arXiv:cond-mat/0305035v2 [cond-mat.str-el] 20 May 2003 Chapter 1 LUTTINGER LIQUIDS: THE BASIC CONCEPTS K.Sch- nhammer o Institut f- r Theoretische Physik, Universit- t G- ttingen, Bunsenstr. 9, D-37073 u a o G- ttingen, Germany o Abstract This chapter reviews the theoretical description of interacting fermions in one dimension. The Luttinger liquid concept is elucidated using the TomonagaLuttinger model as well as integrable lattice models. Weakly coupled chains and the attempts to experimentally verify the theoretical predictions are discussed. Keywords Luttinger liquids, Tomonaga model, bosonization, anomalous power laws, breakdown of Fermi liquid theory, spin-charge separation, spectral functions, coupled chains, quasi-one-dimensional conductors 1. Introduction In this chapter we attempt a simple selfcontained introduction to the main ideas and important computational tools for the description of interacting fermions in one spatial dimension. The reader is expected to have some knowledge of the method of second quantization. As in section 3 we describe a constructive approach to the important concept of bosonization, no quantum fieldtheoretical background is required. After mainly focusing on the Tomonaga- 2 Luttinger model in sections 2 and 3 we present results for integrable lattice models in section 4. In order to make contact to more realistic systems the coupling of strictly 1d systems as well as to the surrounding is addressed in section 5. The attempts to experimentally verify typical Luttinger liquid features like anomalous power laws in various correlation functions are only shortly discussed as this is treated in other chapters of this book. 2. Luttinger liquids - a short history of the ideas As an introduction the basic steps towards the general concept of Luttinger liquids are presented in historical order. In this exposition the ideas are discussed without presenting all technical details. This is done in section 3 by disregarding the historical aspects aiming at a simple presentation of the important practical concepts like the “ bosonization of field operators”. 2.1 Bloch’s method of “sound waves” (1934) In a paper on incoherent x-ray diffraction Bloch [1] realized and used the fact that one-dimensional (d = 1) noninteracting fermions have the same type of low energy excitations as a harmonic chain. The following discussion of this connection is very different from Bloch’s own presentation. The low energy excitations determine e.g. the low temperature specific heat. Debye’s famous T 3 -law for the lattice contribution of three dimensional solids reads in d = 1 kB T π , (1.1) cDebye = kB L 3 h ¯ cs where cs is the sound velocity. At low temperatures the electronic contribution to the specific heat in the “Fermi gas” approximation of Pauli is also linear in T and involves the density of states of the noninteraction electrons at the Fermi energy. This yields for spinless fermions in d = 1 kB T π kB , (1.2) 3 h ¯ vF where vF is the Fermi velocity. With the replacement cs ↔ vF the results are identical. This suggests that apart from a scale factor the (low energy) excitation energies and the degeneracies in the two types of systems are identical. For the harmonic chain the excited states are classified by the numbers nk of phonons in the modes ωk whith nk taking integer values from zero to infinity. h The excitation energy is given by E({nkm }) − E0 = km ¯ ωkm nkm . For small wave numbers km the dispersion is linear ωkm ≈ cs |km |. Therefore the excitations energies are multiples of ¯ cs (2π/L) for periodic boundary conh ditions and multiples of ∆B ≡ ¯ cs π/L for fixed boundary conditions. The h calculation of the partition function is standard textbook material. This is also true for noninteracting electrons but there the calculation involves fermionic cPauli = L 3 occupation numbers nF which take values zero and one. The two textbook calk culations yield Eqs. (1.1) and (1.2), but through the “clever” use of the grand canonical ensemble in order to simplify the fermionic calculation the identity (apart from cs ↔ vF ) remains mysterious. A deeper understanding involves two steps: 1) Linearization of the kinetic energy εk = h2 k2 /(2m) of the free fermions ¯ around the Fermi point kF for fixed boundary conditions or both Fermi points ±kF for periodic boundary conditions. As the argument is simplest for fixed boundary conditions [2] which lead to km = mπ/L we discuss this case for the h moment. Then the energies εkn − εF are integer multiples of ∆F ≡ ¯ vF π/L where vF is the Fermi velocity. 2) Classification of any state of the Fermi system by the number nj of upward shifts by j units of ∆F with respect to the ground state. As the fermions are indistinguishable the construction of the {nj } shown in Fig. 1.1, where the highest occupied level in the excited state is connected with the highest occupied level in the ground state and so forth for the second, third . . . highest levels, completely specifies the excited state. As the nj can run from zero to infinity like bosonic quantum numbers and the excitation energy is given by j (j∆F )nj the canonical partition function for the noninteracting fermions has the same form as the canonical partition function for the harmonic chain apart from ∆F ↔ ∆B if one assumes the Fermi sea to be infinitely deep [3]. As we have linearized ωk for small k as well as εk around kF this equivalence only holds for the low temperature specific heats (kB T ≪ ¯ ωmax , kB T ≪ h εF ). If we denote the creation (annihilation) operator of a fermion with kn = nπ/L by c† (cn ) and assume a strictly linear dispersion εlin = hvF kn for all ¯ n n kn > 0 a more technical formulation of the discussed equivalence can be given by the exact operator identity ∞ T h ¯ vF kn c† cn n = n=1 = h ¯ vF π L ∞ 1 l b† bl + N (N + 1) , l 2 l=1 (1.3) where the operators bl with l ≥ 1 are defined as 1 bl ≡ √ ∞ l m=1 c† cm+l m (1.4) and N ≡ ∞ c† cn is the fermionic particle number operator. The proof of n=1 n the “Kronig identity” (1.3) is simple (see Ref. [4]) . The operators bl obey 4 7 4 4 nF 3 { 1 ∆F 1 0 0 groundstate excited state 20 E0 E0 +20 ∆F Figure 1.1. Example for the classification scheme for the excited states in terms of the numbers nj of upward shifts by j units of ∆F . In the example shown the nonzero nj are n7 = 1, n4 = 2, n3 = 1 and n1 = 2. commutation relations [bl , bl′ ] = 0 and for l ≥ l′ bl , b†′ l 1 =√ ll′ l′ c† cm+l−l′ . m (1.5) m=1 (M ) For all N -particle states |φN ≡ N c†n |Vac in which the M (< N ) lown=1 i est one-particle levels are all occupied one obtains (M ) bl , b†′ |φN l (M ) = δll′ |φN (1.6) for l, l′ ≤ M , i.e. these operators obey boson commutation relations [bl , b†′ ] = l δll′ ˆ in this subspace of all possible N -particle states. 1 ˜ ˜ Later it turns out to be useful to work with T ≡ T − T 0 − µ0 N , where T 0 = ∆F nF (nF + 1)/2 is the ground-state energy, µ0 = ∆F (nF + 1/2) ˜ is the chemical potential of the noninteracting fermions and N ≡ N − nF ˆ 1. ˜ is of the form as the rhs of Eq. (1.3) with N (N + 1) replaced by N 2 . ˜ Then T 5 2.2 Tomonaga (1950): Bloch’s method of sound waves applied to interacting fermions When a two-body interaction between the fermions is switched on, the ground state is no longer the filled Fermi sea but it has admixtures of (multiple) particle-hole pair excitations. In order to simplify the problem Tomonaga studied the high density limit where the range of the interaction is much larger than the interparticle distance, using periodic boundary conditions [5]. Then the Fourier transform v (k) of the two-body interaction is nonzero only for val˜ ues |k| ≤ kc where the cut-off kc is much smaller than the Fermi momentum kc ≪ kF . This implies that for not too strong interaction the ground state and low energy excited states have negligible admixtures of holes deep in the Fermi sea and particles with momenta |k| − kF ≫ kc . In the two intermediate regions around the two Fermi points ±kF , with particle-hole pairs present, the dispersion εk is linearized in order to apply Bloch’s “sound wave method” k ≈ ±kF : εk = εF ± vF (k ∓ kF ). (1.7) Tomonaga realized that the Fourier components of the operator of the density L/2 ρn = ˆ = ρ(x)e−ikn x dx = ˆ −L/2 c† ′ cn′ +n , n n′ L/2 −L/2 ψ † (x)ψ(x)e−ikn x dx (1.8) where c† (cn ) creates (annihilates) a fermion in the state with momentum kn = n 2π L n, plays a central role in the interaction term, as well as the kinetic energy. Apart from an additional term linear in the particle number operator [4], which is usually neglected, the two-body interaction is given by 1 2 1 ˆ v (kn )ˆn ρ−n + ˜ ρ ˆ N v (0) ˜ V = 2L n=0 2L (1.9) Tomonaga’s important step was to split ρn for |kn | ≪ kF into two parts, one ˆ containing operators of “right movers” i.e. involving fermions near the right Fermi point kF with velocity vF and “left movers” involving fermions near −kF with velocity −vF c† ′ cn′ +n + n ρn = ˆ n′ ≥0 n′ <0 ˆ ˆ c† ′ cn′ +n ≡ ρn,+ + ρn,− n (1.10) where the details of the splitting for small |n′ | are irrelevant. Apart from the square root factor the ρn,α are similar to the bl defined in Eq. (1.4) . Their ˆ commutation relations in the low energy subspace are [ˆm,α , ρn,β ] = αmδαβ δm,−n ˆ ρ ˆ 1. (1.11) 6 If one defines the operators bn ≡ 1 |n| ρn,+ ˆ ρn,− ˆ for n > 0 for n < 0 (1.12) and the corresponding adjoint operators b† this leads using ρ† = ρ−n,α to n n,α the bosonic commutation relations [bn , bm ] = 0, [bn , b† ] = δmn ˆ 1. m (1.13) Now the kinetic energy of the right movers as well as that of the left movers ˆ can be “bosonized” as in Eq. (1.3). The interaction V is bilinear in the ρn as ˆ well as the ρn,α. Therefore apart from an additional term containing particle ˆ number operators the Hamiltonian for the interacting fermions is a quadratic form in the boson operators ˜ H = h ¯ kn vF + n>0 v (kn ) ˜ 2π¯ h v (kn ) † † ˜ bn b−n + b−n bn 2π¯ h ≡ HB + HN ,J , ˜ + b† bn + b† b−n −n n + h ¯π ˜ vN N 2 + vJ J 2 2L (1.14) ˜ ˜ ˜ where N ≡ N+ + N− is the total particle number operator relative to the ˜+ − N− the “current operator”, and the velocities are ˜ Fermi sea, J ≡ N given by vN = vF + v (0)/π¯ and vJ = vF . Here vN determines the en˜ h ergy change for adding particles without generating bosons while vJ enters the energy change when the difference in the number of right and left movers is changed. Similar to the discussion at the end of section 1.1 we have defined H H ˜ ˜ ˜ H ≡ H − E0 − (µ0 + v (0)n0 )N , where E0 is the Hartree energy and n0 the ˜ particle density. As the particle number operators Nα commute with the boson † ) the two terms H and H operators bm (bm ˜ B N ,J in the Hamiltonian commute and can be treated separately. Because of the translational invariance the twobody interaction only couples the modes described by b† and b−n . With the n Bogoliubov transformation α† = cn b† − sn b−n the Hamiltonian HB can be n n brought into the form h ¯ ωn α† αn + const., n HB = (1.15) n=0 h ˜ where the ωn = vF |kn | 1 + v (kn )/π¯ vF follow from 2 × 2 eigenvalue problems corresponding to the condition [HB , α† ] = hωn α† . For small kn one ¯ n n obtains for smooth potentials v (k) again a linear dispersion ωn ≈ vc |kn |, with ˜ √ the “charge velocity” vc = vN vJ , which is larger than vF for v (0) > 0 . ˜ 7 The expression for the coefficients cn and sn with c2 − s2 = 1 will be pren n sented later for the generalized model Eq. (1.17) . For fixed particle numbers N+ and N− , the excitation energies of the interacting Fermi system are given h by m ¯ ωm nm with integer occupation numbers 0 ≤ nm < ∞. For small enough excitation energies the only difference of the excitation spectrum for fixed particle numbers with respect to the noninteracting case is the replacement vF ↔ vc . In his seminal paper Tomonaga did not realize the anomalous decay of correlation functions of the model because in his discussion of the density correlation function he missed the 2kF -contribution discussed in section 3. εk / ε F 2.0 0.0 −2.0 −2.0 −1.0 0.0 1.0 2.0 k/k F Figure 1.2. Energy dispersion as a function of momentum. The dashed curve shows the usual “nonrelativistic” dispersion and the full curve the linearized version used (apart from a constant shift) in Eq. (1.3) for k > 0 for fixed boundary conditions. The dot-dashed parts are the additional states for k0 = −1.5kF . The model discussed by Luttinger corresponds to k0 → −∞. 2.3 Luttinger (1963): no discontinuity at the Fermi surface Luttinger, apparently unaware of Tomonaga’s work, treated spinless, massless fermions (in the relativistic sense, but c ↔ vF ) in one dimension, i.e. two infinite branches of right and left moving fermions with dispersion ±vF k [6]. As Luttinger himself made an error with the fact that his Hamiltonian is not bounded from below, it is useful to switch from Tomonaga’s to Luttinger’s model keeping a band cut-off k0 such that k ≥ k0 = 2πm0 /L with m0 < 0 for the right movers and correspondingly for the left movers (see Fig. 8 1.2). Fortunately Luttinger’s error had no influence on his inquiry if a sharp Fermi surface exists in the exact ground state of the interacting model. After a rather complicated calculation using properties of so-called “Toeplitz determinants” Luttinger found that the average occupation nk,+ in the ground state for k ≈ kF behaves as nk,+ − k − kF 1 ∼ 2 kc αL sign(kF − k), (1.16) where αL depends on the interaction strength (see below) [7]. “Thus, in this model, the smallest amount of interaction always destroys the discontinuity of nk at the Fermi surface” [6]. This can be related to the fact that the equal time † correlation functions ψα (x)ψα (0) decay as 1/|x|1+αL in the interacting sys† (x)ψ (0) ∼ 1/|x|d (with d = 1) in the noninteracting tem in contrast to ψα α case. Therefore αL is called the “anomalous dimension”[8]. Apart from the different dispersion Luttinger also used a different interaction. In contrast to Tomonaga he only kept an interaction between the right and left movers but not the term ∼ v (kn )(b† bn + b† b−n ) in Eq. (1.14) . In the ˜ n −n limit of a delta interaction of the right and left movers his model is identical to the massless Thirring model (1958) [9] at that time not well known in the solid state physics community. 2.4 Towards the “Luttinger liquid” concept Luttinger’s treatment of the Dirac sea was corrected in a paper by Mattis and Lieb (1965) [10] which also offered a simpler way to calculate nk,α . The time dependent one-particle Green’s function for the spinless Luttinger model was calculated by Theumann (1967) [11] by generalizing this method. She found power law behaviour in the corresponding spectral function ρ(k, ω), especially ρ(kF , ω) ∼ αL |ω|αL −1 , i.e. no sharp quasiparticle for k = kF consistent with Luttinger’s result for the occupation numbers (Fig.1.3). For a delta interaction her results agreed with an earlier calculation for the massless Thirring model by Johnson (1961) [12]. Later the time dependent one-particle Green’s function was calculated by various other methods, e.g. using Ward identities (Dzylaloshinski and Larkin (1974) [13]) as well as the important method of the ”bosonization of the field operator” (Luther and Peschel (1974) [14]) which will be addressed in detail in section 3. It was first proposed in a different context by Schotte and Schotte (1969) [15]. What is now usually called the “Tomonaga-Luttinger (TL) model” is the following generalization of Eq. (1.14) ˜ HT L = 2π¯ h n L n>0 vF + g4 (kn ) 2π¯ h b† bn + b† b−n n −n 9 g2 (kn ) † † bn b−n + b−n bn 2π¯ h + + h ¯π ˜ vN N 2 + vJ J 2 , (1.17) 2L where vN = vF + (g4 (0) + g2 (0))/2π¯ and vJ = vF + (g4 (0) − g2 (0))/2π¯ . h h The interaction parameters g2 (kn ) and g4 (kn ) are allowed to be different. As Tomonaga’s original model the TL model is exactly solvable, i.e. it can also be brought into the form of Eq. (1.15). The eigenvector components in α† = n cn b† − sn b−n are given by n cn = 1 Kn + √ Kn 1 2 , sn = 1 2 Kn − √ 1 Kn (1.18) with Kn = vJ (kn )/vN (kn ), where vJ(N ) (kn ) ≡ vF +[g4 (kn )∓g2 (kn )]/2π¯ . h The frequencies are given by ωn = |kn | vJ (kn )vN (kn ) ≡ |kn |vc (kn ). The TL-Hamiltonian corresponds to a fermionic Hamiltonian that conserves the number of right and left movers. 1 0.5 0 −2 −1 0 1 2 (k−kF)/kc Figure 1.3. The full line shows the average occupation nk,+ for a TL model with αL = 0.6. The dashed line shows the expectation from Fermi liquid theory, where the discontinuity at kF determines the quasi-particle weight ZkF in ρ+ (kF , ω). As discussed following Eq. (1.48) this can also be realized in a TL model with g2 (0) = 0. There also the details of the interaction are specified. A more general model including spin and terms changing right movers into left movers and vice versa is usually called the “g-ology model”. An important step towards the general Luttinger liquid concept came from the renormalization group (RG) study of this model. It was shown that for repulsive interactions (see section 3) the renormalized interactions flow towards a fixed point Hamiltonian of the TL-type unless in lattice models for commensurate electron 10 fillings strong enough interactions (for the half filled Hubbard model discussed in section 4 this happens for arbitrarily small on-site Coulomb interaction U) destroy the metallic state by opening a Mott-Hubbard gap. The RG approach is o described in detail in reviews by S«lyom (1979) [16] and Shankar (1994) [17]. These results as well as insight from models which allow an exact solution by the Bethe ansatz led Haldane [18, 19] to propose the concept of Luttinger liquids (LL) as a replacement of Fermi liquid theory in one dimension, which “fails because of the infrared divergence of certain vertices it assumes to remain finite” [19] . At least for spinless fermions Haldane was able to show that “the Bogoliubov transformation technique that solves the Luttinger model provides a general method for resumming all infrared divergences present”[19]. Similar to Fermi liquid theory in higher dimensions this new LL phenomenology allows to describe the low energy physics in terms of a few constants, two for the spinless case: the “stiffness constant” K ≡ K0 = vJ /vN (also called √ g in various publications) and the “charge velocity” vc = vJ vN . In his seminal paper Haldane showed explicitly that the LL relations survive in not exactly soluble generalizations of the TL model with a non-linear fermion dispersion. He also gave a clear presentation how to calculate general correlation functions and e.g. the occupancies shown in Fig. 1.3 for the TL model. The technical details are addressed in section 3. Before we do this two additional important aspects of LL-behaviour should be mentioned. The first concerns the strong influence of impurities on the low energy physics [20–25], especially the peculiar modification of the electronic properties of a LL when a single impurity with an arbitrarily weak backscattering potential is present. For a spinless LL with a repulsive two-body interaction, i.e. K < 1 a perturbative bosonic RG calculation [25] shows that the backscattering strength VB is a relevant perturbation which grows as ΛK−1 when the flow parameter Λ is sent to zero. This leads to a breakdown of the perturbative analysis in VB . On the other hand a weak hopping between the open −1 ends of two semi-infinite chains is irrelevant and scales to zero as ΛK −1 . Assuming that the open chain presents the only stable fixed point it was argued that at low energy scales even for a weak impurity physical observables behave as if the system is split into two semi-infinite chains. This leads to a conductance which vanishes with a power law in T at low temperatures [25]. A more technical discussion is presented in section 3. Electrons are spin one-half particles and for their description it is necessary to include the spin degree of freedom in the model. For a fixed quantization axis the two spin states are denoted by σ =↑, ↓. The fermionic creation (annihilation) operators c† n,±,σ (cn,±,σ ) carry an additional spin label as well as the ρn,±,σ and the boson operators bn,σ which in a straightforward way ˆ generalize Eq. (1.12). The interactions gν (k) with ν = 2, 4 in Eq. (1.17) σσ′ σσ′ become matrices gν in the spin labels. If they have the form gν (k) = 11 δσ,σ′ gν (k) + δσ,−σ′ gν⊥ (k) it is useful to switch to new boson operators bn,a with a = c, s bn,c ≡ bn,s ≡ 1 √ (bn,↑ + bn,↓ ) 2 1 √ (bn↑ − bn,↓ ) , 2 (1.19) 1. which obey ba,n , ba′ ,n′ = 0 and ba,n , b† ′ ,n′ = δaa′ δnn′ ˆ The kinetic energy a can be expressed in terms of “charge” (c) and “spin” (s) boson operators using b† bn,↑ + b† bn↓ = b† bn,c + b† bn,s . If one defines the interaction matrix n,c n,s n,↑ n↓ elements gν,a (q) via gν,c (q) ≡ gν (q) + gν⊥ (q) gν,s (q) ≡ gν (q) − gν⊥ (q) , (1.20) √ ˜ ˜ ≡ (Nα,↑ ± Nα,↓ )/ 2 one can write the TL-Hamiltonian ˜ and defines Nα,c(s) ˜ (1/2) for spin one-half fermions as H TL ˜ (1/2) ˜ ˜ HT L = HT L,c + HT L,s , (1.21) ˜ where the HT L,a are of the form Eq. (1.17) but the interaction matrix elements have the additional label a. The two terms on the rhs of Eq. (1.21) commute, i.e. the “charge” and “spin” excitation are completely independent. This is usually called “spin-charge separation”. The “diagonalization” of the two separate parts proceeds exactly as before and the low energy excitations are “massless bosons” ωn,a ≈ va |kn | with the charge velocity vc = (vJc vNc )1/2 and the spin velocity vs = (vJs vNs )1/2 . The corresponding two stiffness constants are given by Kc = (vJc /vNc )1/2 and Ks = (vJs /vNs )1/2 . Because of Eq. (1.21) the dependence of the velocities on the interaction strength (1.20) is obtained using the results for the spinless model following Eq. (1.18). The low temperature thermodynamic properties of the TL model including spin, Eqs. (1.17) and (1.21), can be expressed in terms of the four velocities vNc , vJc , vNs , vJs or the four quantities vc , Kc , vs , Ks . Due to spin-charge separation the specific heat has two additive contributions of the same form as in Eqs. (1.1) and (1.2). If we denote, as usual, the proportionality factor in the linear T -term by γ one obtains 1 γ = γ0 2 vF vF + vc vs , (1.22) where γ0 is the value in the noninteracting limit. To calculate the spin suscep˜ ˜ (1/2) tibility χs one adds a term −hNs to HT L . Then by minimizing the ground 12 ˜ state energy with respect to Ns one obtains Ns ∼ h/vNs , i.e. χs is inversely proportional to vNs . If one denotes the spin susceptibility of the noninteracting system by χs,0 , this yields for the zero temperature susceptibility vF vF χs = Ks . = (1.23) χs,0 vNs vs For spin rotational invariant systems one has Ks = 1 [26]. The zero temperature compressibilty κ is proportional to (∂ 2 E0 /∂N 2 )−1 which using Eqs. L (1.17) and (1.21) leads to vF vF κ = Kc . = (1.24) κ0 vNc vc A simple manifestation of spin-charge separation occurs in the time evolution of a localized perturbation of e.g. the the spin-up density. The time evolution αn,a (t) = αn,a e−iωn,a t for a = c, s implies bn,a (t) = bn,a c2 e−iωn,a t − s2 eiωn,a t −b† cn,a sn,a e−iωn,a t − eiωn,a t n,a n,a −n,a (1.25) If the initial state of the system involves a perturbation of right movers only, i.e. bn,a = 0 for n < 0 and the perturbation is sufficiently smooth ( bn,a = 0 for 0 < n ≪ nc only) the initial perturbation is split into four parts which move with velocities ±vc and ±vs without changing the initial shape. If only the initial expectation values of the bn,↑ are different from zero one obtains for δ ρ↑ (x, 0) ≡ F (x) using Eq. (1.12) δ ρ↑ (x, t) = a 1 − Ka 1 + Ka F (x − va t) + F (x + va t) . 4 4 (1.26) For the spin rotational invariant case Ks = 1 there is no contribution which moves to the left with the spin velocity. Already for the pure g4 -model with Kc = 1 but vc = vs “spin-charge separation” of the distribution occurs. For the spinless model with g2 = 0 the initial distribution splits into one right- and one left-moving part, which is often called “charge fractionalization” [27, 28]. Note that the splitting described in Eq. (1.26) is independent of the details of F (x) like the corresponding total charge. An additional comment should be made: spin-charge separation is often described as the fact that when an electron is injected into the system its spin and charge move independently with different velocities. This is very misleading as it is a collective effect of the total system which produces expectation values like in Eq. (1.26). The easiest way to understand the important manifestation of spin-charge separation in the momentum resolved one-particle spectral functions [29, 30] is to make use of the bosonization of the electronic field operators discussed in the next section. 13 3. Luttinger liquids - computational tools In section 2 many of the important features of LL’s like the absence of a discontinuity at the Fermi surface were presented without giving any details how these properties are actually determined. As the most important tool the bosonization of the field operators is presented in detail in this section. This method is then used to calculate correlation functions like the one-particle Green’s function and the 2kF -density response function. In the second part of this section the TL model with additional interactions and (or) a one particle potential with a “backscattering” contribution is discussed. The model is no longer exactly solvable and RG arguments play an important role [16, 17, 25]. 3.1 Bosonization of the field operator In the following a selfcontained presentation of the bosonization of a single fermion operator including a straightforward construction of the particle number changing part (“Klein factor”) is given. We present the bosonization of the field operator for the right movers described by the cl,+ and just mention the corresponding result for the left movers. The starting point are the commutation relations the cl,+ obey for m > 0 1 [bm , cl,+ ] = − √ cl+m,+ m , 1 [b† , cl,+ ] = − √ cl−m,+ m m . (1.27) If (after taking the limit m0 → −∞) one introduces the 2π-periodic auxiliary ˜ field operator ψ+ (v), where v later will be taken as 2πx/L ˜ ψ+ (v) ≡ ∞ eilv cl,+ , (1.28) l=−∞ it obeys the simple commutation relations 1 1 ˜ ˜ ˜ ˜ [bm , ψ+ (v)] = − √ e−imv ψ+ (v) ; [b† , ψ+ (v)] = − √ eimv ψ+ (v) . m m m (1.29) Products of exponentials of operators linear in the boson operators A+ ≡ λn b† n ; n=0 B− ≡ µ n bn (1.30) n=0 with arbitrary constants λn and µn obey similar commutation relations [bm , eA+ eB− ] = λm eA+ eB− † ; [b† , eA+ eB− ] = −µm eA+ eB− , m (1.31) † which follow from [bm , eλbm ] = λeλbm . We therefore make the ansatz † ˜ ˆ ψ+ (v) = O+ (v)eiφ+ (v) eiφ+ (v) , (1.32) 14 where the operator iφ+ (v) is given by [19] ∞ iφ+ (v) = einv √ bn . n n=1 (1.33) ˆ Then the operator O+ (v) commutes with all the bm and b† . We next construct m ˆ+ (v) such that both sides of Eq. (1.32) yield identical matrix elements. O ˜ ˆ As ψ+ (v) reduces the number of right movers by one, the operator O+ (v) ˆ + (v) we work with the also must have this property. In order to determine O eigenstates of the noninteracting system ( the limit m0 → −∞ is implied and nF is an arbitrary positive integer later related to kF ) ˜ ˜ |{ml }b , N+ , N− ≡ l  (b† )ml  √l ml ! ˜ n F + N− n=m0   c† −n,− ˜ n F + N+ r=m0  c†  |Vac r,+ . (1.34) ˆ ˜ ˜ It is easy to see that O+ (v)|{0}b , N+ , N− has no overlap to excited states ˜ ˜ ˆ ˜ ˜ {ml }b , N+ − 1, N− |O+ (v)|{0}b , N+ , N− = ml (bl ) ˆ ˜ ˜ ˜ ˜ √ O+ (v)|{0}b , N+ , N− . (1.35) {0}b , N+ − 1, N− | ml ! l ˆ As O+ (v) commutes with the bl the rhs of Eq. (1.35) vanishes unless all ml are zero. This implies ˆ ˜ ˜ ˜ ˜ ˜ ˜ O+ (v)|{0}b , N+ , N− = c+ (N+ , N− , v)|{0}b , N+ − 1, N− , (1.36) ˜ ˜ ˜ ˜ ˜ {0}b , N+ − 1, N− |cl,+ |{0}b , N+ , N− = (−1)N− δl,nF +N+ . ˜ (1.37) ˜ ˜ ˜ ˜ where c+ (N+ , N− , v) is a c-number. In order to determine c+ (N+ , N− , v) we ˜ ˜ ˜ ˜ ˜ calculate {0}b , N+ − 1, N− |ψ+ (v)|{0}b , N+ , N− using Eq. (1.28) as well as Eq. (1.32). In the calculation of the matrix element with the fermionic form Eq. (1.28) we use Eq. (1.34) which yields ˜ The factor (−1)N− occurs because we have to commute cl,+ through the prod˜ uct of N− = −m0 + 1 + nF + N− fermionic operators of the left movers if we assume −m0 + nF to be odd. We note that no such factor occurs for the corresponding matrix element of the left movers. The calculation of the ˜ ground state to ground state matrix element of ψ+ (v) using Eq. (1.32) is simple as both exponentials involving the boson operators can be replaced by the ˜ ˜ unit operator and the matrix element is just c+ (N+ , N− , v). The comparison therefore yields ˜ ˜ ˜ ˜ c+ (N+ , N− , v) = (−1)N− eiv(nF +N+ ) (1.38) 15 ˜ ˜ ˜ and c− (N+ , N− , v) = e−iv(nF +N− ) . Together with Eq. (1.34) and the defini˜ tion Nα ≡ Nα − (−m0 + 1 + nF )ˆ this implies 1 ˜ ˜ ˆ ˜ ˜ ˜ ˜ O+ (v)e−i(nF +N+ )v (−1)N− |{0}b , N+ , N− = |{0}b , N+ − 1, N− . (1.39) ˜ ˜ ˆ If we apply the operator O+ (v)e−i(nF +N+ )v (−1)N− to the states in Eq. (1.34) ˆ+ (v) commutes with the boson operators we obtain and use again that O ˜ ˜ ˆ ˜ ˜ ˜ ˜ O+ (v)e−i(nF +N+ )v (−1)N− |{ml }b , N+ , N− = |{ml }b , N+ − 1, N− . (1.40) ˜ ˜ ˆ + (v)e−i(nF +N+ )v (−1)N− is independent This shows that the operator U+ ≡ O of v and given by U+ = ˜ ˜ N+ ,N− {ml } ˜ ˜ ˜ ˜ |{ml }b , N+ − 1, N− {ml }b , N+ , N− | . (1.41) † † It follows immediately that U+ is unitary, i.e. U+ U+ = U+ U+ = ˆ From Eq. 1. ˜ (1.41) one can infer that for arbitrary functions f of the number operator N+ ˜+ ) = f (N+ + 1)U+ . ˜ one has U+ f (N To summarize we have shown that ˜ ˜ ˆ O+ (v) = U+ ei(nF +N+ )v (−1)N− . (1.42) ˜ ˜ ˆ ˆ In O− (u) = U− e−i(nF +N− )u no factor (−1)N+ appears and therefore O+ (v) ˆ− (u) anticommute, which is necessary to enforce anticommutation relaand O ˜ ˜ tions between ψ+ (v) and ψ− (u). It is an easy exercise to show that e.g. the an˜ ˜ ticommutation relations [ψ+ (v), ψ+ (u)]+ = 0 are fulfilled. In the calculation ˆ+ (v) as well as the factor in Eq. (1.32) involving the boson the properties of O ˜ ˆ operators enter. If one replaces the operators Oα (v)e−iαv(Nα +nF ) by “Majorana fermions” ηα which commute with the boson operators and obey the anticommutation relations [ηα , ηβ ]+ = 2δαβ ˆ as often done in the literature, 1, ˜ ˜α (v), ψα (u)]+ = [1 − cos (u − v)]eiα(u+v)(Nα +nF ) , i.e. a viola˜ this yields [ψ tion of the correct anticommutation relations. This implies that the Uα have to ˆ be properly treated. In many publications they are written as Uα = eiθα , where ˆ the phase operators θα are assumed to obey the canonical commutation relaˆ ˜α , θα ] = iˆ [19]. We do not use this concept here because no tions (CCR) [N 1 phase operator can be constructed which obeys the CCR as an operator identity [4, 31–34]. In the following we will always use the “normal ordered” form (all boson annihilation operators to the right of the creation operators) of the bosonization formula Eqs. (1.32, 1.33). Alternatively one introduces a convergence factor e−nλ/2 , whith λ → 0 and works with the Hermitian Bose fields Φα (v) ≡ 16 φα (v) + φ† (v) as well as the fields Φ+ ± Φ− . The derivatives of the latter α fields are related to the total current and the deviation of the total charge density from its average value [35]. As we work with an interaction cut-off kc the introduction of λ is not necessary and because of the space limitation this fieldtheoretic formulation is not used here. 3.2 Calculation of correlation functions for the TL model In order to calculate correlation functions of the TL model with nonzero in˜ teractions it is necessary to express the field operator ψ+ (v) Eq. (1.32) in † instead of the b , b† because the former have a simple terms of the αn , αn n n time dependence and for the temperature dependent expectation values one has α† αn = δmn nB (ωn ), where nB (ω) = 1/(eβω − 1) is the Bose funcm tion. For the ground state calculation all one needs is αn |Φ0 = 0 without using the explicit form of the interacting ground state |Φ0 . For periodic boundary conditions one has bm = cm αm + sm α† where −m † the operators αm and α† commute. Therefore eiφ+ (v) (and eiφ+ (v) ) in Eq. −m (1.32) can be written as a product of two exponentials with the annihilation operators to the right. After once using the Baker-Hausdorff formula, eA+B = 1 eA eB e− 2 [A,B] if the operators A and B commute with [A, B], in order to complete the process of normal ordering one obtains for the physical field operator √ ˜ ψα (x) = ψα (2πx/L)/ L for a system of finite length L with periodic boundary conditions [36] A(L) ˆ 2πx iχ† (x) iχα (x) e α e ψα (x) = √ Oα L L (1.43) with iχα (x) = A(L) ≡ e− θ(αm) cm eikm x αm − sm e−ikm x α−m |m| m=0 ∞ s2 /n n=1 n , (1.44) and θ(x) is the unit step function. This is a very useful formula for the calculation of properties of one-dimensional interacting fermions. For the special choice sn = s(0)e−n/nc where nc = kc L/2π is determined by the interaction cut-off, A(L) can be calculated analytically using ∞ z n /n = − log (1 − z). For nc ≫ 1 this yields A(L) = n=1 2 (4π/kc L)s (0) which shows that the prefactor in Eq.(1.44) has an anomalous 1 2 power law proportional to (1/L) 2 +s (0) . This implies that the cn,α scale like 2 (1/L)s (0) . The time dependent operator ψ+ (x, t) follows from Eq. (1.44) by replacing ˆ αm and α−m by αm e−iωm t and α−m e−iωm t and U+ in O+ by U+ (t). Various 17 kinds of time dependent correlation functions can quite simply be calculated † using this result. Here we begin with iG< (x, t) ≡ ψ+ (0, 0)ψ+ (x, t) . + As U+ commutes with the bosonic operator the particle number changing operators lead to a simple time dependent factor ˜ ˜ ˜ ˜ † ˜ ˜ ˜ ˜ U+ U+ (t)|Φ0 (N+ , N− ) = e−i[E0 (N+ ,N− )−E0 (N+ −1,N− )]t |Φ0 (N+ , N− ) . (1.45) As ψ+ (x) in Eq.(1.43) is normal ordered in the α’s one has to use the Baker† Hausdorff formula only once to normal order ψ+ (0, 0)ψ+ (x, t). This yields with kF = 2πnF /L A2 (L) ikF x [χ+ (0,0),χ† (x,t)] + e e (1.46) L ∞ 1 −i(kn x−ωn t) eikF x +2s2 (cos (kn x)eiωn t −1)] n = e n=1 n [e L ˜ ˜ ˜ ˜ where µ ≡ E0 (N+ , N− ) − E0 (N+ − 1, N− ) is the chemical potential. The analytical evaluation of the sum (integral in the limit L → ∞) in the exponent in Eq. (1.46) is not possible. An approximation which gives the correct large x and t behaviour [37] is to replace ωn by vc kn . This yields for L → ∞ with the exponential cut-off for the sn used earlier [14] ieiµt G< (x, t) = + iµt ie G< (x, t) + eikF x r2 −i = 2π x − vc t − i0 (x − vc t − ir)(x + vc t + ir) s2 (0) , (1.47) † where r = 2/kc . As ψ+ (0, 0)ψ+ (x, 0) for large x decays proportional to 2 (1/x)1+2s (0) the anomalous dimension for the spinless model is given by αL = 2s2 (0) = (K − 1)2 /2K. (1.48) Luttinger’s result for nk,+ follows by performing the Fourier transform with respect to x. The full line in Fig.1.3 was calculated with s2 = 0.3e−2kn /kc , n while the dashed curve corresponds to s2 = 0.6(kn /kc )e−2kn /kc . The latter n example corresponds to an interaction with g2 (k → 0) → 0 which leads to a vanishing anomalous dimension αL . In this case the occupancies nk,+ have a discontinuity at kF as in a Fermi liquid [38]. An efficient numerical algorithm to calculate nk,+ for arbitrary forms of s2 is described in the appendix n of reference [2]. The spectral function ρ< (k, ω) relevant for describing angular resolved photoemission is obtained from Eq. (1.47) by a double Fourier transform ρ< (k, ω) = + = ˜ ˜ c† δ[ω + (H − E0 (N+ − 1, N− ))]ck,+ k,+ ∞ 1 ∞ dteiωt dxe−ikx ieiµt G< (x, t). + 2π −∞ −∞ (1.49) 18 As Eq. (1.47) is reliable in the large x and t limit its use in Eq. (1.49) correctly describes the spectral function for k ≈ kF and ω ≈ 0 [39]. Using the variable substitutions u∓ = x∓vc t the double integral factorizes and with the additional approximation i0 → ir on the rhs of Eq. (1.47) one obtains [11, 14] ˜ ˜ ˜ ρ< (kF + k, ω) ∼ θ(−ω−vc|k|)(−ω+ kvc ) + αL −1 2 ˜ (−ω− kvc ) αL 2 erω/vc . (1.50) Without the additional approximation there is an additional weak dependence ˜ on ω + kvc [29]. The complete spectral function ρ+ (k, ω) = ρ< (k, ω) + + † ρ> (k, ω), where ρ> (k, ω) is defined via iG> (x, t) ≡ ψ+ (x, t)ψ+ (0, 0) can + + + ˜ ˜ be obtained using ρ> (kF + k, ω) = ρ< (kF − k, −ω) which follows from the + + particle-hole symmetry of the model. The absence of a sharp quasi-particle peak is manifest from ρ+ (kF , ω) ∼ αL |ω|αL −1 e−r|ω|/vc . In order to calculate correlation functions of the spin one-half TL model the operators bn,σ which appear in the generalization of the bosonization formula Eqs. (1.32) and (1.33) have to be replaced by the spin and charge bosons √ bn,σ = (bn,c + σbn,s )/ 2. Because of the exponential occurence of the boson operators in Eq. (1.32) and spin-charge separation Eq.(1.21) the Green’s function G< (x, t) factorizes into a spin and a charge part, which both are +,σ of the form as the square root of the function on the rhs of Eq. (1.47) . This √ square root results from the factors 1/ 2 in the expression for the bn,σ . In the spin factor the charge velocity vc is replaced by the spin velocity vs . For the average occupation numbers one again obtains Luttinger’s result Eq. (1.16) with αL = s2 (0) + s2 (0) ≡ αc + αs . The individual contributions can be c s expressed in terms of the Ka ≡ (vJ,a /vN,a )1/2 as αa = (Ka − 1)2 /(4Ka ). (†) As in the spinless model the fermionic (creation) annihilation operators cn,α,σ αL /2 . For spin rotational invariant systems one has K = 1, scale like (1/L) s i.e. no contribution to the anomalous dimension αL from the spin part of the Hamiltonian [26]. For the momentum integrated spectral functions one obtains ρα,σ (ω) ∼ |ω|αL as in the spinless model [40]. The k-resolved spectral functions ρα,σ (k, ω) on the other hand show a drastic difference to the model without spin. The delta peaks of the noninteracting model are broadened into one power law threshold Eq. (1.47) in the model without spin and two power law singularities (see Fig. 1.4) in the model including spin [29, 30, 37] (for αL < 1/2 in the case of a spin independent interaction). The “peaks” disperse linearly with k − kF . It is also straightforward to calculate various response functions for the TL model. We discuss the density response function R(q, z) ≡ − ρq ; ρ−q z /L ˆ ˆ of the spinless model for q ≈ 0 and q ≈ ±2kF , where ˆ ˆ A; B z ≡− i h ¯ ∞ 0 [A(t), B] eizt dt (1.51) 19 ρ+,σ(kF−kc/10,ω)kcvc 15 10 5 0 −0.4 −0.2 0 ω/kcvF 0.2 0.4 ˜ Figure 1.4. Spectral function ρ+,σ (kF + k, ω) as a function of normalized frequency for ˜ k = −kc /10 for the TL- model with a spin independent interaction. The parameters are chosen such that vc = 2vF and αL =1/8. involves the retarded commutator [41] and z is a frequency above the real axis. From the decomposition [42] ψ(x) ≈ ψ+ (x) + ψ− (x) of the field operator ψ(x) in the original Tomonaga model it is obvious that the operator ρ(x) = ˆ † (x)ψ(x) of the density (see Eq. (1.8) ) has two very different contributions ψ † ρ(x) ≈ ρ+ (x) + ρ− (x) + ψ+ (x)ψ− (x) + h.c. ˆ ˆ ˆ (1.52) ≡ ρ0 (x) + ρ2kF (x). ˆ ˆ The spatial Fourier transform of ρ0 is linear in the boson operators Eq. (1.12) ˆ and the q ≈ 0 contribution of the density response function [R(q, z)]0 defined with the operators (ˆ0 )q follows using the (linear) equations of motion for the ρ bn (t) as q 2 vJ (q) 1 (1.53) [R(q, z)]0 = π¯ [qvc (q)]2 − z 2 h This exact result for the q ≈ 0 contribution agrees with the RPA result for the original Tomonaga model. This fact, not mentioned in Tomonaga’s paper [5] as the RPA paper by Bohm and Pines [43] was not yet published, was “discovered” many times in the literature. For the spin 1/2-model [R(q, z)]0 has an additional factor 2 and one has to replace vJ by vJc . The real part of the (q = 0) frequency dependent conductivity σ(ω + i0) follows from [R(q, ω + i0)]0 by multiplication with ie2 ω/q 2 and taking the limit q → 0. This yields for the spinless model (¯ /e2 )Reσ(ω + i0) = vJ δ(ω) = Kvc δ(ω) h (1.54) 20 For the Galilei invariant Tomonaga model Eq. (1.14) one has vJ = vF , i.e. the weight D of the zero frequency “Drude peak” is independent of the interaction, as expected. As D apart from a constant is given by the second derivative of E0 (Φ)/L with respect to the magnetic flux through the 1d ring [44], K (or Kc ) can be obtained from a ground state calculation for microscopic lattice models using K(c) = (Dκ/D0 κ0 )1/2 , where κ is the compressibility discussed in Eq. (1.24). The anomalous decay of the correlation functions for these models, which are more difficult to calculate directly, can then be quantitatively predicted if Haldane’s LL concept is taken for granted. For a weak two-body interaction the result for K(c) − 1 linear in the interaction follows from first order perturbation theory for the ground-state energy, which involves the (nonselfconsistent) Hartree and Fock terms. As they are independent of the magnetic flux, D/D0 has no term linear in v , i.e. Kc ≈ (κ/κ0 )1/2 = (vF /vNc )1/2 , ˜ which holds exactly for Galilei invariant continuum models [45]. Performing (1) the second derivative of E0 (N ) with respect to N yields [46] Kc = 1 − 2˜(0) − v (2kF ) v ˜ + O(˜2 ). v 2π¯ vF h (1.55) In the spinless case the factor 2 in front of v (0) is missing in the result for ˜ K. Instead of D as the second input besides κ one can obtain vc directly by calculating the lowest charge excitation energy (see section 4). The easiest way to calculate the q ≈ ±2kF contribution to the density response is to use the bosonization of the field operators [14]. The first step is to † † normal order ψ+ (x)ψ− (x) using Eq. (1.43) . This gives a factor e[χ+ (x),χ− (x)] which using [χ+ (x), χ† (x)] = −2 m>0 cm sm /m together with the factor − A2 (L) leads to † ψ+ (x)ψ− (x) = a0 L 4π kc L K−1 ˆ ˆ † 2πx )O− ( 2πx )e−i∆χ† (x) e−i∆χ(x) (1.56) O+ ( L L with ∆χ(x) ≡ χ+ (x) − χ− (x) = −i m>0 Km ikm x e αm − e−ikm x α−m . m Here a0 is a dimensionless constant of order unity and the exponent K − 1 of the second factor on the rhs follows using 2s2 + 2cm sm = Km − 1. The m importance of this factor for impurity scattering in Luttinger liquids was first pointed out by Mattis (1974) [21] and will be discussed later. The calculation † † of the two terms of the commutator [ψ+ (x, t)ψ− (x, t), ψ− (0, 0)ψ+ (0, 0)] is then straightforward and one obtains for the spectral function of the q ≈ ±2kF 21 response function the power law behaviour [14] 2 Im[R(±2kF +Q, ω)]2kF ∼ sign(ω)θ(ω 2 −vc Q2 ) 2 ω 2 − vc Q2 2 k2 vc c K−1 (1.57) The static ±2kF + Q response diverges proportional to |Q|2(K−1) which has to be contrasted with the logarithmic singularity in the noninteracting case. In the model including spin the exponent 2K − 2 is replaced by Kc + Ks − 2. ˆ The pair propagator P (q, ω) resulting from the response function for A = † † ˆ = ψ− (0)ψ+ (0) was found by Luther and Peschel to be ψ+ (x)ψ− (x) and B the same as the 2kF -density response, provided the sign of the interaction is reversed [14]. An attractive interaction leads to a power law divergence in P (q = 0, ω = 0) as the temperature is lowered, indicative of large pairing fluctuations. 3.3 The TL model with additional interactions and perturbations The exact solution of the TL model essentially depends on the fact that the numbers of right and left movers are conserved. This symmetry can be destroyed by a one-particle potential with ±2kF -Fourier components or by interaction terms which change the individual particle numbers, like 2kF -“backscattering” terms or Umklapp-terms for a half-filled band. With such additional terms the model is in general no longer exactly solvable. Important insights about the influence of such terms have come from a (perturbational) RG analysis [16, 25]. 3.3.1 Impurity in a spinless TL model We begin with the spinless model with an additional impurity which is described by ˆ VI = ˆ ˆ [VF (x)ˆ0 (x) + VB (x)ˆ2kF (x)] dx ≡ VF + VB , ρ ρ (1.58) ˆ ˆ where VF describes the forward and VB the backward scattering due to the impurity and the two different operators for the densities are defined in Eq. (1.52) . As the forward scattering term is linear in the boson operators it can be treated ˆ ˜ in an exact way. The backscattering term has the property [VB , Nα ] = 0 and the model can no longer be solved exactly (except for K = 1/2 and a special assumption about VB , as discussed below). For a zero range impurity it ˆ ˜ follows directly from Eq. (1.56) that VB scales as (1/L)K while HT L in Eq. ˆB depends crucially on the (1.17) scales as 1/L. Therefore the influence of V 22 sign of the two-body interaction [21, 22]. For repulsive interactions one has ˆ K < 1 which shows that VB is a relevant perturbation. For K > 1, i.e. an ˆ attractive interaction, VB is irrelevant. A detailed RG analysis of the problem was presented in a seminal paper by Kane and Fisher [25]. For a zero range backscattering potential and two-body interaction they mapped the problem to a local bosonic sine-Gordon model [25, 35, 47]. The subsequent RG analysis shows that the backscattering amplitude scales as ΛK−1 when the flow parameter Λ is sent to zero [48], as can be anticipated from Eq. (1.56) . This leads to the breakdown of the perturbational analysis in VB for repulsive interactions. As already mentioned in section 2 this analysis was supplemented by a RG analysis of a weak hopping between two semi-infinite chains. The weak hopping scales to zero like ΛαB for repulsive interactions, where αB = K −1 − 1 is the boundary exponent. It describes e.g. the different scaling ρ(x, ω) ∼ |ω|αB of the local spectral function near a hard wall boundary of a LL [25, 49, 50]. These scaling results together with the asumption mentioned in section 2 leads to the “split chain scenario” in which even for a weak impurity the observables at low energies behave as if the system is split into two chains with fixed boundaries at the ends. Within the bosonic field theory this assumption was verified by quantum Monte Carlo calculations [51] and the thermodynamic Bethe ansatz [52]. This implies e.g. for the local density of states ρ(x, ω) ∼ |ω|αB for small |ω| and x near the impurity like in a LL near a hard wall. The transmission through the impurity vanishes near kF proportional to ∼ |k − kF |2αB which leads to a conductance G(T ) which vanishes with temperature T in power law fashion G(T ) ∼ T 2αB [25]. Additional insight comes from the analysis for the special value K = 1/2 [25, 35, 32] . For VB (x) = VB δ(x) the expression for ∆χ(0) in √ (1.56) can Eq. be written in terms of new boson operators αm ≡ (αm − α−m )/ 2. If one ne˜ glects the momentum dependence of√ m in Eq. (1.56) and puts Km = 1/2 K ˜ one obtains i∆χ(0) = m≥1 αm / m as in the bosonization of a single field operator Eqs. (1.32) and (1.33) . It is then possible to refermionize the K = 1/2-TL model with a zero range impurity. Even the Klein factors can properly be handled [32] and one obtains a model of “shifted noninteracting Fermi oscillators” which can be solved exactly introducing an auxiliary Majorana fermion [35, 32]. Unfortunately the local densities of states cannot be calculated exactly because of the complicated nonlinear relationship between the original fermion operators and the fermion operators which diagonalize the shifted Fermi oscillator problem [32]. Additional results for the transport through a spinless LL containing one impurity were obtained by mapping the problem onto the boundary sine-Gordon model and using its integrability [53]. In order to bridge the two regimes treated by Kane and Fisher one can use a fermionic RG description bearing in mind that it is perturbational in the two- 23 body interaction [54, 55]. It shows that the long range oscillatory effective impurity potential is responsible for the “splitting”, for site impurities as well as for hopping impurities of arbitrary strength. For realistic parameters very large systems are needed to reach the asymptotic open chain regime [55]. Hence only special mesoscopic systems, such as very long carbon nanotubes, are suitable for experimentally observing the impurity induced open boundary physics. For a discussion of the impurity problem in the TL model including spin see also reference [56]. 3.3.2 The TL- model with additional two-body interactions Tomonaga was well aware of the limitations of his approach for more generic two-body interactions (“In the case of force of too short range this method fails”[5]). We therefore first discuss Tomonaga’s continuum model in this short range limit kc ≫ kF opposite to the one considered in section 2. Then low energy scattering processes with momentum transfer ≈ ±2kF are possible and have to be included in the theoretical description of the low energy physics. In the “g-ology” approach one linearizes the nonrelativistic dispersion around the two Fermi points and goes over to right- and left-movers as in section 2. Then the “2kF ”-processes are described by the additional interaction term (1) † † g1 δσ,σ′ + g1⊥ δσ,−σ′ ψ+,σ (x)ψ−,σ′ (x)ψ+,σ′ (x)ψ−,σ (x)dx. Hint = σ,σ′ (1.59) For a spin-independent two particle interaction one has g1 = g1⊥ = g1 . For the zero range interaction assumed in Eq. (1.59) one has to introduce band cutoffs to regularize the interaction term. The RG flow equations for the cut-off dependent interactions on the one-loop level are quite simple [16]. If s runs from zero to infinity in the process of integrating out degrees of freedom one obtains for spin-independent interactions dg1 (s) ds dg2 (s) ds 1 g2 (s) π¯ vF 1 h 1 = − g2 (s) 2π¯ vF 1 h = − (1.60) and g4 is not renormalized. Obviously g1 (s) can be obtained from the first equation only g1 (1.61) g1 (s) = g1 , 1 + s π¯ vF h where g1 is the starting value. It is easy to see that g1 (s) − 2g2 (s) = g1 − 2g2 holds by subtracting twice the second equation from the first in Eq. (1.60) . 24 ∗ In the following we use the notation gν ≡ gν (s → ∞). Now one has to distinguish two cases: ∗ ∗ for g1 ≥ 0 one renormalizes to the fixed line g1 = 0, g2 = g2 − g1 /2 and the fixed point Hamiltonian is a TL model which shows the generic importance of the TL model for repulsive interactions. In this case the g1 -interaction is called marginally irrelevant. For the nonrelativistic continuum model with a ∗ ∗ v ˜ spin independent interaction one has g2c = 2˜(0) − v (2kF ) and g2s = 0 and ∗ − g ∗ )/(2πv + g ∗ + g ∗ )]1/2 ≈ for the stiffness constant Kc = [(2πvF + g4c F 2c 4c 2c 1 − [2˜(0) − v (2kF )]/(2π¯ vF ) and Ks = 1. Due to the approximations made, v ˜ h also here only the result for Kc − 1 linear in v is reliable. The agreement ˜ with the direct calculation Eq. (1.55) shows explicitly to leading order in the interaction that Haldane’s Luttinger liquid concept is consistent. For g1 < 0 the solution (1.61) shows that g1 (s) diverges at a finite value of s. Long before reaching this point the perturbational analysis breaks down and all one can say is that the flow is towards strong coupling. In this case the g1 - interaction is called marginally relevant. In order to to obtain an understanding of the strong coupling regime it is useful to bosonize the additional (1) interaction Hint in Eq. (1.59) [57]. The term proportional to g1 is of the form of a g2 -interaction and therefore bilinear in the boson operators Eq. (1.19) . For the g1⊥ -term one uses the bosonization of the field operators Eqs. (1.32) and (1.33) with additional spin labels. As the g1⊥ -term contains field opera† tors ψα↑ (x)ψα↓ (x) of opposite spin it only involves “spin bosons” Eq. (1.19) , which implies “spin-charge separation” also for this model [58]. The charge part stays trivial with massless charge bosons as the elementary interactions. Luther and Emery showed that for a particular value of g1 the g1⊥ -term can be written as a product of spinless fermion field operators and the exact solution for the spin part of the Hamiltonian is possible using refermionization [57], discussed earlier in connection with the backscattering impurity. The diagonalization of the resulting problem of noninteracting fermions is simple and shows that the spectrum for the spin excitations is gapped. It is generally believed that these properties of Luther-Emery phases are not restricted to the solvable parameter values. Strong coupling phenomena which lead to deviations from LL-properties with gapped phases are discussed in detail in section 4 for lattice models. There in case of commensurate filling Umklapp processes can become important, e.g. for half filling where two left movers from the vicinity of the left Fermi point are scattered into two right movers near the right Fermi point or vice versa. As G = 4kF is a reciprocal lattice vector such a scattering process is a low energy process conserving quasi-momentum. In the g-ology model such processes are 25 described by an additional term (3) Hint = 1 2 σ,σ′ ′ † † σ,σ g3 (x − y) ψ+,σ (x)ψ+,σ′ (y)ψ−,σ′ (y)ψ−,σ (x)e2ikF (x+y) +H.c.] dxdy (1.62) Umklapp processes for σ = σ ′ are only possible for nonzero interaction range. 4. Results for integrable lattice models As mentioned in subsection 2.4, results for integrable models which can be solved exactly by the Bethe ansatz played a central role in the emergence of the general “Luttinger liquid” concept [18]. It is therefore appropriate to shortly present results for the two most important lattice models of this type, the model of spinless fermions with nearest neighbour interaction and the 1dHubbard model. (We put ¯ = 1 in this section.) h 4.1 Spinless fermions with nearest neighbour interaction The one-dimensional single band lattice model of spinless fermions with nearest neighbour hopping matrix element t(> 0), and nearest neighbour interaction U (often called V in the literature) is given by H = −t c† cj+1 + H.c. + U j j j ˆ ˆ nj nj+1 ≡ T + U , ˆ ˆ (1.63) where j denotes the sites and the nj = c† cj are the local occupation number ˆ j operators. In the noninteracting limit U = 0 one obtains for lattice constant a = 1 the well known dispersion ǫk = −2t cos k. For the following discussion of the interacting model (U = 0) we mainly focus on the half filled band case kF = π/2 with vF = 2t. In contrast to the (continuum) Tomonaga model Umklapp terms appear when the interaction term in Eq. (1.63) is written in the k-representation. As discussed below they are irrelevant at the noninteracting (U = 0) fixed point [17]. Therefore the system is a Luttinger liquid for small enough values of |U |. The large U limit of the model is easy to understand: For U ≫ t charge density wave (CDW) order develops in which only every other site is occupied thereby avoiding the “Coulomb penalty”. For large but negative U the fermions want to be as close as possible and phase separation occurs. For the quantitative analysis it is useful that the model in Eq. (1.63) can be exactly mapped to a S = 1/2-Heisenberg chain with uniaxially anisotropic nearest neighbour exchange (“XXZ” model) in a magnetic field by use of the Jordan-Wigner transformation [59]. For U > 0 this model is also called the antiferromagnetic Heisenberg-Ising model. The point U ≡ Uc = 2t corresponds to the isotropic Heisenberg model. For U > 2t the Ising term dominates and 26 the ground state is a well defined doublet separated by a gap from the continuum and long range antiferromagnetic order exists. For −2t < U ≤ 2t there is no long range magnetic order and the spin-excitation spectrum is a gapless continuum. The mapping to the XXZ-model therefore suggests that the spinlesss fermion model Eq. (1.63) in the half filled band case is a Luttinger liquid for |U | < 2t. Before we present the exact results for the Luttinger liquid parameters K and vc from the Bethe ansatz solution [18, 60], we shortly discuss the RG approach to the model. A perturbative RG calculation around the free fermion fixed point is discussed in detail in Shankar’s review article [17]. The first step ˆ is to write the four fermion matrix elements of the interaction U in Eq. (1.63) in the k-representation. This yields for a chain of N sites with periodic boundary condition and values kj = 2πj/N in the first Brillouin zone ˆ k1 , k2 |U |k3 , k4 = 2U cos(k1 − k3 ) δk1 +k2 ,k3 +k4 +2πm N m=0,±1 (1.64) The m = 0 term on the rhs of Eq. (1.64) represents the direct scattering terms and the m = ±1 terms the Umklapp processes. The matrix element antisymmtrized in k3 and k4 is proportional to sin [(k1 − k2 )/2] sin [(k3 − k4 )/2]. Therefore the low energy Umklapp Hamiltonian scales like (1/L)3 which shows that it is strongly irrelevant at the free field fixed point [17]. This analysis confirms the Luttinger liquid behaviour for small values of U , but gives no hint about the critical value Uc for the CDW transition. With the separation ˆ ˆ ˆ U ≡ U0 + UUmklapp implied by Eq. (1.64) one can do better by first treating ˆ + U0 by bosonization and then perform the RG analysis around the correˆ T sponding TL fixed point to get information for which value of U the Umklapp term starts to be a relevant perturbation. For this analysis it is easier to work directly with the unsymmetrized matrix elements in Eq. (1.64). As k1 − k3 ≈ ±π for the low energy Umklapp processes this leads after extending the (linearized) dispersion of the right and left movers from −∞ to ∞ to a g3 -interaction with a range of order r = a. The scaling dimension of the cor(3) responding Hint follows using bosonic normal ordering as in Eq. (1.56). For x − y of order r or smaller one obtains † † ψ+ (x)ψ+ (y)ψ− (y)ψ− (x)L2 ∼ x−y L 2 r L 4(K−1) † 2 ×(U+ )2 U− e2ikF (x+y) eiB (1.65) † (x,y) eiB(x,y) , where B(x, y) = χ− (x) + χ− (y) − χ+ (x) − χ+ (y) with χα (x) defined in Eq. (1.44). The first factor on the rhs is due to the Pauli principle and describes the same physics as the two sin-factors mentioned above for small arguments. 27 Therefore the second factor has to provide more than two powers of L to make the Umklapp term a relevant perturbation, which happens for K < 1/2. As discussed below, the exact Bethe ansatz result for K yields Uc = 2t. If one uses the simple linear approximation for K − 1 in Eq. (1.55) one obtains with lin Eq. (1.64) K lin = 1 − U/(πt) for the critical value Uc /t = π/2, not too bad an approximation. Exact analytical results for the Luttinger liquid parameters for the half filled model can be obtained from the Bethe Ansatz solution [18, 60, 61]. It is not necessarary to address the anomalous decay of the correlation functions directly, but one can use a ground state property and the lowest charge excitation to extract the parameters, as was dicussed in connection with Eq. (1.55). This yields for the stiffness constant K = π/[2 arccos (−U/2t)] and for the charge velocity vc = πt 1 − (U/2t)2 /[π − arccos (−U/2t)]. For repulsive interactions U > 0 the value of K decreases monotonously from the noninteracting value K = 1 to K = 1/2 for U = 2t, which corresponds to an anomalous dimension αL = (K +1/K)/2−1 = 1/4. For attractive interactions K diverges when U approaches −2t, and the charge velocity vc goes to zero. Results for the Luttinger liquid parameter K for less than half filled bands are shown in Fig. 1.5 [62]. The limit a → 0 and n → 0 corresponds to the continuum limit. 1 K 0.75 U=0.5 U=1.0 U=1.5 U=∞ 0.5 0.25 0 0.1 0.2 0.3 0.4 0.5 n Figure 1.5. Luttinger liquid parameter K from the Bethe ansatz solution as a function of the band filling n for different values of U (t = 1). The short dashed curve shows the infinite U result (1/2 + |n − 1/2|)2 . As the interaction goes over to a contact interaction its effect vanishes because of the Pauli principle and K goes to 1. For small enough values of U the linear approximation Eq. (1.55) K lin = 1 − U sin (nπ)/πt provides a good approximation for all values of n, in contrast to the Hubbard model discussed below. 28 In the infinite U limit the Bethe Ansatz equations simplify considerably and the ground-state energy as well as low lying excited states can be calculated analytically [61]. With these results it is easy to show that K = (1 − n)2 holds for 0 < n < 1/2, i.e. K = 1/4, is the lower bound for K in the LL regime of the model [18]. The corresponding upper bound of the anomalous dimension is αL = 9/8. In order to achieve larger values of αL the model in Eq. (1.63) has to be generalized to include longer range interactions [63]. 4.2 The Hubbard model As there exists an excellent review on the LL behaviour in the 1d-Hubbard model [64], the following discussion will be rather short. As the model includes spin the on-site interaction between electrons of opposite spins is not forbidden by the Pauli principle. This is taken as the only interaction in the model. The 1d Hubbard Hamiltonian reads H = −t c† cj+1,σ + H.c. + U j,σ j,σ nj,↑ nj,↓ . ˆ ˆ (1.66) j ˆ ˆ In the extended Hubbard model a next nearest interaction term V j nj nj+1 with nj ≡ nj,↑ + nj,↓ is added [65]. In order to show the important difference ˆ ˆ ˆ to the spinless model Eq. (1.63) we again first discuss the half-filled band case, which is metallic for U = 0. For U ≫ t the “Coulomb penalty” is avoided when each site is singly occpied. Then only the spin degrees of freedom matter. In this limit the Hubbard model can be mapped to a spin-1/2 Heisenberg antiferromagnet with an exchange coupling J = 4t2 /U . In the charge sector there is a large gap ∆c ∼ U while the spin excitations are gapless. The 1d Hubbard model can also be solved exactly using the Bethe ansatz [66] and properties like the charge gap or the ground-state energy can be obtained by solving Lieb and Wu’s integral equation. In contrast to the spinless model described in the previous subsection the charge gap in the Hubbard model is finite for all U > 0. While for U ≫ t it is asymptotically given by U it is exponentially small, ∆c ≈ (8t/π) U/t exp (−2πt/U ), for 0 < U ≪ t. This shows that the Umklapp term is no longer irrelevant at the free field fixed point. The Pauli principle factor of Eq. (1.65) is missing as the interaction is between electrons of opposite spin. The Umklapp term is therefore a marginal perturbation. The RG analysis [16] shows that the Umklapp term is marginally relevant while the 2kF -backscattering (“g1 ”) interaction is marginally irrelevant for U > 0 as discussed following Eq. (1.60). When the band is not half filled Umklapp is not a low energy process and the Hubbard model is a Luttinger liquid with Ks = 1. The LL parameters Kc and va can be obtained by (numerically) solving Lieb an Wu’s integral equation [67]. Even for 0 < U ≪ t the perturbative result Eq. (1.55) works well only for intermediate filling n ≡ Nel /N ≈ 0.5, where Nel is the number of electrons 29 (half filling corresponds to n = 1) . In the limit n → 0 the Fermi velocity vF = 2t sin (πn/2) goes to zero but 2˜(0) − v (2kF ) = U stays finite and the v ˜ correction term increases with decreasing n in contrast to the spinless model. The Bethe ansatz results show that Kc → 1/2 for n → 0 as well as n → 1 for all U > 0. For U → ∞ it leads to Kc → 1/2 for all fillings n different from 1. In this limit the velocities are given by vc = 2t sin (πn) and vs = (2πt2 /U )[1 − sin (2πn)/(2πn)], i.e. the spin velocity goes to zero [64, 67]. The U = ∞ results for vc and Kc can be understood without the Bethe ansatz solution. Double occupancies of the lattice sites are forbidden and the system behaves like a system of noninteracting spinless fermions with kF replaced by 2kF [64]. The spin degrees of freedom play no role and any configuration of the spins gives an eigenfunction of the same energy. This immediately explains the result for vc mentioned above. For a TL model with spin one obtains (for fixed N↑ − N↓ ) from Eqs. (1.17) and (1.21) L(∂ 2 E0 /∂N 2 )L = πvNc /2, while the factor 1/2 is missing in the spinless case. The formula for the spinless case can be used to calculate L(∂ 2 E0 /∂N 2 )L for U = ∞ with vN replaced by vF (2kF ), using the spinless fermion analogy. This yields vNc = 2vc i.e. Kc = 1/2. As the calculation of correlation functions not only requires excitation energies but also many electron matrix elements which are difficult to evaluate using the Bethe ansatz, various numerical methods have been used to study e.g. the manifestation of spin-charge separation in the one-particle spectral function [68, 69]. The Bethe ansatz approach simplifies in the infinite U limit [70]. After earlier work [71, 72] the frequency dependent optical conductivity of the 1d Hubbard model was also studied using Bethe ansatz methods [73, 74], as well as the dynamical density-matrix renormalization group [74]. 5. Weakly coupled chains: the Luttinger to Fermi liquid transition Strictly one-dimensional systems are a theoretical idealization. Apart from this even the coupling to an experimental probe presents a nontrivial disturbance of a Luttinger liquid. Unfortunately the weak coupling of a 1d system to such a probe as well as the coupling between several LLs is theoretically not completely understood [26]. The coupling between the chains in a very anisotropic 3d compound generally, at low enough temperatures, leads to true long-range order. The order develops in the phase for which the algebraic decay of the correponding correlation function of the single chain LL is most slowly [64]. This can lead e.g. to charge-density wave (CDW), spin-density wave (SDW) order or superconductivity. In the following we shortly address some important issues of the coupled chain problem, which are a prerequisite for the theoretical descriptions of the attempts to experimentally verify LL behaviour. In the first part of this section 30 theoretical aspects of the problem of an infinite number of coupled chains are addressed. This is followed by a short discussion of the (approximate) experimental realizations of LLs. As there are other chapters in this book addressing this question the discussion will be rather short. 5.1 Theoretical models We consider a systems of N⊥ coupled chains described by the Hamiltonian N⊥ (ee) Hij Hi + H= i=1 i=j t⊥,ij c† n,(σ),i , cn,(σ),j + (1.67) n,(σ) i,j (ee) where the Hi are the Hamiltonians of the individual chains, the Hij represent the two-body (Coulomb) interaction of electrons on different chains and the last term H (t⊥ ) describes the hopping between the chains with t⊥,ij the (†) transverse hopping matrix elements and the cn,(σ),i the (creation) annihilation operators of one-particle states with quasi-momentum kn along the chain i and spin σ (if spin is included in the model). The individual Hi can be TLHamiltonians Eq. (1.17) or lattice Hamiltonians like in Eqs. (1.63) or (1.66). We address the question if LL physics survives in such a model. The second and the third term on the rhs of Eq. (1.67) describe different types of couplings between the chains. If the transverse hopping is neglected (t⊥ ≡ 0) the model can be solved exactly for special assumptions about the two-body interaction and the Hi . If the individual chains are described by TL-Hamiltonians (ee) Eq. (1.17) and the interaction Hij can be expressed in terms of the densities ρn,(a),α,i the exact solution is possible by bosonization [75, 76]. This ˆ is important when the long range Coulomb interaction is taken into account. For a single chain the corresponding one-dimensional Fourier transform v (q) ˜ (which has to be regularized at short distances) has a logarithmic singularity for q → 0. This leads to K(c) = 0 and the divergence of the anomalous dimension, i.e. the system is not a LL. The 4kF harmonic of the density-density correlation function shows a very slow decay almost like in a Wigner crystal [77]. The Coulomb coupling between the chains removes this singularity and a three-dimensional extended system of coupled chains is a LL [75]. The corresponding anomalous dimension can be calculated and leads to values of order unity for realistic values of the coupling constant e2 /(π¯ vF ) [76]. If 2kF h scattering terms of the interaction are kept the model can no longer be solved exactly and a more complicated scenario emerges in the parquet approximation [78]. The inclusion of the transverse hopping presents a difficult problem even if the inter-chain two-body interactions are neglected. This is related to the fact that the transverse hopping is a relevant perturbation for αL < 1[79–81]. This 31 can easily be seen if the individual chains are described by TL-Hamiltonions (†) Eq. (1.17), scaling like 1/L. As discussed in section 3 the cn,(σ),i scale like (1/L)αL /2 . As H (t⊥ ) involves products of creation and annihilation operators on different chains no further boson normal ordering is necessary and H (t⊥ ) scales as (1/L)αL . This suggests “confinement” for αL > 1: if an extra electron is put on the j-th chain it stays there with probability close to 1 even in the long time limit. This conclusion can be questioned as RG calculations perturbative in t⊥ demonstrate that the hopping term generates new and relevant interchain two-particle hoppings. These calculations show that the system flows to a strong-coupling fixed point which cannot be determined within the approach [81, 82]. If inter-chain two-body interactions are included the relevance of hopping terms can be different. When only density-density and current-current interactions between the wires are included, as discussed above [75, 76], the possible relevance around this Gaussian model, recently called sliding LL [83–85], can be different. If the single chains are in the spin-gapped Luther-Emery regime [57] single-particle hopping between the chains is irrelevant and the coupled system can show power-law correlations characteristic of a 1d-LL [83, 85]. For the spinless model single particle and pair hoppings can be irrelevant for strong enough forward interactions [84]. In the following we concentrate on the Luttinger to Fermi liquid crossover. In order to get a quantitative picture it is desirable to study models which allow controlled approximations. The simple perturbative calculation in t⊥ for the calculation of the one-particle Green’s function by Wen [80] discussed below is unfortunately only controlled in the rather unphysical limit when the transverse hopping is independent of the distances of the chains (t⊥,ij ≡ t⊥ ) [86]. The (retarded) one-particle Green’s function G is expressed in terms of the selfenergy Σ G(k , k⊥ , z; t⊥ ) = 1 z − ǫk , k⊥ − Σ(k , k⊥ , z; t⊥ ) , (1.68) where ǫk ,k denotes the energy dispersion for the noninteracting model and ⊥ z = ω + i0 is the frequency above the real axis. For small t⊥ the dispersion can be linearized around k = ±kF near the open noninteracting Fermi surface. This yields ǫk ,k ≈ ±vF (k ∓ kF ) + t⊥ (k⊥ ). In the context of Fermi ⊥ liquid theory the selfenergy is studied in (all orders) perturbation theory in the two-body interaction v around the noninteracting limit. This can be done using standard Feynman diagrams. In the present context one wants to study how the LL behaviour for finite two body interaction and finite anomalous dimension is modified by the transverse hopping. Similar to perturbation theory for the Hubbard model around the atomic limit nonstandard techniques have to be 32 used [87]. The simplest approximation, which corresponds to the “Hubbard I” approximation for the Hubbard model, is to replace Σ in Eq. (1.68) in zeroth order in t⊥ by the selfenergy Σ(chain) (k , z) of a single chain [80]. This approximation first used by Wen reads for k ≈ kF 1 G(k , k⊥ , z; t⊥ )Wen = G+ (k , z) −1 , (1.69) − t⊥ (k⊥ ) where G+ is determined by the spectral function ρ+ discussed following Eq. (1.50) via a Hilbert transform. In the asymptotic low-energy regime this yields ˜ ˜ ˜ G+ (kF +k , z) = A0 [(k /kc )2 −(z/ωc )2 ]αL /2 /(z−vc k ) for spinless fermions, with ωc ≡ kc vc and A0 = παL /[2 sin (παL /2)]. Wen’s approximate Green’s function leads to a spectral function with the same range of continua as ρ+ (k , ω). In addition there can be poles at ωk ,k⊥ , determined by setting the denominator in Eq. (1.69) equal to zero. The poles located at ωk ,k = 0 determine the ⊥ ˜ Fermi surface k (k⊥ ) of the interacting coupled system. From Eq. (1.69) and ˜ the simple form of G+ one obtains A0 (k /kc )(1−αL ) = t⊥ (k⊥ ), which shows that the reduction of warping of the Fermi surface (FS) by the interaction is proportional to [t⊥ (k⊥ )/ωc ]αL /(1−αL ) . This is shown in Fig. 1.6 for a two dimensional system of coupled chains. If one writes t⊥ (k⊥ ) ≡ t⊥ c(k⊥ ), with ky/πay 1 0 −1 −2 −1 0 1 2 kx/kF Figure 1.6. Fermi surface “flattening” in Wen’s approximation for coupled chains for different values of the the anomalous dimension αL for a single chain. The dotted lines show the noninteracting FS, the long dashed curves correspond to αL = 0.125 and the full ones to αL = 0.6. At αL = 1 the FS degenerates to two parallel lines as without interchain coupling, called the “confinement transition”. c(k⊥ ) a dimensionless function, the new effective low energy scale is given 33 by teff = ωc (t⊥ /ωc )1/(1−αL ) . The weights Zk of the poles for k values on ⊥ the Fermi surface are also proportinal to [t⊥ (k⊥ )/ωc ]αL /(1−αL ) . Wen’s approximate solution has the Fermi liquid type property of quasi-particle poles with nonzero weight on the Fermi surface, except at the special points where t⊥ (k⊥ ) vanishes. The improved treatment by Arrigoni [88] shows that this peculiar vanishing of the quasi-particle weigths is an artefact of Wen’s approximation. The new idea involved is to let the number of “perpendicular” dimensions “d − 1” go to infinity. This extends the original idea of the “dynamical mean field theory” (DMFT) [89], where one treats the Hubbard model in infinite dimensions as an effective impurity problem to the case of a chain embedded in an effective medium. Results are obtained by carrying out a resummation of all diagrams in the t⊥ -expansion which contribute in this large dimension limit [88]. This approach shows explicitly how the leading order Wen approximation is uncontrolled at low energies. For the case of weakly coupled one-dimensional Mott insulators one expects the approximation to be better controlled [90]. Despite the Fermi liquid like properties at energy scales much smaller than teff the coupled chain system can nevertheless show LL like properties for energy scales larger than teff if there is a large enough energy window to the high energy cutoff ωc which describes the regime where the asymptotic LL ˜ power laws hold for a single chain. Then for temperatures lower than ωc but ˜ higher than teff the system behaves like a LL. The integrated spectral functions ρ< (ω) probed by angular integrated photoemission, for example, show apα,(σ) proximate power law behaviour ∼ (−ω)αL for temperatures larger than teff in the energy window kB T < −ω < ωc . Unfortunately little is known about the ˜ value of ωc for microscopic models. An exception is the Tomonaga model Eq. ˜ (1.14) with a constant v (k) up to the cutoff kc , where the high energy cutoff ωc ˜ ˜ equals ωc = min(vc , vs )kc [4]. This implies for the integrated spectral function for the very large U Hubbard model with periodic boundary conditions that the power law |ω|αL only holds in a narrow energy window ∼ vs , which vanishes proportional to 1/U in the U → ∞ limit [65]. Another example is the Hubbard model at boundaries where ωc can be very small for small U [50]. ˜ As an alternative way to treat the “anisotropic large dimension model” [88] one can try to solve the resulting chain-DMFT equations numerically, using e.g. a quantum Monte Carlo algorithm [91]. In this reference the Hi were chosen as Hubbard Hamiltonians (1.66) with chain lengths of 16 and 32 sites. The results for a partly filled band as a function of temperature indicate in fact a crossover from a LL to a FL at the estimated crossover scale as the temperature is lowered. In agreement with Arrigoni [88] the authors find that the quasi-particle weight is more uniform along the Fermi surface than suggested by Wen’s approximation Eq. (1.69). At half filling and low but finite temper- 34 atures the crossover from the Mott insulator to FL crossover was examined (the intermediate LL regime was too narrow to be visible). In the future it is to be expected that this method applied to longer chains and additional nearest neighbour interaction will provide important results which allow a more realistic comparison with experimental work. Because of space limitations the interesting field of a finite number of coupled chains cannot be discussed here [92]. 5.2 On the experimental verification of LL behaviour There exist several types of experimental systems were a predominantly 1d character can be hoped to lead to an (approximate) verification of the physics of Luttinger liquids. In the following we present a short list of the most promising systems and discuss some of the experimental techniques which have been used. As these topics are also discussed in other chapters of this book we do not attempt a complete list of references but only refer to most recent papers or to review articles on the subject. The following systems look promising: Highly anisotropic “quasi-one-dimensional” conductors There has been extensive work on organic conductors, like the Bechgaard salts [93, 94], as well as inorganic materials [95, 96]. Artificial quantum wires Two important types of realizations are quantum wires in semiconductor heterostructures [97, 98] or quantum wires on surface substrates [99, 100]. Carbon nanotubes The long cylindrical fullerenes called quantum nanotubes are also quantum wires but have been listed separately because of their special importance in future applications like “molecular electronics” [101, 102]. Using the peculiar band structure of the π-electrons of a single graphite plane it was shown that single wall “armchair” nanotubes should show LL behaviour with Kc ∼ 0.2 − 0.3 down to very low temperatures [103, 104], despite the fact that two low energy channels are present. Fractional quantum Hall fluids Electrons at the edges of a two-dimensional fractional quantum Hall system can be described as a chiral Luttinger liquid [105]. The power law tunneling density of states observable in the tunneling current-voltage characteristics shows power laws of extraordinary quality [106]. The theoretical predictions for general filling factors between the Laughlin states ν = 1 and ν = 1/3 [107, 108] are not bourne out by experiment [109]. As in these 35 chiral LLs the right- and left-movers are spatially separated the edge state transport is quite different from the case of quantum wires and FQH fluids are not further discussed in the following. Promising experimental techniques to verify LL behaviour are: High resolution photoemission One of the earliest claims of possible verification of Luttinger liquid behaviour was from angular integrated photoemission of the Bechgaard salt (TMTSF)2 PF6 , which showed a power law supression at the chemical potential with an exponent of order 1 over an energy range of almost one eV [110]. There are serious doubts that this suppression can be simply explained by the LL power law behaviour [94]. Therefore a large number of other quasi-one-dimensional conductors were examined [94–96, 111]. In addition periodic arrays of quantum wires on surface substrates were studied by angular resolved photoemisssion (ARPES), but the interpretation of a two peak structure as spin-charge separation [99] was questioned [100]. Spin-charge separation was shown to occur in the 1d Hubbard model also at higher energies on the scale of the conduction band width [69, 70, 73]. Recent ARPES spectra of TTF-TCNQ were interpreted with the 1d Hubbard model at finite doping to show signatures of spin-charge separation over an energy scale of the conduction band width. As for the Hubbard model Kc > 1/2 for n = 1 which implies αL < 1/8 for the anomalous dimension the experimentally found nearly linear spectral onset at low energies cannot be explained within the same model. ARPES data for the “Li purple bronze” seem to compare favorably to the LL lineshape [96]. For the quasi-one-dimensional antiferromagnetic insulators SrCuO2 and Sr2 CuO3 ARPES spectra have been interpreted to show evidence of spin-charge separation [112]. For a more in depth discussion see the chapter by Grioni in this book. Transport As discussed in section 3 even a single impurity has a drastic effect on the conductance of a LL, which vanishes as a power law with temperature. Another issue is the “conductance puzzle” of a clean LL. There has been an extended discussion whether the quantized value e2 /h for noninteracting electrons in a single channel is modified by the interaction to Kc e2 /h [113, 114]. Apparently the answer depends sensitively on the assumptions made about the contacts, a very delicate theoretical as well as experimental problem [115]. Experimental results are available for cleaved edge overgrowth quantum wires [97] as well as carbon nanotubes [116–118]. In the nanotubes the authors observe approximate power laws of the conductance which seem to be consistent with LL behaviour. A detailed dicussion of 36 transport through quantum wires is presented in the chapter by Yacoby. For a recent theoretical discussion of experimental results on the interchain transport in the Bechgaard salts see references [119, 120]. There the question of energy scales and the importance of the proximity of the incipient Mott insulator are addressed. Optical properties Optical properties have long been used to investigate electronic properties of quasi-one-dimensional systems [121]. The optical behaviour of different Bechgaard salts was analyzed recently using LL concepts [122]. At low energies, smaller than about ten times the Mott gap, the importance of dimerization and interchain hopping was pointed out [123]. As there is a separate chapter about the optical response in chains and ladders it will not be discussed further here. Obviously neither the list of systems nor that of methods is coming close to being complete. They were presented to show that there are intensive experimental activities in the attempt to verify the elegant LL concept put forward by theoreticians. Further work on both sides is necessary to come to unambiguous conclusions. Acknowledgements: For useful comments on the manuscript the author would like to thank J. Allen, E. Arrigoni, D. Baeriswyl, L. Bartosch, J. von Delft, R. Egger, F. Essler, F. Gebhard, A. 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