discussed by Andergassen et al. (2006a) and Andergassen et al. (2006b). δ10000 * T 6 δ100 δ6 B 0 -1 -2 -3 0 -0.5 -1 -4 10 10 -3 10 -2 10 -1 10 0 10 1 T 5. Resonant tunneling We next review the results on resonant transport through a double barrier–defining an interacting quantum dot embedded in a LL (Enss et al., 2005; Meden et al., 2005). The setup is sketched in Fig. 17. The linear conductance g is characterized by a hierarchy of energy scales. The functional RG is a unique tool to access this problem as it provides reliable results on all scales. For a fixed dot size ND and fixed barriers Vl/r (or tl/r ) the dot can be tuned to resonance varying Vg . Only for symmetric dots with Vl = Vr (or tl = tr ) the peak conductance becomes “perfect” gp = 1/(2π). For asymmetric barriers a backscattering component of the single-particle inhomogeneity remains, leading to a reduced gp . Due to the interaction backscattering grows during a RG procedure and on asymptotic scales the conductance vanishes with scaling exponent γo . The same holds away from resonance regardless of the ratio Vl /Vr (or tl /tr ). Thus the non-interacting resonance of finite width either disappears (asymmetric barriers) or turns into a resonance of zero width (symmetric barriers). A rich T -dependence is found on resonance and for symmetric barriers on which we now focus. Without loss of generality we only consider site impurities as barriers. The functional RG procedure can directly be applied to the double barrier problem. The dot parameters only enter via the non-interacting propagator. Figure 21 shows the peak conductance gp (T ) for a dot with high barriers and two different dot sizes. The relevant ∗ energy scales B, δND = vF /ND , TND (see below), and T the δN are indicated by the arrows. For δND two barriers behave as independent impurities. Using scattering theory one can show that in this case gp is obtained by adding the resistances of the two barriers (Enss et al., 2005; Jakobs et al., 2007b). This explains why for ND = 100, for which this temperature regime is clearly developed, gp (T ) agrees to the solid line obtained by taking g(T )/2 of a single site impurity of equal heigth as used for the double barrier. Note that it is a non-trivial fact that in this temperature regime the individual resistances can be added to give the total resistance. In the presence of inelastic processes one would of course expect this result (resistors in series) FIG. 21 Upper panel: gp (T ) for U = 0.5, N = 104 , Vl/r = 10, n = 1/2, and N = 6 (squares), 100 (diamonds). The arrows ∗ indicate the relevant energy scales B, δND , TND and δN . The solid curve shows g(T )/2 for a single barrier with V = 10 and U = 0.5, N = 104 . Lower panel: Logarithmic derivative of the conductance. Dashed line: γo ; dashed-dotted line: γo /2 − 1. (Data taken from Enss et al. (2005).) but they are absent in the mesoscopic setup–the ones resulting from the electron-electron interaction are suppressed by the approximations. In fact, the case of three barriers constitutes an example for which adding resistances does no longer hold (Jakobs et al., 2007b). For T δND the width of −∂f /∂ǫ is smaller than δND and only the resonance peak around ǫ = 0 of T (ǫ, T ) contributes to the integral in Eq. (142). The width w of this peak vanishes as T γo /2 /ND (Enss et al., 2005) leading to gp (T ) ∝ T γo /2−1 . The lower bound of this scaling regime, first discussed using bosonization (Furusaki, 1998; Furusaki and Nagaosa, 1993b), is reached when T −1/(1−γo /2) ∗ ∗ equals w, i.e. at TND ∝ ND . For T < TND , 2πgp approaches 1. For T reaching δN any power-law scaling in T with an interaction dependent exponent is cut off by the finite size of the interacting part of the quantum wire. In addition of identifying the different temperature regimes the functional RG approach allows to (i) quantify the size of the crossover regime–typically half an order of magnitude–and to (ii) obtain results for “nonuniversal” regimes as e.g. realized for ND = 6 and δND < T ≪ B. For dots with weak barriers and sufficiently large ND only the regime with scaling exponent γo /2 − 1 is realized and for weak barriers and small ND none of the above power-law regimes emerges (Enss et al., 2005; Meden et al., 2005). Resonant transport in LLs was also studied by bosonization (Furusaki, 1998; Furusaki and Nagaosa, 1993b), poor man’s fermionic RG (Nazarov and Glazman, 2003; Polyakov and Gornyi, 2003), and numerically (H¨ gle and Egger, 2004). u The temperature dependence of the peak conductance of resonant tunneling nicely exemplifies that the functional RG approach provides sensible results on all energy scales even for problems with a hierachy of scales. 45 6. Y-junctions The power of the functional RG approach to uncover unconventional fixed points and the related interesting physics was exemplified by discussing a specific junction of three 1D wires, a so-called Y-junction. The three LL wires (index ν = 1, 2, 3) each of length N and coupled to a non-interacting semi-infinite lead via a “perfect” contact are described by the basic model discussed in Sec. VI.C.3. The symmetric junction pierced by a magnetic flux φ, is sketched in Fig. 22 and given by 2 φ t∆ V 3 FIG. 22 Sketch of the symmetric Y-junction of three quantum wires (Barnab´-Th´riault et al., 2005). e e 3 3 HY = −tY −t△ c† c0,ν + h.c. + V 1,ν n0,ν ν=1 ν=1 3 eiφ/3 c† c0,ν+1 + h.c. 0,ν , (159) ν=1 where the wire indices 4 and 1 are identified. The junction is characterized by the three parameters tY , V , and t△ . Using scattering theory (Barnab´-Th´riault et al., e e 2005; Barnab´-Th´riault et al., 2005; Enss et al., 2005), e e the U = 0 conductance from wire ν to wire ν ′ can be written as 2 2πgν,ν ′ = 4 (Im κ) e−iφ − κ 2 2 |κ3 − 3κ + 2 cos φ|2 , (160) ˆ with a single complex parameter κ = (−V −t2 G0 )/|t△ |. Y 1,1 ˆ 0 is obtained for one of the equivaThe Green function G ˆ lent disconnected (tY = 0) wires and G0 ∈ C denotes its 1,1 diagonal matrix element taken at the first site j = 1. It is evaluated at energy ǫ + i0 with ǫ → 0. Equation (160) holds if ν and ν ′ are in cyclic order and is independent of the pair considered; gν ′ ,ν follows by replacing φ → −φ. If φ does not correspond to an integer multiple of π and for generic junction parameters the conductance from ν to ν ′ differs from the one with reversed indices indicating the breaking of time-reversal symmetry. This constitues the 1 0.8 1.5 0.6 1 0.4 0.5 0 −3 tY 1 most interesting situation and we focus on such fluxes. In Fig. 23 the conductance from wire ν to ν ′ (cyclic) for φ = 0.4π is shown for the upper half of the complex κplane. For restored time-reversal symmetry the largest conductance allowed by the unitarity of the scattering matrix is 2πgν,ν ′ = 4/9 (denoted the “perfect junction value” in the following); even for optimized parameters a reflection of 1/9 is unavoidable. Im κ Other examples from this class are situations in which the wire-lead contacts are not modeled as being “perfect” (Jakobs et al., 2007b) and models in which the leads and contacts are described in a more realistic way (W¨chter et al., 2009). a 0.2 −2 −1 0 Re κ 1 2 0 3 2π g ν,ν′ FIG. 23 (Color online) The non-interacting conductance 2πgν,ν ′ (cyclic indices) as a function of the complex parameter κ which in turn is a function of the junction parameters tY , V , and t△ . The flux is φ = 0.4π. For U = 0 the Y-junction can straightforwardly be treated within the functional RG based approximation scheme (Barnab´-Th´riault et al., 2005). We here focus e e on T = 0. To compute the conductance from Eq. (160), ˆ ˆ G0 must be replaced by the auxiliary Green function G obtained by considering Σ (at the end of the RG flow for the full system) as an effective potential for a single disconnected wire setting tY = 0 (Barnab´-Th´riault et al., e e ˆ develops a dependence 2005). Via the RG flow of Σ, G on (tY , t△ , V ), U , and δN = vF /N . The latter energy scale is a natural infrared cutoff–in contrast to the flow parameter Λ which is artificial and sent to 0. A comprehencive picture of the low-energy physics is obtained from the dependence of κ on δN . In Fig. 24 each line is for a fixed set of junction parameters and δN as a variable. The flux is chosen as φ = 0.4π and the arrows indicate the direction of decreasing δN . As Im κ has the opposite ˆ sign of Im G1,1 < 0 it is restricted to positive values. Equation (160) allows for four distinguished conductance situations (see Fig. 23): (i) on the line Im κ = 0, gν,ν ′ = gν ′ ,ν = 0 for almost all Re κ; (ii) it is interrupted by three points having flux-dependent positions with the conductance 2πgν,ν ′ = 2πgν ′ ,ν = 4/9; (iii) for a specific flux-dependent κ one finds 2πgν,ν ′ = 1 and 2πgν ′ ,ν = 0; (iv) gν,ν ′ = gν ′ ,ν = 0 is also reached for |κ| → ∞. These are the fixed points of the RG flow as is evident from Fig. 24. (i) is an interupted line of decoupled chain fixed points with vanishing conductances which is stable for U > 0 and unstable in the opposite case. Analyzing the dependence of gν,ν ′ on δN in its vicinity for different U one finds that the scaling exponent is independent of 46 0.8 2 exponent Im κ 1.5 1 0.5 max. asym. fixed point 2(∆-1) perf. junct. fixed point U/(3π) 0.4 0 -0.4 -1.5 0 −3 −2 −1 0 Re κ 1 2 3 FIG. 24 (Color online) Flow of κ for U = −1, n = 1/2, and φ = 0.4π. Arrows indicate the direction for U < 0. For U > 0 it is reversed. For details see the text (Barnab´-Th´riault et al., 2005). e e φ given by γo as obtained for the single impurity. (ii) constitute three perfect junction fixed points (circles in Fig. 24). For U > 0 each of the three fixed points has a basin of attraction given by one of the three parts of the curve C(φ) (curved thick line in Fig. 24 interrupted by the square) on which the reflection 1 − 2πgν,ν ′ − 2πgν ′ ,ν takes a local minimum. The U dependence of the scaling exponent when approaching one of the fixed points along its corresponding line is shown in Fig. 25 (circles). It is independent of φ and for small |U | it can be fitted by U/(3π). These fixed points have not been found by any method which is based on bosonization and the exact dependence of their scaling exponent on K is presently unknown. Because of the factor 1/3 in the leading order it must be different from the K-dependence of any of the exponents discussed so far. (iii) The basins of attraction are separated by the maximal asymmetry fixed point (maximal breaking of time-reversal symmetry; square in Fig. 24). For φ = π/2 this fixed point was identified by a bosonization based approach (Chamon et al., 2003), and it was conjectured that the behavior found holds for all fluxes different from integer multiples of π. The functional RG results indeed confirm this–at least for small to intermediate |U |–as one finds this fixed point for all such φ and obtains a flux-independent scaling exponent which to leading order agrees to the bosonization result γY = 2(∆ − 1) with ∆ = 4K/(3 + K 2 ) (see Fig. 25). The bosonization exponent shows a non-monotonic dependence on K and thus U , which the approximate functional RG approach does not capture. This implies that the maximal asymmetry fixed point is unstable for repulsive interactions, and stable for sufficiently small attractive ones but turns unstable again for larger attractive interactions. (iv) In the mapping of the complex plane onto the Riemann sphere the g = ∞ fixed point (north pole) is part of the projected line of decoupled chain fixed points and shows the same stability properties and scaling dimension. -1 -0.5 U 0 0.5 1 1.5 FIG. 25 Scaling exponents of the Y-junction close to the fixed points (Barnab´-Th´riault et al., 2005). e e The most interesting physics is associated with the perfect junction fixed points which for U > 0 each have one stable direction. If the junction parameters of a noninteracting system at fixed φ = mπ, m ∈ N0 are chosen such that the resulting κ lies on C(φ), but not on one of the three special points (ii), gν,ν ′ = gν ′ ,ν and the conductance indicates the breaking of time-reversal symmetry as expected. Turning on an interaction U > 0 the “finetuned” system flows to one of the perfect chain fixed points with equal “perfect” conductances 2πgν,ν ′ = 4/9 and 2πgν ′ ,ν = 4/9. Therefore, at small energy scales the junction conductance does no longer indicate the explicit breaking of time-reversal symmetry. For generic junction parameters away from C(φ) one finds related behavior. Close to the line of decoupled chain fixed points the relative difference |gν,ν ′ −gν ′ ,ν |/(gν,ν ′ +gν ′ ,ν ) scales as a power law in δN with an exponent given by γo /2 and thus vanishes if U > 0. This implies that gν,ν ′ and gν ′ ,ν become equal faster than they go to zero. In that sense for U > 0 and up to the unstable maximal asymmetry fixed point, on small scales the conductance does not show the breaking of time-reversal symmetry–time reversal symmetry is “restored” by the interaction. Other types of junctions of an arbitrary number of LL wires were studied using functional RG (Barnab´-Th´riault et al., 2005) as well as by the poor e e man’s fermionic RG (Aristov et al., 2010; Lal et al., 2002) and bosonization based approaches (Chen et al., 2002; Nayak et al., 1999). 7. Non-equilibrium transport through a contacted wire Non-equilibrium functional RG was used to study a finite bias transport geometry with an impurity-free N site interacting wire contacted to two non-interacting semiinfinite leads by tunnel barriers modeled by reduced hopping matrix elements as introduced in subsection VI.C.3: t0,1 = (tL − 1) and tN,N +1 = (tR − 1) (Jakobs et al., 2007a). In equilibrium this model features a local singleparticle spectral function ρj (ω) which close to the chemical potential, in the vicinity of the contacts, and for repul- 47 fL/R (ω) = eβ(ω−µL/R) + 1 −1 + = β −1 ωn eiωn 0 iωn − ω + µL/R (161) is replaced by + Λ fL/R (ω) = β −1 ωn Θ(|ωn | − Λ)eiωn 0 iωn − ω + µL/R . (162) Details of this procedure including a discussion of the initial conditions and its relation to the temperature flow scheme (Honerkamp and Salmhofer, 2001a) are presented by Jakobs (2010). Within the lowest-order truncation and after taking the equilibrium limit this cutoff implemented for Keldysh Green functions yields the same flow equations as the Matsubara functional RG with the frequency cutoff Eq. (143) (Jakobs, 2010; Jakobs et al., 2007a, 2010b). In the presence of a finite bias voltage the level-1 truncation scheme (bare two-particle vertex) with the cutoff procedure (162) was applied. As discussed in Sec. VI.C.2, in equilibrium this is sufficient to obtain scaling exponents correctly to leading order in U . For weak tunneling ΓL/R = πt2 ρ0 ≪ 1, with ρ0 the density of states L/R of the disconnected, non-interacting leads taken at the last lattice site, the flow of the retarded non-equilibrium self-energy matrix Σret,Λ is given by a weighted sum of two equilibrium flows d ret,Λ Σ = dΛ λ=L,R d eq,Λ Γλ Σ ΓL + ΓR dΛ , (163) µ=µλ where the terms inside the brackets on the right hand side are given by Eq. (158) with the chemical potential set to µL or µR respectively (and U Λ → U ). As discussed in subsection VI.C.4 each such term leads to an oscillatory slowly decaying self-energy originating at the inhomogeneity–the tunnel barriers in the present case–and extending into the interacting part of the wire. The two chemical potentials µL/R imply two different (L/R) wave numbers 2kF . Because of the weighting factor Γλ /(ΓL + ΓR ) the amplitudes of the two superimposed decaying oscillations are generically different and depend on the strength of the bare inhomogeneity. The resulting non-equilibrium effect of two different and Γλ -dependent exponents characterizing the scaling of the spectral function close to µL and µR goes beyond the naive expectation that the bias voltage plays the role of an infrared cutoff scale only (see e.g. Schoeller (2009)). In Fig. 26 the local spectral function near the left contact and for a restricted energy range around µL/R is shown. Due to the finite temperature (T = 10−4 ) and the finite size of the interacting wire (N = 2 × 104 ) the suppression at µL/R is incomplete (cut off by max{T, δN }), but the difference in the exponent is still apparent. A detailed analysis shows that the exponents at µL/R are given by γL/R = ΓL/R γo (µL/R )/(ΓL + ΓR ) where the argument in the open boundary exponent γo indicates that it depends on the band filling and thus the chemical potential. After adding a third probe lead these “non-universal” exponents can be measured in a transport experiment (Jakobs, 2010; Jakobs et al., 2007a). spectral weight (arbitrary units) sive interactions is suppressed: ρj (ω) ∼ ω γo (Enss et al., 2005). The linear conductance behaves as g(T ) ∼ T γo which can be understood from viewing transport as an end-to-end tunneling between a LL and a Fermi liquid lead and using the sum of two resistances as discussed in subsection VI.C.5. A cutoff scheme which conserves causality to any truncation order (Jakobs, 2010) is given by an imaginary frequency cutoff. The Fermi function of the two leads which can be written as a Matsubara sum 2 1 γL=γo/5 γR=4γo/5 0 -0.1 µR 0 ω µL 0.1 FIG. 26 Suppression of the local one-particle spectral weight as a function of energy near the left contact (at site 5) of an interacting wire driven out of equilibrium by a finite bias current. The parameters are T = 10−4 , N = 24 , U = 0.5, tL = 0.075, tR = 0.15, and µL/R = ±0.05. (Data taken from Jakobs et al. (2007a).) D. Quantum dots A spatially confined system featuring a few energy levels is called a quantum dot. In a transport geometry the dot is coupled to at least two leads. Quantum dots show interesting physics if all relevant energy scales (e.g. levellead couplings and T ) are smaller than the level spacing of the isolated system. Due to the strong confinement the two-particle interaction on the dot cannot be neglected and leads to phenomena such as Coulomb blockade and the Kondo effect. 1. Spin fluctuations In the Kondo regime the physics is dominated by spin fluctuations. The virtues and limitations of the functional RG approach to describe aspects of Kondo physics in and out off equilibrium were 48 extensively studied within the single-impurity Anderson model and more complex variants of the latter (Andergassen et al., 2006a; Bartosch et al., 2009a; C. Karrasch and Meden, 2008; Eichler et al., 2009; Gezzi et al., 2007; Goldstein et al., 2009; Hedden et al., 2004; Isidori et al., 2010; Jakobs, 2010; Jakobs et al., 2010a; Karrasch, 2010; Karrasch et al., 2006, 2007a,b, 2008; Karrasch and Meden, 2009; Kashcheyevs et al., 2009; Meden and Marquardt, 2006; Schmidt and W¨lfle, o 2010; Weyrauch and Sibold, 2008; Xu et al., 2008, 2010). As the number of correlated degrees of freedom in quantum dots is small the static truncation used for LLs was extended to contain all second order processes including a frequency dependent two-particle vertex and self-energy, capturing the full real-space as well as spin structure (Hedden et al., 2004; Jakobs, 2010; Jakobs et al., 2010a; Karrasch, 2010; Karrasch et al., 2008, 2010c). In fact, the studies of the single-impurity Anderson model constitute one of the rare examples of the functional RG approach to correlated Fermi systems using a complete level-2 truncation (even supplemented by parts of the six-point vertex through the replacement discussed in the second part of Sec. II.C.2). Including the frequency dependence clearly improves the results beyond bare perturbation theory of the same (that is second) order but it is presently not possible to reach the strong coupling regime in a controlled way. A discussion of the problems yet to be solved was presented by Karrasch et al. (2008), Karrasch et al. (2010c), Karrasch (2010), Jakobs et al. (2010a), and Jakobs (2010). An alternative way to include the full frequency dependence was recently introduced for another impurity model by Schmidt and Enss (2011). In the following a simple quantum dot model dominated by charge fluctuations is discussed. 2. Charge fluctuations in non-equilibrium The quantum dot model belongs to the class of spinless models introduced in subsection VI.C.3. For a three site interacting chain (N = 3) with U1,2 = UL and U2,3 = UR two hopping impurities are located at bonds (1, 2) and (2, 3): t1,2 = tL − 1, t2,3 = tR − 1. Lattice site 2 constitutes a single-level dot which can be adjusted in energy by a gate voltage Vg (see Fig. 17). An electron on this site interacts with lead electrons via a nearest-neighbor coupling UL/R which are otherwise noninteracting. Choosing ν = 1/2 in Eq. (152), Vg =0 corresponds to the particle-hole symmetric point with dot occupation n2 = 1/2. This model is a lattice realization of the interacting resonant level model (IRLM). The use of a variety of analytical as well as numerical methods led to a rather complete understanding of the physics of this model in equilibrium (see e.g. Borda et al. (2007) and references therein). In addition, the current under a finite bias voltage µL = Vb /2 and µR = −Vb /2 was investigated (Borda et al., 2007; Boulat et al., 2008; Doyon, 2007). Field theoretical methods were applied in the scaling limit in which all energy scales are much smaller than the band width B. In the following the focus is on this limit. Functional RG results for the equilibrium and non-equilibrium properties beyond the scaling limit including a favorable comparison with recent numerical time-dependent density-matrix renormalization group data (Boulat et al., 2008) are presented by Karrasch et al. (2010c), Karrasch (2010), and Karrasch et al. (2010a). First order perturbation theory in UL/R leads to logarithmic terms in the self-energy of the form UL/R ln (tL/R /B) which in the scaling limit become large. They indicate the appearance of power laws in tL/R with UL/R dependent exponents. To uncover them requires a treatment which goes beyond perturbation theory. In the limit of weak to intermediate two-particle interactions a Keldysh functional RG approach to the IRLM in the level-1 truncation leads to a comprehensive picture of the physics in and out of equilibrium. In particular, it allows to identify the relevant energy scales. For the present model instead of Eq. (162) another cutoff scheme suitable for non-equilibrium (Jakobs, 2010; Jakobs et al., 2010a) was implemented and tested (Karrasch, 2010; Karrasch et al., 2010c). In this approach each of the three interacting sites is coupled to its own auxiliary lead–in addition to the coupling of sites 1 and 3 to the physical leads. The local density of states at the contact points of the auxiliary leads is assumed to be energy independent (wide band limit) such that the hybridization is energy independent and forms an additional onsite “energy” iΛ on each of the three sites. The auxiliary couplings are then considered as the cutoff and flow from Λ = ∞, at which regularization is achieved, down to Λ = 0, at which the auxiliary leads are decoupled and the original problem is restored. One can show that in the lowest order truncation and in the equilibrium limit the Keldysh contour flow equations become equal to the equilibrium ones obtained using the Matsubara formalism with the (at T = 0) sharp energy cutoff Eq. (143). Similarly to the imaginary frequency cutoff of subsection VI.C.7 it conserves causality even after truncation of the functional RG flow equation hierarchy. In addition, in the equilibrium limit this so-called reservoir cutoff scheme obeys the KMS relation in any truncation order (Jakobs, 2010; Jakobs et al., 2010a,b). In the scaling limit and to lowest order in U only flow equations for the hydridizations ΓΛ with initial values λ Γini = πρ0 t2 appear (λ = L/R); the flow of the level λ λ energies of the sites 1 to 3 is of order U 2 . The renormalized hybridizations set the width of the resonance at Vg = 0. For Λ being smaller than the band width the flow equations for the rates read (ΓΛ = ΓΛ + ΓΛ ) R L dΓΛ Λ + ΓΛ λ . = −2ρ0 Uλ ΓΛ λ dΛ (µλ − Vg )2 + (Λ + ΓΛ )2 (164) 49 They have the approximate solutions Γλ ≈ Γini λ Λ0 max{|µλ − Vg |, Γ/2} 2ρ0 Uλ . (165) The scale Λ0 is of the order of the band width. Within the static approximation the current takes the form of the non-interacting expression with the bare hybridizations Γini replaced by the renormalized ones λ 1 ΓL ΓR Vb /2 − Vg Vb /2 + Vg I= arctan . + arctan π Γ Γ Γ (166) It turns out to be useful (Andergassen et al., 2011; λ Karrasch et al., 2010a) to introduce the two scales Tu = L R Γini Λ0 /Tu , with Tu = Tu + Tu , and the asymmetry paλ L R rameter c2 = Tu /Tu . The same flow equation can be derived using the so-called real-time RG in frequency space (Andergassen et al., 2011; Karrasch et al., 2010a; Schoeller, 2009). Within this approach also the relaxation into the steady state was analyzed in detail (Andergassen et al., 2011; Karrasch et al., 2010a). We first review the results obtained for left-right symmetric model with tL = tR = t′ and UL = UR = U as well as particle-hole symmetry Vg = 0 (Andergassen et al., 2011; Karrasch, 2010; Karrasch et al., 2010a). From Eq. (165) it follows that in this case the maximum of either |µλ | = |Vb |/2 or Γ itself cuts off the RG flow. The charge susceptibility χ = − d n2 /dVg |Vg =0 is directly given by the renormalized width χ−1 = πΓ, which at Vb = 0 and to leading order in U (Γλ → Γini on the right hand side λ of Eq. (165)) gives the scaling relation χ ∼ (Γini )αχ −1 , αχ = 2ρ0 U + O(U 2 ) . (167) In the non-interacting case χ ∼ (Γini )−1 as expected. λ From Eq. (166) it follows that the current for Tu = Γλ ≪ Vb ≪ B is given by I ∼ Γ ∼ Vb−αI , αI = 2ρ0 U + O(U 2 ) . (168) Equations (167) and (168) were also obtained using other approaches (Borda et al., 2007; Boulat et al., 2008; Doyon, 2007) and suggest that the bias voltage merely plays the role of an additional infrared cutoff, besides e.g. Γ or temperature (Borda and Zawadowski, 2010). That this is in general not the case is nicely shown by a functional RG treatment away from particle-hole and/or left-right symmetry (Andergassen et al., 2011; Karrasch, 2010; Karrasch et al., 2010a). The differential conductance g = dI/dVb has a maximum when Vg crosses the chemical potential at Vg = ±Vb /2 (Andergassen et al., 2011; Karrasch et al., 2010a). In the on-resonance case the current for V ≫ Γ reads ΓL ΓR I(Vb ) ≈ = Tu 2Γ Tu 2ρ0 UL Γ c Tu 2ρ0 UL Γ + Tu Vb 1 c 2ρ0 UR Tu Vb 2ρ0 UR c . 1 + c2 (169) The bias voltage dependence is clearly more involved than in Eq. (168). In particular, simple power-law scaling with exponent −2ρ0 UR is only recovered in the extreme limits of either Vb ≫ Tu or c ≫ 1 (Andergassen et al., 2011) as the exponent of the second term in the denominator 2ρ0 UR is small. Off resonance (e.g. at Vg = 0) and for Vb ≫ Γ the current is given by I(Vb ) ≈ Tu Tu |Vb /2−Vg | c Tu |Vb /2−Vg | 2ρ0 UL 2ρ0 UL + Tb |Vb /2+Vg | 1 c 2ρ0 UR Tu |Vb /2+Vg | 2ρ0 UR 2c . 1 + c2 (170) The more involved role of Vb is again apparent. A power law is obtained in the above studied left-right symmetric case or for very strong left-right asymmtery (c ≪ 1 or c ≫ 1) (Andergassen et al., 2011). This concludes the analysis of the IRLM which shows that the functional RG can be a tool to obtain a comprehensive picture of the equilibrium and steady-state nonequilibrium physics of a dot model dominated by charge fluctuations. The approach allows for an unbiased analysis of the non-equilibrium rates and cutoff scales. VII. CONCLUSION A. Summary The functional RG has proven to be a valuable source of new approximation schemes for interacting fermion systems. The heart of the method is an exact flow equation, which describes the flow of the effective action as a function of a suitable flow parameter. The flow provides a smooth evolution from the bare action to the final effective action from which all properties of the systems can be obtained. Approximations are obtained by truncating the effective action. In many cases, rather simple truncations turned out to capture rather complex many-body phenomena. Compared to the traditional resummations of perturbation theory these approximations have the advantage that infrared singularities are treated properly due to the built-in RG structure. Approximations derived in the functional RG framework can be applied directly to microscopic models, not only to renormalizable effective field theories. Remarkably, the functional RG reviewed here as a computational tool is very similar to RG approaches used by mathematicians to derive general rigorous results for interacting fermion systems. We have dedicated a large portion of this review to general features of the functional RG method for interacting Fermi systems (Sec. II). After defining the relevant generating functionals, we have presented a selfcontained derivation of the exact functional flow equation and its expansion leading to an exact hierarchy of flow equations for vertex functions. We have reviewed the different choices of flow parameters used so far, along with their advantages and disadvantages. Truncations and their justification by power-counting have been dis- 50 cussed in detail for translation invariant bulk systems, with links to the closely related mathematical literature. In Secs. III-VI we have reviewed applications of the functional RG to specific systems. Most of the approximations used in these sections are based on relatively simple truncations involving only the flow of the twoparticle vertex and/or the self-energy. Nevertheless a rich variety of phenomena associated with low energy singularities and instabilities is captured. Instead of summarizing the content of each section, let us merely highlight some distinctive features. Sec. III reviews functional RG work on the stability analysis of two-dimensional electron systems with competing instabilities. The main advantage of the functional RG based one-loop computation of the two-particle vertex, compared to other weak-coupling approximations, is that particle-particle and particle-hole channels are treated on equal footing, such that there is no artificial bias due to a selection or a different treatment of channels. In the conventional many-body framework a summation of all parquet diagrams would be required to achieve this, but a solution of the parquet equations is extremely difficult. Spontaneous symmetry breaking, the topic of Sec. IV, can be treated either by a purely fermionic flow, or by coupled flow equations for the fermions and a Hubbard-Stratonovich field for the order parameter. It seems that a comprehensive treatment of all relevant fluctuation effects related to symmetry breaking can be achieved. Applications of the fRG to quantum criticality, reviewed in Sec. V, have begun only recently. Approximations beyond Hertz-Millis theory can be obtained from non-perturbative truncations of the effective order parameter action, or by treating fermions and order parameter fluctuations in a coupled flow instead of integrating the fermionic degrees of freedom at once. While the applications reviewed in Secs. III-V address translation invariant bulk systems, the purpose of Sec. VI is to show how the functional RG can be fruitfully applied to inhomogeneous systems such as quantum wires and quantum dots – in thermal equilibrium and also in a non-equilibrium steady state. A strikingly simple truncation of the flow equation hierarchy turned out to describe a wealth of non-trivial quantum transport properties characterized by low-energy power-laws and complex crossover phenomena. B. Future directions The number of functional RG based works on interacting Fermi systems has increased steadily over the last decade, but the possibilities opened by this approach are far from being exhausted. There are many opportunities and challenges concerning both fundamental developments of the method and the extension to a broader range of systems. On the methodological side there are a number of open issues. In systems with an instability of the normal metallic state, the flow of the effective interactions is not yet fully understood, even on the level of truncations involving only the two-particle vertex and the self-energy, since a faithful parametrization of singular momentum and frequency dependences of the vertex is not easy. The most outstanding challenge is probably to identify accurate and computable truncations of the exact flow equation for strongly interacting systems such as systems close to a Mott metal-insulator transition. It is clear that three-particle and higher order vertices cannot be discarded in a strongly interacting system. However, they will usually not lead to qualitative changes such as new singularities. Hence, there is hope that the contribution from many-body vertices can be absorbed in the structure appearing already on the two-particle level. After all, many strong coupling phenomena, including the Mott transition, consist essentially in the formation of two-particle bound states. To capture effects related to strong local correlations, such as the Mott transition, one may also try to treat higher order vertices in a local approximation. This would make a link to the dynamical mean-field theory (DMFT), where all vertices, including the self-energy, are approximated by local functions (Georges et al., 1996). For systems with strongly interacting order parameter fluctuations there are already a number of nonperturbative approximations for bosonic actions on the market. The local potential approximation presented in Sec. V is only the simplest one. It can be extended by taking non-local contributions into account, either in a derivative expansion (Berges et al., 2002), or by including the full momentum or frequency dependence up to a certain level in the hierarchy (Blaizot et al., 2005). Such approximations may be very useful for studying incommensurate density wave instabilities in cases where the modulation vector of the density wave can be determined only after taking fluctuations into account. Recently, the functional RG was extended to a real time (or real frequency) Keldysh functional RG which can be used for studying correlated Fermi systems in non-equilibrium (Gezzi et al., 2007; Jakobs, 2003, 2010; Karrasch, 2010). First applications, partly reviewed in Sec. VI, indicate that also for these type of problems the functional RG constitutes a useful tool of outstanding flexibility. So far only non-equilibrium steady states were studied. To investigate a time evolution is technically straightforward but requires a significantly increased computational effort or additional approximations. With few exceptions, applications of the functional RG to interacting Fermi systems have so far been limited to purely fermionic one-band systems. There are many extensions of this restricted class of systems, where the flexibility of the functional RG can be fruitfully used in the future. Multi-band models have been studied already for the pnictide superconductors, but there are many more and qualitatively different models for transition metal oxides with orbital degrees of freedom. One may include phonons and analyze the electron-phonon interaction ef- 51 fects beyond the Eliashberg theory. Allowing for disorder one may study the complex interplay of interaction and disorder effects. It is not hard to generalize the exact flow equations for the extensions listed above. The interesting task is then to devise suitable truncations. Last but not least, the functional RG is an ideal manybody tool to be combined with ab initio band structure calculations. A lot of work in the last 15 years has been dedicated to the ab initio calculation of correlated electron materials with the DMFT (Anisimov and Izyumov, 2010; Kotliar et al., 2006). As in DMFT, an arbitrary band structure can be used as input for a functional RG calculation. Furthermore, one can easily implement nonlocal potentials and non-local two-particle interactions. obtained as follows. Using the definition of W Λ and the flow equation for V Λ , one can write d d ∆GΛ Λ WΛ = e ¯0 V dΛ dΛ ∆ ¯Λ = ∆ ¯ Λ e G0 V Λ ˙ G0 + e ∆GΛ ¯ 0 − ∆GΛ V Λ + ˙ ¯ 0 = 1 diff Λ Λ ∆ ¯Λ V V ˙ 2 G0 1 ∆GΛ diff Λ Λ e ¯0 ∆ ¯Λ V V . ˙ G0 2 Using the decomposition ∆GΛ = ∆factor1 + ∆factor2 + ¯ ¯ ¯ GΛ GΛ 0 0 0 ∆diff (when acting on a product), this yields ∂Λ W Λ = ¯ GΛ 0 ∆diff ¯Λ e G0 1 2 Acknowledgments We are very grateful for fruitful collaborations and/or discussions with S. Andergassen, J. Bauer, D. Baeriswyl, C. Castellani, A. Chubukov, C. Di Castro, J. von Delft, A. Eberlein, T. Enss, J. Feldman, R. Gersch, H. Gies, W. Hanke, C. Husemann, S. Jakobs, P. Jakubczyk, C. Karrasch, A. Katanin, H. Kn¨rrer, P. Kopietz, D.-H. Lee, B. o Obert, J. Pawlowski, C. Platt, M. Pletyukhov, M. Rice, D. Rohe, A. Rosch, H. Schoeller, P. Strack, S. Takei, R. Thomale, E. Trubowitz, C. Wetterich, P. W¨lfle, and o H. Yamase. All of us greatly benefitted from the DFG research group Functional renormalization group in correlated fermion systems (FOR 723). Appendix A: Wick-ordered flow equations In this appendix we present a derivation of Wick ordered flow equations for fermions, which have been used for calculations of instabilities and symmetry-breaking in the two-dimensional Hubbard model. Λ Wick ordered m-particle functions Wm are generated from the Wick-ordered effective interaction (Salmhofer, 1999) W Λ [χ, χ] = e ¯ ∆GΛ ¯ 0 V Λ [χ, χ] . ¯ (A1) The exponent in the Wick-ordering factor is the func¯ ¯ ¯ tional Laplacian ∆GΛ = (∂χ , GΛ ∂χ ) with GΛ = G0 − GΛ . ¯ 0 0 0 0 The Wick-ordered interaction converges to V for Λ → 0, ¯ since GΛ vanishes in that limit. However, the flow equa0 tions for W Λ and the corresponding m-particle functions differ from those for V Λ . The flow equation for the generating functional W Λ reads (Salmhofer, 1999) d 1 ∆diff ¯Λ W Λ = e G0 ∆diff W Λ W Λ , ˙ ¯Λ G0 dΛ 2 (A2) where the superscript ”diff” indicates that the Laplacian takes one derivate on the first, and the other on the second factor W Λ on the right-hand side. This equation is ∆diff e ˙ ¯Λ ∆GΛ ¯ 0 G0 VΛ e ∆GΛ ¯ 0 V Λ and thus Eq. (A2). Expanding in powers of Grassmann fields and comparing coefficients, one obtains a hierarchy of flow equations for the m-particle functions W (2m)Λ , which is illustrated diagrammatically in Fig. 27. The line with the dash W d dΛ (2m)Λ W = (2n)Λ 2 Σ W 1 n,j (2m−2n+2j)Λ . . . . j FIG. 27 Diagrammatic representation of the flow equations for the effective m-particle interactions W (2m)Λ in the Wickordered version of the functional RG; the internal line with ¯0 ¯0 a dash corresponds to ∂Λ GΛ , the others to GΛ ; all possible pairings leaving m ingoing and m outgoing external legs have to be summed. is due to contractions generated by ∆diff in Eq. (A2), ˙ ¯Λ G0 the other lines are generated by the exponential of ∆diff . ¯ GΛ 0 Note that the right-hand side of the Wick-ordered flow equations is bilinear in the effective interactions, and no tadpole terms appear. Note also that the propagators connecting the vertices have support for energies at and below the cutoff scale Λ, such that the integration region shrinks as Λ decreases. One might worry that the low-energy propagators lead to infrared divergences even for Λ > 0. This is not the case, as can be seen from the general infrared power-counting analysis presented by Salmhofer (1999). Appendix B: Details of power counting 1. Propagator bounds Here we show, using properties of the dispersion function and the Fermi surface, that sΛ ≤ a + b log Λ0 Λ and GΛ ≤ c Λ−1 (B1) 52 where a, b and c are constants that do not depend on Λ. In absence of Van Hove singularities, b = 0. We first consider the case where the self-energy effects are not taken into account (they are discussed in Subsection II.E.4). k2 +ξ 2 Then one can simply take χΛ (k) = χ> 0Λ2 k where χ> (η) is a fixed (i.e. Λ-independent) increasing function that vanishes at least linearly at η = 0, tends to 1 as η → ∞ and satisfies χ′ (η) ≤ η −2 for large η. We can > then verify (B1) by scaling, as follows. The full propagac tor is GΛ (k) = (ik0 − ξk )−1 χΛ (k), so |GΛ (k)| ≤ Λ where −1 2 c = max{η χ> (η ) : η > 0} is finite. The single-scale propagator is S Λ (k) = − 2 (ik0 + ξk ) χ′ > Λ3 2 2 k0 +ξk Λ2 . 4 Λ3 dk0 dE N (E) 2 k0 + E 2 χ′ > 2 k0 +E 2 Λ2 4 Λ3 ∞ 0 ρ2 dρ χ′ > ρ2 Λ2 , (B3) 2π dϕ N (ρ cos ϕ). (B4) 0 If the density of states N is bounded, using N (E) ≤ N0 and scaling out Λ implies sΛ ≤ a, with a = 8πN0 ρ2 χ′ (ρ2 )dρ < ∞. In presence of a Van Hove > point on the Fermi surface, N stays bounded in dimensions d ≥ 3, but diverges logarithmically for d = 2. In this case, the ϕ-integral contributes an additional factor log Λ. Thus (B1) holds. The hypotheses on χ> are satisfied in particular for the standard strict cutoff functions that vanish identically near η = 0, and which are identically 1 for η ≥ 1. For such cutoffs, the single-scale propagator is nonvanishing only in a “momentum shell” of thickness Λ around the Fermi surface, and the above bounds can also be obtained by estimating the k-space volume of this shell (see also Section B.3). 2. Power counting Here we prove (70) to all orders in the running coupling expansion. All terms on the right hand side of the flow (2m)Λ equation for Γr are of the form 1 1 tr SΛ PΛ (k, σ) = 2 2 dl ¯ α,α′ l0 Λ ˆΛ Sα,α′ (l) Pα,α′ (k, σ; l, −l), (B5) ˆ ˆ (2m+2)Λ in the first term of (67) and given where P Λ = Γr by the other summands in (67) for the other terms, and dd l (2π)d contains both frequency and momen- tum summations. Taking the absolute values inside the ˆ sum and estimating the factor P Λ by its maximum P Λ , we obtain 1 2 tr (SΛ PΛ ) ≤ sΛ P Λ α (B6) dk |Sα,α′ (k)|. The second sim¯ ˆΛ with sΛ = max α′ ple inequality that we shall use is that P1 . . . Pn ≤ P1 . . . Pn . It implies bounds for all LΛ -contributions p (2mq )Λ in terms of GΛ and Γrq bounded by sΛ where the 4 instead of 2 gives a (crude) bound for the change from the Matsubara sum to the integral for large 1 2 enough β. Changing variables to ρ = (k0 + E 2 ) 2 and a polar angle ϕ, we obtain sΛ ≤ 1 β (B2) Using that the Matsubara sum is a Riemann sum for the convergent integral of S Λ over k0 and introducing the density of states N (E) = dd k δ(ξk − E), we get sΛ ≤ dl = ¯ (2m+2)Λ Γr + VΛ GΛ + p−1 , so that GΛ d (2m)Λ dΛ Γr (2m)Λ Γr−1 (2m Γr1 1 )Λ . . . Γ(2mp )Λ rp ′ is . (B7) p≥2 The power counting is now determined by sΛ and GΛ . Given (B7) and (B1), the proof of (70) is an effortless induction argument. The inductive scheme proceeds in the standard way of (Polchinski, 1984), namely upwards in r ≥ 1 and at fixed r, downwards in m, starting at (2m+2)Λ = 0. The induction start r = 1 is m = r, where Γr trivial. Let r ≥ 2 and assume (70) to hold for all (r′ , m′ ) with r′ < r and for r′ = r, m′ > m. The right hand side of (B7) contains only terms to which the inductive hypothesis (70) applies. Inserting it, using (B1), and collecting powers in the form 1 − p + q (2 − mq ) = 1 − m and p (rq − mq + 1) = r − m, we obtain q=1 d (2m)Λ dΛ Γr ≤ γr sΛ r−m+1 fΛ r Λ1−m ˜ (2m) (B8) (2m) where the constant γr ˜ is a weighted sum of products of (2mq ) (2m)Λ the γrq . We now use the initial condition Γr =0 Λ (2m)Λ d (2m)ℓ = − Λ 0 dℓ dℓ Γr to write Γr , take the norm of this equation, and use (B8). This gives Λ0 (2m)Λ Γr ≤ γr ˜ (2m) dℓ sℓ r−m+1 fℓ r ℓ1−m (B9) Λ By definition, fΛ ≥ fΛ′ if Λ ≤ Λ′ , so fℓ ≤ fΛ for all ℓ in the integration interval. Thus the last integral is Λ (2m) bounded by γr fΛ r Λ 0 dℓ sℓ r−m+1 ℓ1−m . Because sℓ is ˜ Λ at most logarithmic in ℓ, and m ≥ 3, Λ 0 dℓ sℓ α ℓ1−m ≤ 2−m α KΛ sΛ with a constant K that depends on α and (2m) m. This, together with an appropriate choice of γr , completes the induction step. For m = 2, doing the last integral increases the power of the logarithm by one. This case is discussed in more detail in Subsection B.3. For m = 1, the self-energy term, the same simple d bound gives dΛ ΣΛ ≤ sΛ fΛ , so the integral gives a contribution of order fΛ . When a counterterm is used to keep the Fermi surface fixed, the initial condition for ΣΛ 53 at Λ = Λ0 is given by the counterterm, which needs to be adjusted such that at low scales Λ, ΣΛ (0, k) = O(Λ) whenever ξk = 0. This leads to the self-consistency relation mentioned in section II.E.4. A similar proof can be given in the Wick ordered scheme (Salmhofer, 1998b); it is even simpler because the double induction used here is replaced by single induction on r. A crucial point in obtaining (B8) is that all the dependence on p and on the mq drops out when the power of Λ is collected. It is this property that makes many-fermion models with short-range interactions renormalizable in the strict quantum-field-theoretical sense. The classification in relevant, marginal and irrelevant terms now also (2m)Λ becomes apparent because for m ≥ 3, the Γr grow as Λ decreases: suppose we add an additional (2m ≥ 6)˜ point interaction vertex V (2m)Λ0 of order 1 to the initial interaction at Λ0 . Its insertion at a lower scale Λ is a facΛ tor ( Λ0 )m−2 smaller than that of the effective 2m-pointvertex created by the two-particle interaction. Thus the ˜ influence of V (2m)Λ0 wanes at low scales – it is irrelevant. A simple adaptation of the above inductive argument indeed shows that the inclusion of such additional terms with m ≥ 3 in the interaction at Λ0 changes only prefactors in the power counting bounds. For m = 2, this suppressing factor is absent, so that these terms are marginal (the more detailed analysis of Section B.3 shows how to separate the marginally relevant from the marginally irrelevant terms). Moreover, it is clear that this power counting breaks down when Γ(4)Λ develops singularities as a function of k and ω, because then fΛ = ∞. Finally, for m = 1, the scale derivative of the self-energy obtained by the above argument is of order Λ1−m , as in (B8), but since m = 1, this integrates to O(1) instead of O(Λ2−m ) – this term is relevant. To get its size back to O(Λ2−m ) in the momentum shell where |ξ(k)| ∼ Λ, one needs a cancellation by a counterterm, as described briefly in Section II.E.4. In the Taylor expansion required to do the cancellation, the derivative of the self-energy appears. By the above power counting, this is a marginal term. In the Luttinger model, it is really marginal and causes the anomalous exponents. For curved Fermi surfaces in d ≥ 2 dimensions, it is seen to be irrelevant by the arguments discussed in Section B.3.b. There is a hard problem hidden in the recursion of (2m) the constants γr . In the recursion described above, the number of terms that gets added corresponds to the number of Feynman graphs with r vertices, which grows factorially in r, so that the bound obtained in (2m) (2m) is of order r!. If satuand cr this way for γr rated, it would lead to a convergence problem. However, due to the fermionic antisymmetry, sign cancellations in the sum over Feynman diagrams prevent this factorial from arising. For proofs, we refer interested readers to the literature (see Disertori and Rivasseau (2000); Feldman et al. (1998a, 2002, 1992); Pedra and Salmhofer (2008); Salmhofer and Wieczerkowski (2000) and refer- ences therein). In their application to propagators with Fermi surfaces, these proofs also provide a rigorous basis for the use of Fermi surface patches (first used in (Feldman et al., 1992) and there called “sectors”). Patching the Fermi surface has become an essential tool also in applications, see Section III. In a typical lattice model, the kinetic energy per particle is bounded, so that the flow is usually started at the highest value (the bandwidth) of the kinetic energy, Λ0 . The χΛ we used here also cuts off large frequencies. Thus the starting interaction is in this case one where the degrees of freedom with frequencies |k0 | above Λ0 have already been integrated over. This starting action can be obtained by convergent perturbation theory, see Pedra and Salmhofer (2008). 3. Improved power counting This is a refinement of power counting, valid in a large class of bulk fermion systems in d ≥ 2 (Feldman et al., 1996, 1998b, 1999, 2000; Feldman and Trubowitz, 1990; Shankar, 1994). It is the deeper reason behind the emergence of Fermi liquid behaviour and of dominant Cooper pairing tencencies in weakly coupled standard fermion systems, and it provides a precise link between Fermi surface geometry and scaling properties of the effective m-particle vertices in general. We discuss this in the absence of self-energy effects, to bring out the main effects as clearly as possible. (The self-energy changes the Fermi surface; if the interacting Fermi surface is regular, the following analysis remains essentially unchanged.) We also assume a strict cutoff function, i.e. χ> (η) = 0 for η ≤ (1 − δ)2 , where 0 < δ < 1/2 is fixed, and χ> (η) = 1 for η ≥ 1. Again, this choice is not essential; it just simplifies the discussion. a. Effects of curvature on power counting The integral IΛ (k) = dp0 dd p|S Λ (p)| |GΛ (±p + k)| arises from the trace on the right hand side of the RG equation when all effective vertices and all but one of the propagators GΛ have been estimated by their maximal values. It thus determines the maximal possible value of a term on the right hand side of the RG equation, where the dependence on one external momentum is kept. In particular, IΛ is directly relevant for the oneloop contributions to the flowing four-point vertex. The power counting done above corresponds to the estimate IΛ (k) ≤ GΛ sΛ ≤ Λ−1 sΛ , so that Λ Iλ dλ grows logarithmically in Λ for small Λ. Since Σλ = 0, S Λ (k) = (ik0 − ξk )−1 ∂Λ χΛ (k) and GΛ (k) = χΛ (k) = GΛ0 (k) − ik0 − ξk Λ0 dλ S λ (k). (B10) Λ The term GΛ0 is nonvanishing at large frequencies, but not important ( GΛ0 ≤ Λ−1 , hence a factor 0 54 Λ/Λ0 smaller than GΛ when Λ gets small. Thus dp0 dd p|S Λ (p)| |GΛ0 (p + k)| ≤ sΛ Λ−1 , hence its inte0 gral over Λ is bounded by a constant). The derivative of the strict cutoff function vanishes unless Λ(1 − δ) ≤ |ik0 − ξk | ≤ Λ, so S Λ (p) vanishes unless |p0 | ≤ Λ and p is in the momentum space shell FΛ = {k : |ξk | ≤ Λ}, 1 and there, |S Λ (p)| ≤ Λ2 . The p0 -sum in IΛ gives at most 2Λ, and the p-integral gives the d-dimensional volume of the intersection FΛ ∩ (k ± Fλ ) of two momentum space shells, where one is shifted by k. It follows that 2 Λ Λ0 dλ vold (FΛ ∩ (k ± Fλ )) . λ2 Λ (B11) This links the scaling behaviour of terms in the RG equation to the geometric properties of the Fermi surface. Obviously, the volume of the intersection is at most as large as the volume of FΛ itself: vold (FΛ ∩ (k ± Fλ )) ≤ vold FΛ ≤ const.Λ. Using this in (B11) gives the general power counting bound mentioned at the beginning of this section, IΛ (k) ≤ const.Λ−1 . Assuming that ξ−k = ξk , this bound is always saturated for k = 0, and also for those k for which the shift by k makes the two shells overlap over a significant region of the Fermi surface, that is, when k is a nesting vector of the Fermi surface. For other values of k, the intersection volume can be much smaller than that of FΛ . A general definition of non-nesting was given, and power counting bounds were derived when it is satisfied, by Feldman et al. (1996), and extended to the case with Van Hove singularities in (Feldman and Salmhofer, 2008a,b). Here we only cite the result for the case of a strictly convex and positively curved Fermi surface without Van Hove singularities, discussed also in the Appendix of (Salmhofer, 1999). In that case, and for Λ ≤ λ ≤ vF,min |k|, one can show that the volume ratio vold (FΛ ∩(k±Fλ )) is proportional to vold FΛ IΛ (k) ≤ O(Λ−1 ) + 0 λ |k|vF,min κ λ κ d−1 2 FIG. 28 Intersections of a momentum shell around the Fermi surface with its translate, as arising in loop integrals on the right hand side of the RG equation. When the Fermi surface is curved, the intersection volume decreases strongly unless the translating momentum is small. in the Hubbard model near to half-filling and at small next-to-nearest hopping, one thus has an effective nesting at those scales and at those k where the quotients in (B12) and (B13) are so big that they give a bound that is bigger than the trivial bound 1 for the volume ratio. Eqs. (B12) and (B13) imply that for small |k|, Λ0 0 IΛ (k)dΛ ≤ const. log Λ0 |k|vF,min (B14) (where the constant depends on the curvature of the Fermi surface) and that the function remains bounded for |k| not close to zero (for details, see Feldman et al. (1996); Salmhofer (1998a, 1999)). Thus, for convex curved Fermi surfaces, the four-point function can diverge only at k = 0 and there, only logarithmically (by a similar argument, one can see that it can diverge only at k0 = 0). The particle-particle correction to the vertex function has exactly this behaviour. In the particle-hole term, there is an additional sign cancellation that removes the logarithm. The same argument shows that in general, divergences can occur only at nesting vectors of the Fermi surface. (2) (B12) b. Uniform improvement from overlapping loops (2) (B13) An extension of these geometric estimates to two-loop ′ ′′ integrals of the type dp dq S Λ (p) S Λ (q)S Λ (p±q±k) is very useful for d ≥ 2: it is shown in (Feldman et al., 1996) that in absence of nesting and Van Hove singularities, such integrals contain a scaling improvement independently of k. Such two-loop integrals associated to graphs with overlapping loops arise when the RG equation gets iterated; the graph classification of (Feldman et al., 1996, 1999) shows that in a precise sense, the overwhelming majority of graphs in the Feynman graph expansion contains one or even two such subintegrals, hence becomes subleading at low scales. As is explained in detail in (Feldman et al., 1996; Salmhofer, 1998a), in absence of nesting and Van Hove singularities, this justifies the particle-particle-ladder aproximation, it singles out the Hartree-Fock type contributions to the self-energy by scaling arguments, and it allows to show that the derivative of the self-energy is RG-irrelevant instead of marginal. if k ∈ Fλ if k ∈ Fλ . Here vF,min is the smallest value of |∇e| on the (2) Fermi surface, Fλ is a O(λ)-neighbourhood of the set {2k : ξk = 0} (note that 2k is taken modulo reciprocal lattice vectors), and κ denotes the minimal curvature on the Fermi surface. This is illustrated for λ = Λ in Fig. 28. In the first case, the intersection is transversal, which decreases the intersection volume by the factor in (B12). The second case corresponds to a 2kF -intersection, where √ the curvature in a region of size λ determines the intersection volume, corresponding to (B13). 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