epl draft arXiv:0709.1562v2 [cond-mat.str-el] 15 Feb 2008 Transport of interacting electrons through a potential barrier: nonperturbative RG approach D.N. Aristov1,2 1 2 3 (a) ¨ and P. Wolfle1,2,3 Institut f¨r Theorie der Kondensierten Materie, Universit¨t Karlsruhe, 76128 Karlsruhe, Germany u a Center for Functional Nanostructures, Universit¨t Karlsruhe, 76128 Karlsruhe, Germany a Institut f¨r Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany u PACS PACS PACS 71.10.Pm – Fermions in reduced dimensions (anyons, composite fermions, Luttinger liquid, etc.) 73.63.Nm – Quantum wires (Electronic transport in nanoscale materials and structures) 71.10.-w – Theories and models of many-electron systems Abstract. - We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g2 , and subject to a potential barrier of arbitrary strength, as a function of temperature. We first map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. Analyzing the perturbation theory in the fermionic representation the diagrams contributing to the renormalization group (RG) β-function are identified. A universal part of the β-function is given by a ladder series and summed to all orders in g2 . First non-universal corrections beyond the ladder series are discussed. The RG-equation for the temperature dependent conductance is solved analytically. Our result agrees with known limiting cases. Electron transport in nanowires has been studied theoretically for more than 20 years. Initially it was found that electron-electron interaction affects even the conductance of a clean wire [1, 2] . In the case of realistic boundary conditions, namely attaching ideal leads to the interacting quantum wire, the two-point conductance of a clean wire is that of the leads, equal to one conductance quantum per channel, irrespective of the (forward scattering) interaction [3,4]. Alternatively, it has been argued that the screening of the external field by the interacting electron system leads to a renormalization of the conductance to its ideal value of unity [5]. The work of Kane and Fisher [2] and Furusaki and Nagaosa [6] showed, that interaction has a dramatic effect on the conductance in the presence of a potential barrier. For repulsive interaction these authors found that the conductance tends to zero as the temperature, T , or more generally, the excitation energy of the electrons approaches zero. This was shown at low temperature in the limits of weak potential barrier and strong potential barrier (tunneling limit), and at special values of 1 the interaction parameter, K = 2 and K = 1 , for all tem3 peratures [2, 7, 8]. It has been argued, that these results (a) On leave from Petersburg Nuclear Physics Institute, Gatchina 188300, Russia apply to a contact free four-point measurement, which is best realized by measuring the absorption losses of an a.c. field in the limit ω → 0 [3, 4] (for a different view, see [5]). We recall that the reason for the strong suppression of the conductance by a repulsive interaction is that the Friedel oscillations of the charge density around the potential barrier act as a spatially increasingly extended effective potential as the temperature is lowered. A proper treatment of the two-point conductance in the limit of weak interaction, taking into account the gradual build-up of the Friedel oscillations as the infrared cutoff is lowered has been given by Yue, Matveev and Glazman [9]. These authors used the perturbative RG for fermions to derive the conductance for an arbitrary (but short) potential barrier. A generalization of that approach to the case of two barriers has been given in [10]. In this letter we propose to extend the approach of Yue et al. to arbitrary strength of interaction. We argue that the β-function of the RG-equation for the conductance can be obtained in very good approximation by summing a class of contributions in all orders of the interaction. As we show below, this is possible in this case, at least at low temperature, since the class of diagrams with the maximum number of loops in any order, which are the ones D.N. Aristov et al. contributing to the β-function, form a ladder series. At intermediate temperatures additional diagrams contribute small corrections to the β-function. The result of the solution of the RG-equation for the conductance, using the approximate β-function thus obtained, is found to agree with all known results on the scaling behavior of the conductance, including the case K = 1 , but goes far beyond: 2 it is valid for any interaction strength K and any potential scattering strength. A careful analysis of the dependence of the result on the cutoff procedure chosen shows that certain terms in the β-function are not universal and depend on the cutoff scheme. The Model. We consider a one-dimensional system (coordinate x) of spinless electrons subject to a potential barrier at x = 0. The barrier is characterized by ˜ transmission and reflection amplitudes t = t = cos θ, r = −˜∗ = i sin θeiφ with negligible energy dependence r in the energy range of interest (width of order of temperature, T , around the Fermi energy, ǫF ). We assume the extension of the barrier, a, to be narrow, akF ≪ 1, and neglect all interaction processes both close to the barrier |x| < a (here kF is the Fermi wave number) and in the leads, |x| > L. Beyond the large scale L the system is assumed non-interacting, which allows for an asymptotic single-particle scattering states representation. If c+ and c+ are operators creating electrons in right1k 2k moving and left-moving single particle scattering states of the barrier (k > 0), we may define the electron creation operator ψ + (x) as ψ + (x) = + + ˜ + Θ(−x) ψ1 (x) + rψ1 (−x) + tψ2 (x) + Θ(x) + tψ1 (x) + rψ2 (−x) ˜ + + + ψ2 (x) , 1) ( ∞ + dk where ψ1,2 (x) = 0 2π e±ikx c+ . 1,2k It is convenient to define current operators Jµ (x), µ = µ + 0, 1, 2, 3 by Jµ (x) = 1 Σα,β=1,2 ψα (αx)ταβ ψβ (βx), where 2 + τ µ are the Pauli matrices plus the unit matrix, ψα (αx) = + + ψ1 (x) or ψ2 (−x) for α = 1, 2. We call J0 the isocharge current and the vector J = (J1 , J2 , J3 ) the isospin current; these operators obey U (1) and SU (2) Kac-Moody algebras, respectively [11, 12]. In this paper we will not make use of these relations, but instead will work in the fermion representation. Nonetheless, the representation allows one to work in the chiral fermion representation. The particle density operators for incoming (i) and outgoing (o) particles in terms of the Jµ ’s are given by (here x>0) ρiR,L (∓x) = ρoR (x) = = ρoL (−x) = = + ψ1,2 (∓x)ψ1,2 (∓x) = + tψ1 (x) + r ψ2 (−x) ˜ + ˜ J0 (x) + J3 (x) + ˜ rψ1 (x) + tψ2 (−x) ˜ J0 (x) − J3 (x) . with R33 = |t|2 − |r|2 = cos 2θ, R32 = Im{tr∗ + tr∗ } = − sin 2θ cos φ, R31 = Re{tr∗ − tr∗ } = sin 2θ sin φ. We consider a model with interaction constant g2 (no backscattering, no Umklapp processes). The Hamiltonian is given by H = H0 + H1 , with ∞ H0 = 2πvF 0 L H1 = 2g2 a 2 2 2 ˜2 dx J0 (−x) + J0 (x) + J3 (−x) + J3 (x) , ˜ dx J0 (−x)J0 (x) − J3 (−x)J3 (x) , (3) where vF is the Fermi velocity. The forward scattering interaction g4 of like-movers may be absorbed into redefinition of vF in the usual way [11]. In Fig. 1 we show in a pictorial way our parametrization of the fermionic densities and the interaction, H1 . In Fig. 1a the g2 -interaction processes are shown in the usual scattering configuration. The representation in terms of the currents Jµ is in the chiral basis (all particles moving to the right, see Fig. 1b), which leads to a seemingly nonlocal interaction. It should be clear that electrons in the left half space (x < 0) are not affected by the barrier yet, whereas electrons on the right (x > 0) are. Note that in the case of perfect reflec˜ tion, t = 0, we have J3 = −J3 and the observable densities (see below) form an Abelian U (1) sub-algebra of SU (2). This is the case of “open boundary bosonization”, which allows a complete and rather simple analysis. [13] One can also show that the J3 part of Eq. (3) can be reduced by a canonical transformation H ′ = U † HU to the Hamiltonian with the interaction part ′ H1 = vF B.J(x = 0) − 2g2 L dx J3 (−x)J3 (x) , (4) a ∞ here U = exp i 0 dxB.J(x) and B = 2θ(cos φ, sin φ, 0). [14] The first term in (4) corresponds to the rotation of the incoming isospin current J by the “magnetic field”, B, at the origin, Fig. 1b. Eq. (4) thus resembles the Hamiltonian for the Kondo problem in the current algebra approach. [12] The major simplification in our case is the classical nature of B, as opposed to the quantum Kondo spin S, see [12]. Current and conductance. The total electron density ρ(x) is given by ρ(x) = [ρiR (x) + ρoL (x)] Θ(−x) + [ρoR (x) + ρiL (x)] Θ(x), = ˜ J0 (−x) + J0 (x) + sgn(x) −J3 (−|x|) + J3 (|x|) , = ρc (x) + ρs (x) , (5) J0 (−x) ± J3 (−x) , where subscript c(s) refers to isocharge (isospin) compo[t∗ ψ1 (x) + r∗ ψ2 (−x)] nents. From the continuity equation ∂t ρ(x) = −∂x j(x) = ˜ −i[ρ(x), H] one finds the current ˜ r∗ ψ1 (x) + t∗ ψ2 (−x) (2) j(x) = ˜ vF J0 (x) − J0 (−x) + J3 (−|x|) + J3 (|x|) = jc (x) + js (x) . (6) ˜ Here J3 = (RJ)3 is the third component of the isospin We now consider the linear response to an applied voltage 1 current vector J rotated by the orthogonal matrix Rµν V (x, t) = 2 V (t)sgn(x), which is seen to couple only to ¡¢£ Transport of interacting electrons through a potential barrier 1 −x 1 −x −z 1 g2 g2 2 g2 (c) z z y y Fig. 2: Three Feynman diagrams, depicting the first order in g2 contribution to the conductance. The other three diagrams are obtained from these by reversing the direction of fermionic propagation. 1 2 J 1 (b) −z (b) z y (a) 2 1 (a) 2 −x −z 2 J G(1) = − 2 g2 Fig. 1: (a)The interaction of the left- and right-going densities in the basis of scattered waves. (b) The interaction of the leftand right-going densities in the chiral basis, corresponding to non-local interaction, Eq. (3). the isospin components, ρs , of the density operator. The conductance is then given by (in units of e2 /2π ) ∞ G(x, t) = −2πiΘ(t) js (x, t) , dy ρs (y, 0) is ∝ Ω and will be dropped. One finds , (7) 0 where we used the fact that correlation functions mixing the isocharge and isospin sectors vanish. g2 T0 sin2 (2θ) ln , 4π T (10) in agreement with [9]. The ultraviolet cutoff, T0 , is determined by the width of the potential barrier a as T0 = vF /(2πa), the infrared cutoff arises at finite T > vF /L through the Green’s function at time t = 0: G(2a ; t = 0) ∼ T / sinh(2πT a/vF ). In the limit of zero temperature the finite-T logarithm Λ ≡ ln(T0 /T ) is replaced by the zero temperature expression Λ0 ≡ ln(L/a). In n-th order the diagrams with only one loop contribute the scale dependent terms [ln T0 /T ]n . Our prinicipal observation here is that the diagrams with the maximum number of loops (n loops) contribute linearly in logarithm ∝ ln(T0 /T ). They form a set of ladder diagrams. The diagrams linear in ln(T0 /T ) but not contained in this ladder series will be discussed below. The sum of all ladder diagrams (see Fig. 3) may be calculated, and will be denoted ¯ by L(x1 , x2 ; ωn ). Later we will need the integrated quan∞ ¯ tity L(x1 ; ωn ) = 0 dx2 e−ωn x2 L(x1 , x2 ; ωn ), which obeys the integral equation Perturbation theory in g2 . The contributions to G(Ωm ) in n-th order of g2 may be calculated with the help of Feynman diagrams in the position-energy representation (Ωm is the external Matsubara frequency). We draw n vertical wavy lines in parallel, the upper endpoint of the 1 3 ∞ i-th line at −xi with isospin matrix 2 ταβ , the lower one at g dze−ωz L(z; ω) L(x; ω) = −ge−ωx 4π + ω(Y + ) µ 1 xi with matrix 2 R3µ ταβ attached and carrying the factor 2 0 ∞ −2g2 ; α, (β) are isospin indices of ingoing, (outgoing) 1 2 g ω dz e−ω|x−z|L(z; ω) . (11) + fermion lines. The external vertices are at −x with matrix 2 0 1 1 τ3 and at y with matrix 2 (Rτ )3 . The vertex points are 2 connected by Green’s functions Here we defined Y = cos 2θ, g = g2 /2π and we use units with vF = 1. The Eq. (11) is of Wiener-Hopf type and its i (8) solution is L(x; ωn ) = −4πgbe−ωnpx where p = 1 − g 2 G(x ; ωn ) = − sign(ωn )Θ(ωn x)e−ωn x/vF , vF and b = (1+p)/(1+p+gY ). The conductance contribution where the ωn are Matsubara fequencies ωn = (2n + 1)πT . G(L) in linear order in ln T0 , summed to all orders in g2 , T All internal x -variables are integrated on the positive semi is given by axis. The trace over the product of all isospin matrices in ∞ 2 each fermion loop is taken and a factor of 1/n! is applied G(L) = 1 − Y T 2 ¯ dx1 dx2 dy L(x1 , x2 ; ω)G(x1 − x; ǫ) 4 to each n-th order diagram. The limit Ωm → +0 is taken ǫ,ω 0 at the end. × G(−x1 − x2 ; ǫ − ω)G(x2 − y; ǫ)G(y + x; ǫ + Ω) . (12) The incoming component of ρs (y), J3 (−y), only con(0) tributes in zeroth order: Gi = 1 . Adding the contri- Taking the limit Ω → 0 at x ≫ vF /T one finds 2 bution from the outgoing component one finds T0 T0 −g(1 − Y 2 ) ln ≡ −g(1 − Y 2 ) ln . G(L) = 1 2 (0) 2 2 + gY T T 1+ 1−g G = (1 + cos 2θ) = cos θ = |t| . (9) 2 (13) The diagrams of first order in g2 are shown in Fig. 2. Comparing eq.(13) with eq.(10) we see that the resummaNote that the “vertex correction type” diagram, Fig. 2c, tion corresponds to dressing of the interaction, g → g. D.N. Aristov et al. −x1 −x1 −x1 −x2 −x1 −z −x2 −x1 −z −x2 ¡¢£ ¤¥¦ §¨ + ¡ ¢ £ ¤ ¥ = x2 + x1 + x1 x2 + x1 z x2 x1 z + ··· + x2 ¯ Fig. 3: Ladder series, L(x1 , x2 ; ωn ), describing the combined effect of interaction and barrier, and leading to a linear-inlogarithm ln(T0 /T ) contribution to conductance. Renormalization group approach. In perturbation theory the n-th order contribution in g2 is a polynomial in Λ of degree n. If the theory is renormalizable, all terms with higher powers of Λ should be generated by a renormalization group equation for the scaled conductance, G(Λ), or equivalently for Y (Λ) = 2G(Λ) − 1. Our approximation to the β-function of the RG equation is given by the prefactor of ln T0 in the perturbation theory result (13), with Y T replaced by Y (Λ): dY 2g(1 − Y 2 ) =− = βL (Y ) . dΛ 1 + 1 − g 2 + gY (14) Introducing the Luttinger parameter, K = [(1 − g)/(1 + g)]1/2 , we can rewrite (14) as 1 1 dΛ =− − , (15) dY 2(Y + 1)(K −1 − 1) 2(1 − Y )(1 − K) + +  +  + + +  symm.  Fig. 4: Skeleton Feynman graphs, leading to lowest order logarithmic contribution, g 3 Λ, beyond ladder series. Fig. 3, g → g, the ladder result (14) thus corresponds to only one non-trivial Matsubara frequency summation, which amounts to a ”one-loop” RG correction in the usual classification. In this case the scale invariant linear logarithmic contribution is not sensitive to the cutoff regularization, i.e. when L exceeds the inverse temperature ln L/a → ln(vF /2πT a) and the prefactor of the logarithm is preserved. In other words, these contributions are universal in the RG sense. In the order g 2 , non-trivial ”twoloop” corrections to (14) are absent. In the order g 3 the ”three-loop” contributions are divided into two groups. The first group consists of the first diagram in Fig. 4 and its symmetry-related partners: its contribution to G is π2 3 2 2 24 Λg (1 − Y ) ; this linear-in-Λ correction is again universal. The additional diagrams in Fig. 4 contain both Λ3 and Λ contributions. In this situation the linear-in-Λ terms contributing to the β−function are dependent on the cutoff scheme, i.e. are non-universal. The above cited value c3 = π 2 /12 corresponds to T = 0 , i.e. the hard infrared cutoff L in real space. If we calculate corrections at T ≫ vF L−1 , then we obtain the soft cutoff result c3 = 1/4 instead, in agreement with [16] ; we use this latter value below. Inverting (16) we get dΛ/dY = (βL (Y ))−1 − g h(g, Y ) where h(g, Y ) = c3 − gY O(1)+ . . . for small g has been defined. Notice that in the non-ladder corrections g h(g, Y ) we may substitute g by its renormalized value g → g. Truncating at lowest level beyond the ladder series, we get −1 dΛ/dY ≃ [βL (Y )] − c3 g . (17) which is easily integrated (see below). We note that (15) has a kind of duality symmetry: it is invariant under K → K −1 , Y → −Y . For weak interaction, we expand K ±1 ≃ 1 ∓ g and recover the result by [9]. In the limiting cases of nearly transparent and nearly perfectly reflecting barrier, 1 we recover the results by [2, 6], with G ∼ (T /T0 )2( K −1) 2(1−K) and 1 − G ∼ (T0 /T ) , respectively. A remaining question concerns the existence of additional terms in the RG β-function, not contained in the ladder series (15) (cf. [15]). To answer it, we have calculated G using computer algebra up to fourth order (∼ 40, 000 diagrams). [14] The results are summarized as follows. In higher orders we find both leading (∼ g n Λn ) and subleading (∼ g n Λk , k < n) contributions. Most of these terms correspond to Eq. (14). However, starting from third order of g we find also subleading contributions linear in Λ, which are explicitly different from the form Integrating the latter equation we have (15) and arise from diagrams depicted in Fig. 4. Adding (T /T0 )2(1−K) = Φ(G)/Φ(G0 ) , (18) these terms to the β-function of (14) we obtain K G 4c (1−K) dY (K + G(1 − K)) 3 , Φ(G) = = −g(1−Y 2 )+g 3 (1−Y 2 )2 (c3 −gY c4 +O(g 2 )) , (16) 1−G dΛ with g the above result of the ladder resummation, c3 = π 2 /12 and c4 = 0.238 . . .. Notice the different power of (1 − Y 2 ) in front of the extra terms in (16), which renders these terms irrelevant in the limits G → 0, G → 1. Before proceeding further we should discuss the issue of universality of the logarithmic corrections, i.e. their dependence on the cutoff regularization scheme. Interpreting the ladder summation as a dressing of the interaction, with c3 = 1/4 and the assumed initial condition G = G0 at T = T0 . The equation (18) is the central result of this paper. Let us discuss it in more detail. The above ladder approximation(14) would correspond to setting c3 = 0 in (18). It is seen that the above cited scaling-law dependences of G on T remain asymptotically exact at G → 0 and G → 1. The existence of terms in the β-function beyond the ladder Transport of interacting electrons through a potential barrier G= 4τ 2 √ , 1 + 4τ 2 + 1 + 16τ 2 τ= T |t| T0 1− |r| 2 1 2 2 |r| , (19) where |t|, |r| are the bare transmission and reflection amplitudes. At high temperatures the result reported in [2, 7] tends to the clean limit value for a four-point measurement, 1 G = 2 ; here we assume that the two-point conductance is obtained by multiplying the latter result by a factor of 1/K equal to 2 in order to make contact with our theory. Fixing the overall scale T ∗ in both solutions (τ ≡ T /T ∗) by their high-temperature behavior, G ≃ 1−τ −1 , we show the overall picture in Fig. 5. It is seen that the result of the ladder summation overestimates the renormalization of the barrier, predicting smaller conductance at low T . At the same time the adjustment of the ladder summation by c3 = 1/4, Eqs.(17), (18), (19), gives excellent agreement with the exact solution [2,7], with a relative deviation not exceeding 4 % in the whole temperature range. In case K = 1/3 (g = 4/5), relevant for the description of point conductance between the quantum Hall edge states, [8] we do not have a closed analytic expression for G(T ). It is however possible to make a comparison to our theory. We fix the overall temperature scale by adopting G ≃ 1−τ −4/3 at high temperatures, then we have G ≃ 9τ 4 from our Eq. (18). Multiplying again the result reported in [8] by a factor of 1/K = 3, we have G ≃ 10.0638τ 4 at low T , whose prefactor is within 12 % from our value. We may thus conclude that our result (18) provides a very good approximation even in the strongly interacting case g ∼ 1, i.e. it is sufficient for all practical purposes. Conclusion. In this letter we presented a theory of transport of interacting electrons through a potential barrier, in the linear response regime and at all temperatures, for any short range barrier and for any forward scattering interaction. We employed a representation in terms of chiral fermions, which greatly simplifies the perturbation theory in the interaction parameter g2 . In this way the scale dependent contributions to the conductance may be studied systematically. At low energies long range corre- 1 0.1 2 G, e / h series modifies the behavior of the conductance at intermediate values G ∼ 1/2, and the role of h(g, Y ) ∼ 1/4 can hence be largely viewed as a redefinition of the cutoff energy ln T0 → ln T0 + O(gG0 ), when going from higher to lower T . The duality symmetry, Eq. (15), between the −1 scaling exponents is preserved, T 2(K−1) → T 2(K −1) , in contrast to the recent claim in [17] ; the breaking of duality reported in [17] might be connected to the approximate character of the solution for the set of flow equations there. Let us also compare our findings to exact expressions available from the thermodynamic Bethe Ansatz method. In the particular case of K = 1/2 (g = 3/5) the conductance G is obtained as a closed analytic function of T . [2,7] Our Eq. (18) reduces to a quadratic equation, with relevant solution 0.01 exact solution ladder summation "adjusted" ladder 0.001 0.0001 0.01 1 0.1 10 100 T /T* Fig. 5: Conductance as a function of temperature for K = 1/2. The exact result available from TBA (multiplied by factor 2) is shown by pluses. [2,7] The results of ladder summation, (18) with c3 = 0, are shown by dashed line ; the result of Eq. (18) with the adjusting term c3 = 1/4 is shown by solid line. lations lead to logarithmically diverging terms in the form of powers of Λ = ln T0 . T In particular, the terms linear in Λ may be summed to all orders in g2 , and the prefactor may be identified with the β-function of the renormalization group equation for the conductance as a function of the scaling variable Λ. At intermediate temperatures additional small corrections to the β-function were found in third and higher orders of perturbation theory. Approximating these additional terms by the lowest order (in g2 ) gave excellent agreement with the known exact result at K = 1 . This ap2 pears to be one of the few cases where the β-function can be determined beyond perturbation theory. Our results are in agreement with all known results, where applicable, but go far beyond. The RG-equation may be integrated analytically to give the conductance as an implicit function of the temperature. 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