Transport through asymmetric two-lead junctions of Luttinger liquid wires D.N. Aristov1, 2, 3 and P. W¨lfle2, 4 o 1 Petersburg Nuclear Physics Institute, Gatchina 188300, Russia Institut for Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany 3 Department of Physics, St.Petersburg State University, Ulianovskaya 1, St.Petersburg 198504, Russia 4 Institute for Condensed Matter Theory, and Center for Functional Nanostructures, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany (Dated: June 25, 2018) arXiv:1207.4646v1 [cond-mat.str-el] 19 Jul 2012 2 We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g1 , g2 , connected through a short junction, within the scattering state formalism. Following earlier work we formulate the problem in current algebra language, and calculate the scale dependent contribution to the conductance in perturbation theory keeping the leading universal contributions to all orders in the interaction strength. From that we derive a renormalization group (RG) equation for the conductance. The analytical solution of the RG-equation is discussed in dependence on g1 , g2 . The regions of stability of the two fixed points corresponding to conductance G = 0 and G = 1,respectively, are determined. Introduction. Charge or spin transport in threedimensional metallic systems may be described in terms of Landau quasi-particles. In strictly one-dimensional quantum wires quasi-particles are found to be unstable on account of the interaction between electrons. An important part of that physics is captured by the exactly solvable Tomonaga-Luttinger liquid (TLL) model.1,2 As is well known, transport through junctions of TLL wires is strongly affected by the interaction in the wires, in some cases leading to a complete blocking of transmission. The latter behavior can be traced back to the formation of Friedel oscillations of the charge density around the barrier, leading to an infinitely extended effective barrier potential in the limit of low energy. The suitable language to describe this situation is the renormalization group (RG) method, allowing to calculate the conductance as a function of (length or energy) scale. Quite generally, the transport behavior at low energy/temperature is dominated by only a few fixed points of the RG flow. In the neighborhood of these fixed points the conductance is found to obey power-law behavior as a function of temperature T for the infinite system or as a function of system length L at zero temperature. The problem of the two-wire junction has been studied first in the pioneering works3,4 , using the method of bosonization. Later, their results have been confirmed by many other authors, see5 and references therein. For special values of the interaction exact results have been obtained6,7 . A purely fermionic formulation of the problem has been introduced in the limit of weak interaction by Yue, Glazman and Matveev8 . We have extended that theory to the regime of strong coupling by summing up an infinite series of terms in perturbation theory, identified as the leading and universal contributions9 . The results obtained with our method for the two-lead junction as well as the three lead junction with time-reversal symmetry10,11 and with magnetic flux12 are in agreement with exact theoretical results, where available. However, our results go beyond what has been obtained by other authors in various ways. The majority of the previous works considered the symmetric case of equal interaction strength in both half wires, with exception of the works13,14 discussed below. In this paper we generalize our previous treatment9 to the case of two wires with different interaction strength g1 , g2 . This includes the case of a barrier at the end of a TLL wire, for which one of the interaction parameters is zero, e.g. g2 = 0. As before, we confine our considerations to spinless fermions. The model. To illustrate the system we are interested in, we first consider a tight binding Hamiltonian Htb of free spin-less fermions describing two quantum wires connected at a single junction by tunneling amplitudes: 2 Htb = N t0 c+ cj,n+1 + tb c+ c1,0 + h.c. 2,0 j,n (1) j=1 n=0 Here c+ creates a fermion in wire j at site n, and tb is j,n the tunneling amplitude connecting the sites (j, n = 0) at the junction. The 2×2 S-matrix characterizing the scattering at this junction has the structure (up to overall phase factors in the individual wires) S= r t t r = sin θ i cos θe−iφ iφ i cos θe sin θ (2) We choose this parametrization in terms of the transmission and reflection amplitudes t, r , since it is readily generalizable to the case of multi-wire junctions (n wires, n > 2 ). The above form of the S-matrix is completely general (up to irrelevant phase factors) and, in fact, defines our model. Passing to the continuum limit, linearizing the spectrum at the Fermi energy and adding forward scattering interaction of strength gj in wire j , we may write the TLL Hamiltonian H in the representation of 2 incoming and outgoing waves as 2 0 H = 0 Hj int Hj dx −∞ j=1 0 int [Hj + Hj Θ(−L < x < −l)] , (3) = † vF ψj,in i † ψj,in − vF ψj,out i ψj,out , = † † 2πvF gj ψj,in ψj,in ψj,out ψj,out . (4) (5) We put vF = 1 from now on. The range of the interaction lies within the interval (l, L), where l > 0 serves as a ultraviolet cutoff and separates the domains of interaction and potential scattering on the junction; non-interacting leads correspond to large |x| beyond L. In terms of the doublet of incoming fermions Ψ = (ψ1,in , ψ2,in ) the outgoing fermion operators may be expressed with the aid of the S-matrix as Ψ(x) = S · Ψ(−x) . We express the interaction term of the Hamiltonian in terms of density † operators ρj,in = ψj,in ψj,in = Ψ+ ρj Ψ = ρj , and ρj,out = † ψj,out ψj,out = Ψ+ ρj Ψ = ρj , where ρj = S + · ρj · S , as int Hj = 2πg1 ρ1 ρ1 + 2πg2 ρ2 ρ2 . (6) The matrices are given by (ρj )αβ = δαβ δαj and (ρj )αβ = + Sαj Sjβ . A convenient representation of 2×2-matrices is in terms of Pauli matrices σj , j = 1, 2, 3, the generators of SU (2) (see9 ). Notice that the interaction operator only involves σ3 (besides the unit operator (σ0 )αβ = δαβ ). We note T r(σj ) = 0, T r(σj σk ) = 2δjk , j = 0, 1, 2, 3. Defining a two-component vector s = (σ3 , σ0 ), we have ρj = 1/2 µ Rjµ sµ , where the 2 × 2 matrix R has the properties R−1 =√ T , det R = 1, and R11 = R12 = R −R21 = R22 = 1/ 2. The outgoing amplitudes will be expressed in terms of σj = S + · σj · S. With the aid of the σj the S-matrix may be parametrized by three angular (Euler) variables (see9 ), S = eiα1 σ3 /2 eiα2 σ1 /2 eiα3 σ3 /2 . For the case under consideration only two of these, θ, φ, are relevant: S = e−iφσ3 /2 ei(π−2θ)σ1 /2 eiφσ3 /2 . The corresponding elements of the S-matrix have been given in Eq. (2). Parametrization of conductance matrix. We may define a 2 × 2 matrix of conductances Gjk relating the current Ij in lead j (flowing towards the junction) to the electrical potential Vk in lead k : Ij = k Gjk Vk . It follows from the conservation of charge that j Gjk = 0 and from invariance under a shift of the zero of energy that k Gjk = 0. Therefore, the conductance matrix has only one independent element G = (1 − a)/2, which 1 relates the net current I = 2 (I1 − I2 ) to the bias voltage V = (V1 − V2 ), I = GV . We note the relation 1 G = 2 (RT GR)11 , while all other elements of RT GR are zero. In the linear response regime the conductances are related to the S-matrix by Gjk = δjk − T r(˜r ρk ) = ρj r δjk − |Sjk |2 , where the label r indicates that the quantity is fully renormalized by interactions. Defining Yjk = ˜µ ˜j T r(˜r ρk ) and using the above relation of ρr and sr , we ρj see that the conductance components may be expressed 1 in terms of Y µν = 2 T r(˜r sν ) as Yjk = (RYRT )jk . sµ Here and in the following bold faced quantities marked with overbar are matrices in the transformed space (indices µ, ν ). It follows from the analysis of the transformed matrix RT GR given above that the matrix Y has block structure, the nonzero elements being given by the conductance parameter introduced above and by unity, Y 11 = a, Y 22 = 1, Y 12 = Y 21 = 0. From the above relations we see that the parameter a may be expressed by the angle θ in the above parametrization of the S-matrix as a = − cos 2θ. We find therefore that a is confined within the region a ∈ [−1, 1]. Perturbation theory. We now calculate the conductance in perturbation theory in the interaction. In first order we have to evaluate the diagrams depicted in Fig. 1. Here solid lines denote Green’s functions in position-energy representation (ωn are fermionic Matsubara frequencies), G(x; ωn ) = −isign(ωn )θ(xωn )e−xωn . The double wavy lines denote the interaction operator, which in the lowest order is given by the matrix g µν = (RT gR)µν , with gjk = δjk gj , in the transformed or initial representations, respectively. At the ends of the interaction lines operators σ3 and σ3 are attached depending on whether x < 0 or x > 0. As a result, one finds in lowest order in the interaction (1) 1 Y µν = − 2 Tr W µν W µ ν gν µ Λ 1 = − 2 δµ1 δν1 (a2 − 1)(g1 + g2 )Λ . (7) 1 Here the W µν = (RT WR)µν = 2 [sµ , sν ], and Wjk = [ρj , ρk ] , are 2×2 matrices for each pair of µν (or jk) and the trace operation Tr is defined with respect to that matrix space, whereas the g µν = (RT gR)µν , with gjk = δjk gj are scalars. Notice that only one matrix element, W 11 , is nonzero. We may extend this analysis into the strong coupling regime by summing up infinite classes of contributions in perturbation theory. It has been found in9 that the diagrams shown in Figs. 1 and 2 provide the leading terms in the neighborhood of the fixed points. Their contribution is universal in contrast to other higher order terms (see below). These may be interpreted as a renormalization of the bare interaction, 2πgkl δ(x − y) → Lkl (x, y; ωn ). The contribution to conductance stemming from the first two diagrams (a) and (b) in Fig. 1 in this ladder approximation is given by 3 −y −y −x2 −y −x2 −x2 (b) (a) (c) x1 x1 x x1 x x FIG. 1: Feynman diagrams depicting the corrections to conductance. Two first diagrams, (a) and (b), correspond to Eq. (8), the third diagram (“vertex correction”) vanishes in the static limit, external Ω → 0. 1 G(a+b) = − T 2 4 ∞ L dx1 dx2 ,ω l dy L11 (x1 , x2 ; ω)G(y + x; + Ω) (8) L × (G(x1 − x; )G(−x1 − x2 ; − ω)G(x2 − y; ) + cos 4θ G(−x2 − x; )G(x1 + x2 ; + ω)G(−x1 − y; )) , Here only the (1, 1) element of L enters, corresponding to the fact that operators σ3 and σ3 are attached to the ends of the renormalized interaction line. The factor cos 4θ appears as Tr(σ3 σ3 σ3 σ3 )/2. In the limit of zero temperature we may convert the summation over Matsubara frequencies to an integration along the imaginary axis. Another contribution to G is obtained by reverting the arrows on the fermion lines in Fig. 1a, Fig. 1b, and doubles the above result. Performing the integrations on and on y from L to ∞ and taking the limit Ω → 0 we find G(L) = (1 − a2 ) dx1 dx2 dω ¯ L11 (x1 , x2 ; ω)θ(ω)e−ω(x1 +x2 ) , 2π It is useful to first calculate L in the initial representation, where the interaction matrix g is diagonal, but the matrix Y is nondiagonal. Then L is found to satisfy the following linear integral equation for ω > 0 (Ref. [9], note that the definitions of Y and for the strength of interaction there are different): L(x, y; ω) L2 (x, y; ω) = 2πgδ(x − y) L 1 − 2π 0 dz l gYΠ(x + z, ω), gΠ(x − z, ω) gΠ(z − x, ω), 0 L(z, y; ω) L2 (z, y; ω) with the fermionic loop Π(x, ωn ) = (2π)−1 (δ(x) − |ωn |θ(xωn )e−xωn ). Expressing L2 (x, y; ω) via L(x, y; ω) we have L L(x, y; ω) = 2πgδ(x − y) + ωg with gj = gj /d2 and dj = j l 1 dz (Y − 2 g)e−ω(x+z) − 1 ge−ω|x−z| L(z, y; ω), 2 2 1 − gj . In order to solve this integral equation we first calculate a partial summation, defined by C(x, y; ω) = limY →0 L(x, y; ω). Then L(x, y; ω) will be the solution of L(x, y; ω) = C(x, y; ω) + ω 2π L dz1 dz2 C(x, z1 ; ω)Ye−ω(z1 +z2 ) L(z2 , y; ω) , (9) l and C(x, y; ω) satisfies the integral equation L 1 C(x, y; ω) = 2πgδ(x − y) − 2 ωgg dz e−ω(x+z) + e−ω|x−z| C(z, y; ω) . l These integral equations are shown diagrammatically in Fig. 2. (10) 4 We now define C(x, y; ω) = 2πgδ(x − y) − C1 (x, y; ω), so that the inhomogeneity in the integral equation for C1 (x, y; ω) is differentiable L C1 (x, y; ω) = πωg2 g e−ω(x+y) + e−ω|x−y| − 1 ωgg 2 dz e−ω(x+z) + e−ω|x−z| C1 (z, y; ω) , l The integral equation for C1 (x, y; ω) may be converted into a second order differential equation −y L −y = −y x −y = x −z1 L(x, y; ω) = C(x, y; ω) ω + V(x; ω)Y(1 − Q−1 Y)−1 V(y; ω) 2π + x C −y z2 x −y −x z x + x FIG. 2: Feynman diagrams depicting the integral equations for the renormalized interaction, Eqs.(9) and (10). ∂2 1 − ω 2 (1 + gg) C1 (x, y; ω) = −2πω 2 g2 gδ(x − y), ∂x2 Since the matrix g is diagonal, C1 is diagonal and we have two uncoupled differential equations for the components C1,j , j = 1, 2 . Taking into account the boundedness of C1,j , the general solution is given by 2 C1,j (x, y; ω) = πωj gj gj (Aj (y)e−ωj x + e−ωj |x−y| ) , where ωj = ω/dj . It follows from the boundary conditions at x = 0 that Aj (y) = e−ωj y . The quantity C is thus a diagonal matrix given by C(x, y; ω) = 2πd−2 gδ(x − y) − πωd−3 g3 e−ω(x+y)/d + e−ω|x−y|/d We now return to the integral equation for L(x, y; ω). Since the kernel is separable it is easily solved. We define the auxiliary matrix functions U(y; ω) = dxe−ωx L(x, y; ω), V(y; ω) = dxe−ωx C(x, y; ω) = 2πd−1 ge−ωy/d as well as the matrix Q−1 = ω dxdye−ω(x+y) C(x, y; ω) = g(1 + d)−1 . Multiplying 2π the above integral equation (9) by e−ωx and integrating over x we find U = V + Q−1 YU = (1 − Q−1 Y)−1 V and we finally get This result is now substituted into the expression for the conductance. Performing the integration over y first we get dye−ωy L(x, y; ω) = V(x; ω)[1 + Y(1 − Q−1 Y)−1 Q−1 ] = V(x; ω)[1 − YQ−1 ]−1 . Next integrating over positive ω yields (dω/2π)e−ωx V(x; ω)[1 − YQ−1 ]−1 = (1/x)g(1 + d)−1 [1 − YQ−1 ]−1 = (1/x)[Q − Y]−1 . Finally, the integration over x produces the scale dependent logarithm Λ = ln(L/l) . The conductance in the ladder approximation is found after transforming to the rotated basis (quantities with overbar) and taking the (1, 1)-element of the matrix, 1 G(L) = 2 (1 − a) − (1 − a2 ) (Q − Y)−1 11 Λ Renormalization group equations. The renormalization of the conductances by the interaction is determined from the scale dependent contributions in perturbation theory. Differentiating these results with respect to Λ (and then putting Λ = 0) we find the RG equation for the quantity a = 1 − 2G in the ladder approximation da = 2(1 − a2 ) (Q − Y)−1 dΛ 11 (11) Here Y = diag(a, 1) and Q = RT QR , such that Q11 = Q22 = Q+ = (q1 + q2 )/2 and Q12 = Q21 = Q− = (q1 − q2 )/2 where qj = (1 + 2 1 − gj )/gj = (1 + Kj )/(1 − Kj ), and Kj = (1 − gj )/(1 + gj ) is the usual Luttinger liquid parameter for wire j. We define 2 γ = Q+ − Q− /(Q+ − 1) = −1 −1 K1 + K2 + 2 , −1 −1 K1 + K2 − 2 (12) (note that |γ| > 1 for any K1,2 > 0 ), then the RGequation takes the explicit form da a2 − 1 =2 . dΛ a−γ (13) RG flow and conductance. The fixed points of the above RG equation are labelled N , at a = 1, G = 0, (complete separation of the wires) and A at a = −1, G = 1 (ideal conductance through the junction). In order 5 4 3 K2 A 2 1 N 0 0 1 2 3 K1 4 FIG. 3: Stability regions, labeled by the corresponding fixed point are shown in K1 -K2 plane. The hyperbola separating the regions is shown by solid line, the non-interacting values K1,2 = 1 are shown by dashed lines. to discuss the stability and to calculate the conductance we rewrite the RG-equation in terms of the conductance K2 = K we recover the well-known3,4 exponents 2(K −1 − 1), 2(1 − K). Of special interest is the case g1 = g, g2 = 0, when we have γ = 2q − 1 = (1 + 3K)/(1 − K) and the −1 power laws are given by G = cN (L/l)−(K −1) at N and −2(K−1)/(K+1) 1 − G = cA (L/l) at A. We may compare these findings with results obtained in Ref. [13] for a nonsymmetric position of an impurity in a TLL wire. When taking the impurity position sufficiently close to the interface between the TLL wire and a non-interacting lead, scaling exponents identical to the above ones were found in13 , as can be read off from Eqs. (10) and (13) there. Thus we recover the correct exponents in the scaling law for the conductance. We obtained them by summation of the ladder sequence for the renormalized interaction in the presence of the junction. As was shown in9 , the first contributions to conductance beyond the ladder series appear in the third order of interaction and are of three-loop type (the two-loop RG contributions are absent). These latter terms do not contribute to above scaling exponents, but define the relation between the prefactors cN and cA . The Eq. (15) implies that both cN and cA depend on the initial conditions, encoded in the amplitude c. It is easily seen, that the ladder approximation corresponds to the relation c1−γ /c1+γ = 1 , N A G(1 − G) dG = −4 = β(G) dΛ 2G − 1 + γ (14) Stability of a fixed point requires that the derivative of β(G) at the fixed point is negative. At fixed point N this translates into γ − 1 > 0, or more explicitly, q1 + q2 > 2, −1 −1 and in terms of the Luttinger parameters K1 +K2 > 2 i.e. either both interactions should be repulsive, or one of them is attractive, but weak relative to the repulsive one. At fixed point A the condition is γ + 1 < 0, or −1 −1 q1 + q2 < 2, and therefore K1 + K2 < 2 , meaning that the interaction is predominantly attractive, but a weaker repulsive interaction in one of the wires is possible. The line separating the stability regions in the K1 -K2 -plane is the hyperbola, (K1 − 1 )(K2 − 1 ) = 1 , passing through 2 2 4 the (no interaction) point K1 = K2 = 1. This hyperbola and corresponding stability regions are shown in Fig. 3 The RG-equation may be integrated to give G1−γ (1 − G)1+γ = c(L/l)4 (15) In the vicinity of fixed points N and A , respectively, we then find the power laws G = cN (l/L)4/(γ−1) , G = 1 − cA (L/l) 4/(γ+1) at N , at A (16) (17) −1 −1 Explicitly we have 4/(γ − 1) = (K1 + K2 − 2), and 4/(γ + 1) = 2(K1 + K2 − 2K1 K2 )/(K1 + K2 ). At K1 = It was shown9 that the three-loop corrections change this ratio to values, different from unity and depending on the strength of interaction and regularization scheme, i.e. non-universal. Our expression (12) shows that the boundary expo−1 −1 nents depend only on the sum K1 +K2 . This is in pre14 cise agreement with Eq. (13) in Ref. , where the scaling exponent of the point contact between two chiral (Hall edge) states was derived. On the basis of this observation it was suggested there, that the combination of two chiral states, K1 = 1/3 and K2 = 1, is equivalent to the well-known problem of impurity in TLL wire with K1 = K2 = 1/2, which can be fully solved.3 We note here that the coincidence of the scaling exponents at two fixed points N and A might not necessarily mean the coincidence of the full scaling form for the conductance. The non-universal three-loop terms (which are not discussed in the standard bosonization approach) may be different in cases K1 = 1/3, K2 = 1 and K1 = K2 = 1/2. Conclusion. In this paper we employed a fermionic description of a general two-wire junction of two TLL-wires to derive the renormalization group equation for the conductance, using the approach developed by us earlier9 . We used an infinite summation of perturbation theory in the form of a ladder approximation, allowing for an analytical solution for arbitrary junction parameters and interactions in the wires. As demonstrated earlier the approximation is asymptotically exact in the vicinity of the fixed points. As in the well-studied case of a symmetric junction there exist two fixed points of the RG 6 flow. Fixed point N corresponds to a complete separation of the wires, i.e. the conductance vanishes. It is stable in a region of the K1 -K2 −coupling constant plane which is predominantly repulsive, meaning that a weakly attractive component, say K2 > 1 is permitted. At fixed point A the conductance assumes its maximum value. It is stable in the complementary part of the coupling constant plane. The two stability regions in the K1 K2 −plane are separated by a hyperbolic boundary curve. The representation chosen is maximally general and may therefore be easily extended to junctions connecting more 1 2 3 4 5 6 7 8 T. Giamarchi, Quantum Physics in One Dimension (Clarendon Press, Oxford, 2003). A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization and Strongly Correlated Systems (Cambridge University Press, Cambridge, 1998). C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46, 15233 (1992). A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631 (1993). I. Safi and H. J. Schulz, Phys. Rev. B 59, 3040 (1999). U. Weiss, R. Egger, and M. Sassetti, Phys. Rev. B 52, 16707 (1995). P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. B 52, 8934 (1995). D. Yue, L. I. Glazman, and K. A. Matveev, Phys. Rev. B 49, 1966 (1994). than two wires. The 1-2-symmetric three wire junction (“Y-junction”) has been considered by us in Refs.10–12,15 . Work on the four-wire junction is in progress. We thank D.G. Polyakov for helpful comments on the manuscript and A.P. Dmitriev, I.V. Gornyi and V.Yu. Kachorovskii for useful discussions. The work of D.A. was supported by the German-Israeli-Foundation (GIF), the Dynasty foundation. The work of D.A. and P.W. was supported by the DFG-Center for Functional Nanostructures. 9 10 11 12 13 14 15 D. N. Aristov and P. W¨lfle, Phys. Rev. B 80, 045109 o (pages 22) (2009). D. N. Aristov, A. P. Dmitriev, I. V. Gornyi, V. Y. Kachorovskii, D. G. Polyakov, and P. W¨lfle, Phys. Rev. Lett. o 105, 266404 (2010). D. N. Aristov and P. W¨lfle, Phys. Rev. B 84, 155426 o (2011). D. N. Aristov and P. W¨lfle, arXiv:1110.1159 . o A. Furusaki and N. Nagaosa, Phys. Rev. B 54, R5239 (1996). C. de C. Chamon and E. Fradkin, Phys. Rev. B 56, 2012 (1997). D. N. Aristov, Phys. Rev. B 83, 115446 (2011).