Transport of interacting electrons through a double barrier in quantum wires 1 arXiv:cond-mat/0212355v2 [cond-mat.str-el] 14 May 2003 2 D. G. Polyakov1,∗ and I. V. Gornyi1,2,∗ Institut f¨r Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany u Institut f¨r Theorie der kondensierten Materie, Universit¨t Karlsruhe, 76128 Karlsruhe, Germany u a We generalize the fermionic renormalization group method to describe analytically transport through a double barrier structure in a one-dimensional system. Focusing on the case of weakly interacting electrons, we investigate thoroughly the dependence of the conductance on the strength and the shape of the double barrier for arbitrary temperature T . Our approach allows us to systematically analyze the contributions to renormalized scattering amplitudes from different characteristic scales absent in the case of a single impurity, without restricting the consideration to the model of a single resonant level. Both a sequential resonant tunneling for high T and a resonant transmission for T smaller than the resonance width are studied within the unified treatment of transport through strong barriers. For weak barriers, we show that two different regimes are possible. Moderately weak impurities may get strong due to a renormalization by interacting electrons, so that transport is described in terms of theory for initially strong barriers. The renormalization of very weak impurities does not yield any peak in the transmission probability; however, remarkably, the interaction gives rise to a sharp peak in the conductance, provided asymmetry is not too high. PACS numbers: 71.10.Pm, 73.21.-b, 73.23.Hk, 73.63.-b I. INTRODUCTION Effects related to the Coulomb interaction between electrons become increasingly prominent in systems of lower spatial dimensionality as their size is made smaller. Recent experimental progress in controlled preparation of nanoscale devices has led to a revival of interest in the transport properties of one-dimensional (1D) quantum wires. Owing to the particular geometry of the Fermi surface, systems of dimensionality one are unique in that the Coulomb correlations in 1D change a noninteracting picture completely and thus play a pivotal role in lowtemperature physics. A remarkable example of a correlated 1D electron phase is the Luttinger-liquid model (for a review see, e.g., Refs. 1,2). In this model, arbitrarily weak interactions ruin the conventional Fermi liquid phenomenology by essentially modifying low-energy excitations across the Fermi surface. As a result, the tunneling density of states develops power-law singularities on the Fermi surface. Moreover, interactions between oppositely moving electrons generate charge- and spin-density wave correlations that lead to striking transport properties of a Luttinger liquid in the presence of impurities. In particular, even a single impurity yields a complete pinning of a Luttinger liquid with repulsive interactions, which results in a vanishing conductance at zero temperature.3,4 In addition to quantum wires, largely similar ideas apply to edge modes in a Hall bar geometry in the fractional quantum Hall regime, which are thought to behave as spatially separated chiral Luttinger liquids.5 Evidence has recently emerged pointing towards the existence of the Luttinger liquid in metallic single-wall carbon nanotubes.6,7 The Luttinger liquid behavior was observed via the power-law temperature and bias-voltage dependence of the current through tunneling contacts attached to the nanotubes. Further technological advances have made possible the fabrication of low-resistance contacts between nanotubes and metallic leads (see, e.g., Refs. 8,9,10,11 and references therein). These recent developments have paved the way for systematic transport measurements in Luttinger liquids with impurities. In this paper, we study electron transport through a double barrier in a 1D liquid. In the 1D geometry, two impurities in effect create a quantum dot inside the system. Resonant tunneling through the two impurities is a particularly attractive setup to investigate the correlated transport in an inhomogeneous Luttinger liquid. Due to the resonant behavior of the current, the interplay of Luttinger-liquid correlations and impurityinduced backscattering should be more easily accessible to transport measurements. Two striking experimental observations have been reported recently. In Ref. 12, a resonant structure of the conductance of a semiconductor single-mode quantum wire was attributed to the formation (with reduction of electron density by changing gate voltage) of a single disorder-induced quantum dot. In Ref. 13, two barriers were created inside a carbon nanotube in a controlled way with an atomic force microscope. In both cases, the amplitude of a conductance peak Gp as a function of temperature T showed power-law behavior Gp ∝ T −γ with the exponent γ noticeably different from γ = 1. The latter is the value of γ expected in the absence of interactions (for a review, see Refs. 14,15) provided T lies in the range Γ ≪ T ≪ ∆, where Γ is the width of a resonance in the transmission coefficient and ∆ is the single-particle level spacing. The width of a conductance peak w followed a linear temperature dependence w ∝ T in both experiments. On the theoretical side, resonant tunneling in a Luttinger liquid was studied previously in a number of papers.3,16,17,18,19,20 In particular, the width Γ ∝ T αe was shown16,19 to shrink with decreasing temperature. The exponent αe depends on the strength of interaction 2 and describes tunneling into the end of a semi-infinite liquid. The dimensionless peak conductance (in units of e2 /h) obeys Gp ∼ Γ/T in the above range of T , which indeed leads to a smaller value of γ = 1 − αe . The reduced exponent γ reported in Ref. 12 was positive (and different for different conductance peaks, in the range γ ∼ 0.5 − 0.8), whereas in Ref. 13 the reported value of γ ≃ −0.7 was negative. More specifically, in Ref. 13, the conductance as a function of the gate voltage showed certain traces of periodicity characteristic to the Coulomb blockade regime.14,15,21 Surprisingly, both the amplitude Gp and the width w were reported to vanish with decreasing T , in sharp contrast to the noninteracting case. While such behavior is known to be possible for very strong repulsive interaction,16,19 the required strength of interaction would then be much larger than expected and indeed reported (see Refs. 6,7,22,23 and references therein) in carbon nanotubes. Roughly a doubling (or even a larger factor) of the expected6,7,22,23 exponent αe , which is αe ∼ 0.6 − 1.0, would be necessary to fit the experimental data. Although the modeling22,23 of carbon nanotubes as a four-channel Luttinger liquid has its share of complications, the observations13 appear to present a puzzle. It was suggested in Ref. 13 that a certain novel mechanism of “correlated tunneling” dominates over the conventional sequential tunneling for T ≫ Γ, leading to a doubling of the exponent α, which might explain the experiment. In the subsequent works24,25 the basic ideas relevant to the resonant tunneling have been questioned. In particular, the lowest-order contribution to the resonance peak conductance Gp for ∆ ≫ T ≫ Γ has been argued24,25 to come from processes of second order in the end-tunneling density of states. While being a characteristic feature of the cotunneling regime far in the wings of the conductance peak, this, however, disagrees with the sequential tunneling picture inside the conductance peak.16,19 This also poses a problem with the persistence3 of perfect transmission through symmetric barriers in the case of weak interaction. Moreover, the main suggestion24 is that taking a finite-range (but falling off fast beyond this range) interaction changes the conventional picture completely, as compared to the standard treatment of a zero-range interaction in the Luttinger liquid model. According to Ref. 24, higher-order tunneling processes in combination with the effects of a non-zero range of interaction dominate the dependence of Gp on T even for T ≫ Γ. A nearly perfect agreement with the experiment data13 has been claimed (for a range of interaction far smaller than the distance between the barriers). However, no explicit formula for the conductance has been given in Ref. 24, while we see no ground for the low-energy long-distance physics to be essentially different if the radius of interaction is made finite. In another recent attempt26 to explain the experiment,13 the observed power law was attributed to the contact resistance. Namely it was noted that the resistance of tunneling contacts to the leads, Rc , and the resistance of the quantum dot add up (if one applies Kirchhoff’s law), so that the anomalous T dependence in Ref. 13 might be explained if Rc ≫ G−1 , where Gp p is understood as the resonance peak conductance of the dot. This is a legitimate proposition, although the measured contact resistance in Ref. 13 was reported to be relatively low.13,24 It is thus desirable to examine the resonant tunneling in a Luttinger liquid in a broad range of temperature down to T = 0 and for various parameters of the barriers. Our purpose in this paper is to analyze transport through a double barrier of arbitrary strength, strong or weak, symmetric or asymmetric, within a general framework of an analytical method applicable to all these situations. There are a variety of techniques to construct the low-energy transport theory.2,27,28 The method we develop here is valid for weak interaction and is based on the renormalization group (RG) approach of Refs. 29,30, which was applied earlier in a variety of contexts.31,32,33 One of the appeals of this kind of theory is that it allows one to treat weak and strong scatterers on an equal footing, which is technically significantly less straightforward in the bosonization method.27 Thus it would be of interest to apply this method to a 1D mesoscopic interacting system with many impurities (previously, the problem of transport in a disordered Luttinger liquid was studied by perturbative in disorder methods based on bosonic field theories in, e.g., Refs. 4,34,35). A double barrier is the simplest “many-impurity” system which exhibits effects essential to the physics of disordered interacting 1D liquids. Very recently, a generalization of the RG approach (initially developed for a structureless barrier29,30 ) has been proposed in Ref. 33 for the model of a single impurity with an energy-dependent scattering matrix. This model is relevant to the resonant tunneling of weakly interacting electrons through a double barrier and accounts properly for the interaction processes within an energy band of width ∆ around the Fermi level. In particular, a nonmonotonic behavior of the conductance peak as a function of T , caused by left/right asymmetry of scattering amplitudes, was investigated in Ref. 33 within the singleresonance model. The fermionic RG33 has been shown to be in accord with the results3,16,19 for the resonant tunneling in Luttinger liquids, obtained earlier by different methods. However, to describe microscopically the spatial structure of a system of two or more impurities, a more systematic analysis is needed. Specifically, one has to develop an approach that would include contributions to the renormalized scattering amplitudes due to interaction processes involving energy transfers larger than ∆. Also, the single-resonance model33 does not describe contributions from tunneling through multiple levels inside the quantum dot (multi-level quantum dots represent a typical experimental situation). Finally, it would be interesting to study transport through weak impurities (and through strongly asymmetric structures) for which the 3 bare transmission coefficient exhibits no pronounced resonant structure. All this adds to our motivation to study the resonant tunneling through a double barrier by generalizing the RG approach of Refs. 29,30. The RG approach enables us to investigate in detail the resonant transport of weakly interacting spinless electrons. Within the fermionic RG approach, we confirm earlier results3,16,19 obtained within bosonic field theories. We examine the conductance through a double barrier for arbitrary strength and an arbitrary shape of the barrier, not restricting ourselves to the model of a single resonant level. In particular, we demonstrate the existence of narrow conductance peaks for two weak impurities, which is in sharp contrast to the noninteracting case. We do not find any trace of the correlated tunneling mechanism proposed in Refs. 13,24,25. We also clarify the relationship between the RG method29,30 and Hartree-Fock (HF) approaches. The paper is organized as follows. First, in Sec. II A, we briefly outline the fermionic RG approach to transport through a single impurity. In Sec. II B, we discuss a HF treatment of transmission through a single impurity and show its inadequacy to the problem. In Sec. III, we turn to a double barrier. We start with a perturbative expansion in Sec. III A and derive the RG equation for transport through the double barrier in Sec. III B. We then analyze contributions to scattering amplitudes from different energy scales, compared to the level spacing inside the quantum dot and the resonance width, in Secs. III C–III E. In Sec. III F, we concentrate on the case of weak impurities. Finally, in Sec. IV we calculate the amplitude and the shape of the resonance conductance peaks. II. A. SINGLE IMPURITY Renormalization group: Basic results We begin with a brief description of transport through a single structureless impurity in the spirit of the RG approach29,30 . Without interaction, the impurity is characterized by a transmission coefficient t0 and reflection coefficients rL0 and rR0 , from the left and from the right respectively (we put the impurity at the center of coordinates, x = 0). Suppose that the energy dependence of the bare scattering matrix can be neglected far from the boundaries of an energy band (−D0 , D0 ) around the Fermi level. The energy scale D0 serves as the ultraviolet cutoff of RG transformations and, physically, is of the order of vF /d (throughout the paper we put = 1) or the Fermi energy ǫF , whichever is smaller.36 Here d is the radius of interaction and vF is the Fermi velocity. Deep inside the band (−D0 , D0 ), we linearize the energy spectrum around the Fermi level. The differential RG equations29,30 read ∂t/∂L = −αt R , ∂rL,R /∂L = αrL,R T , (1) where L = ln(D0 /|ǫ|), the energy ǫ is measured from the Fermi level, the transmission probability T = |t|2 , and R = 1 − T. The boundary conditions at L = 0 set the scattering amplitudes at their noninteracting values t0 , rL0 , and rR0 . Throughout the paper we consider spinless electrons, for which the interaction constant is α = (Vf − Vb )/2πvF , (2) where Vf and Vb are the Fourier transforms of a pairwise interaction potential yielding forward (Vf ) and backward (Vb ) scattering. The forward scattering does not lead to transitions between two branches of right- and leftmovers, whereas the backscattering does. We assume that α > 0. Note that, for spinless electrons, the interactioninduced backward scattering and forward scattering relate to each other as direct and exchange processes, so that the backscattering only appears in the combination Vf − Vb and thus merely redefines parameters of the Luttinger model (formulated1,2 in terms of forwardscattering amplitudes only). In particular, the backscattering does not lead to any RG flow for α. For spinful electrons this is valid only to one-loop order.1,2 It is also worth mentioning that for a point interaction Vf = Vb , so that α = 0, hence for spinless electrons one has to start with a finite-range interaction. However, the RG flow for the scattering matrix (1) occurs for |ǫ| vF /d and is governed solely by the constant α. It follows that on low-energy scales one can effectively consider the interaction as local, Veff (x − x′ ) = 2παvF δ(x − x′ ), and formally deal exclusively with forward scattering. A non-zero range of interaction for kF d ≫ 1 can manifest itself only in the boundary conditions to Eqs. (1) at |ǫ| ∼ D0 = vF /d and therefore does not affect the singular behavior of the renormalized scattering matrix at ǫ → 0. We assume that the Coulomb interaction between electrons is screened by external charges (e.g., by metallic gates, in which case d is given by the distance to the gates) and that a resulting α ≪ 1. For a treatment of the unscreened Coulomb interaction, see, e.g., Refs. 18,37. Integration of Eqs. (1) gives29,30 R R0 = T T0 D0 |ǫ| 2α . (3) The phases of the scattering amplitudes are not affected by the renormalization. Equations (1) are equivalent to a one-loop renormalization, so that Eq. (3) is valid to first order in interaction ∼ O(α) in the exponent of the power-law scaling. As follows from Eq. (3), whatever the initial values of T0 , at α > 0 they all flow to the fixed point of Eqs. (1) at zero transmission,3 T = 0 at ǫ = 0, see Fig. 1. In the limits of a weak impurity (both R0 ≪ 1 and R ≪ 1) and a strong tunneling barrier (T0 ≪ 1), Eq. (3) coincides with the RG results obtained by bosonization,3 provided α ≪ 1. Equation (3) gives the transmission probability for electrons with energy ǫ at temperature T = 0. For finite T , the renormalization stops at |ǫ| ∼ T . 4 the regimes II and III become −2α → 2[(1 + 2α)−1/2 − 1] and 2α → 2αe = 2[(1 + 2α)1/2 − 1], respectively.2 To make a connection with experiment, it is worth noting that in the Luttinger liquid model (Vb = 0) for N channels with the same Fermi velocity and no interchannel transitions, the constant α is related to the exponent αe which describes tunneling into the end of the liquid (see Sec. I) as follows: 1 T0 I II T III 0 αe = N −1 [ (1 + 2N α)1/2 − 1 ] . Dr D0 Dp ε impurity non−perturbative interaction non−pertubative FIG. 1: Schematic behavior of the transmission coefficient T(ǫ) for weak interaction and a single weak impurity with a bare coefficient T0 ≃ 1 in three different regions, according to Eqs. (3),(4): (I) a small logarithmic correction δT; (II) powerlaw scaling of the correction δT; (III) power-law vanishing of T. The Fermi level is at ǫ = 0. Beyond the microscopic scale D0 , it is instructive to introduce two more energetic scales that characterize the renormalization of the transmission coefficient by weak interaction, Dp and Dr (Fig. 1): ln(Dp /D0 ) = −1/α , 1/2α Dr /D0 = R0 . (4) The energy Dp defines the scale on which a perturbation theory in interaction breaks down. If |ǫ| Dp , the interaction requires a non-perturbative treatment. The scale Dp does not depend on R0 and is much smaller than D0 for α ≪ 1. The energy Dr defines the scale on which a perturbation theory in the impurity strength breaks down. If |ǫ| Dr , a weak impurity with R0 ≪ 1 yields strong reflection, R ∼ 1. Provided R0 ≪ 1, the scales Dp and Dr are parametrically different and, for any α, Dr ≪ Dp . We will see in Sec. III that the scale Dr is of central importance in RG theory for a double-barrier structure. To summarize this section, there are three different types of behavior of T with ǫ, as illustrated in Fig. 1: I. Dp |ǫ| D0 , a small logarithmic correction δT ∼ αR0 ln(|ǫ|/D0 ) is perturbative in both interaction and the impurity strength; II. Dr |ǫ| Dp , power-law scaling of the correction δT ∼ −R0 (|ǫ|/D0 )−2α is perturbative in the impurity strength but non-perturbative in interaction; III. |ǫ| Dr , power-law vanishing of T ∼ (|ǫ|/Dr )2α is non-perturbative in both the impurity strength and interaction. If the interaction is not weak (i.e., α is not small), the regime I shrinks to zero while the scaling exponents in (5) E.g., N = 1 for spinless and N = 2 for spin-degenerate electrons in a quantum wire with a single mode of transverse quantization,2 N = 4 in the Luttinger-liquid model of the armchair carbon nanotube.22,23 Note that αe = α for any N in the limit of weak interaction (α → 0). For a discussion of the tunneling into the end of a multi-channel Luttinger liquid with non-equivalent channels see Ref. 38. B. Hartree-Fock versus renormalization group We now turn to a conceptually important point that is highly relevant to our calculation of the resonant tunneling in Sec. III. The basic idea of Refs. 29,30 was to relate the RG transformations (1) to the scattering of an electron off Friedel oscillations created by the impurity screened by other electrons.39 Indeed, to first order in α, when the interaction-induced correction to the density distribution may be neglected, the HF correction to the scattering amplitudes is logarithmically divergent ∝ ln(D0 /|ǫ|). The essence of this singularity is a slow decay of the Friedel oscillations, which behave in 1D, in the absence of interaction, as |x|−1 sin(2kF |x| + φF ). Here kF is the Fermi wavevector, |x| the distance to the impurity, and φF a constant phase shift. This observation suggests that a HF treatment of the problem would yield a series of leading logarithms αn Ln . In the HF problem, it would be sufficient to solve iteratively in α a nonlinear Schroedinger equation (H0 + UHF {ψ} − ǫ)ψ = 0, where H0 is the noninteracting Hamiltonian. The static selfconsistent scattering potential is Ui + UHF , where Ui is the bare impurity potential and the nonlocal HF term reads UHF (x, x′ ) = − V (x − x′ )ρ(x, x′ ) (6) + δ(x − x′ ) dx1 V (x − x1 )ρ(x1 , x1 ) . Here V (x) is a potential of pairwise interaction between ∗ particles, ρ(x, x′ ) = q ψq (x′ )ψq (x) is a sum over all occupied states, and ρ(x1 , x1 ) in the Hartree term is the total electron density which includes the Friedel oscillations. Self-consistency requires that both the wave functions ψq and the scattering potential UHF be corrected on every step of the HF iterative procedure. In effect, Ref. 30 (see also Ref. 32) suggests that the HF procedure, being carried out in the “leading-log” approximation, would reproduce Eq. (3). In particular, the HF calculation30 of 5 second order in α gives a correction ∼ α2 L2 and, according to Ref. 30, this HF correction does coincide with the one obtained from an expansion of Eq. (3) up to terms ∼ O (α2 ). Following this logic, the RG (1) is essentially an effective method of solving the HF problem self-consistently. In fact, however, while the HF theory is indeed a “logarithmic theory” in the sense that it can be expanded in a series of leading logarithms αn Ln , the series is very much different from the RG solution (3), even to one-loop order. To demonstrate this, let us represent HF solutions for waves ψ± (x) which are incident on the scatterer from the left (+) and from the right (−) as ψ± (x) = a±+ (x)e+ (x) + a±− (x)e− (x), where e± (x) = exp[±i(kF + ǫ/vF )x], aµν (x) are amplitudes −1 varying slowly on a scale of kF for kF |x| ≫ 1 and satisfying the scattering boundary conditions a++ (−∞) = a−− (∞) = 1, a+− (−∞) = a−+ (∞) = 0. Separating the slow and fast variables, the HF equations for ǫF /|ǫ| ≫ kF |x| ≫ 1 are written in the leading-log approximation as ∂a±,+ /∂Lx = (α/2) η a±,− , ∂a±,− /∂Lx = (α/2) η ∗ a±,+ , (7) where Lx = ln(D0 |x|/vF ) and the function η(x) describes the envelope n(x) = |η(x)|/2π|x| of the Friedel oscillations. The latter are given for kF |x| ≫ 1 by 1 Im {η(x)e2ikF x } = 2πx 0 1 dǫ Re {(a∗ aµ− )e−2i(ǫF +ǫ)x/vF } . (8) µ+ πvF µ=± −D0 Solutions of Eqs. (7) for x > 0 and x < 0 are matched onto each other by means of the bare scattering matrix. Equivalently, Eqs. (7) can be cast in the form ∂ ln η/∂Lx = αI/2 , ∂I/∂Lx = 2α|η|2 (9) with η = a++ a∗ + a−+ a∗ and I = µν |aµν |2 . The +− −− function I(x) is a contribution of states close to the Fermi energy to a smooth part of electron density. Equations (7),(9) can be solved exactly. The scattering amplitudes and the shape of the Friedel oscillations are then found self-consistently. However, the rather intricate HF solution appears to be a qualitatively wrong approximation in our problem of impurity scattering in an interacting 1D liquid. The difference between the RG and HF solutions is already seen for a weak impurity with R0 ≪ 1. While the RG gives for |ǫ| ≫ Dr a power-law behavior of the reflection amplitude rL,R = rL0,R0 eαL , (10) the HF expansion yields a geometric progression rL,R = rL0,R0 1 . 1 − αL (11) Equation (11) can be obtained also by summing up ladder diagrams in the particle-hole channel. Note a pole characteristic to HF approaches. We thus see that even for weak interaction the HF solution correctly describes the scattering by a screened impurity only for |ǫ| ≫ Dp , where it gives a small perturbative correction ∼ αL. At the next order in α, the HF expansion does generate a term ∼ rL0,R0 α2 L2 in rL,R but, in contrast to Ref. 30, the numerical coefficient in front of it is a factor of 2 larger than in Eq. (3), which signifies a breakdown of the HF approach beyond the simplest Born approximation. It is worth noting that in Ref. 32 another HF-type (“non-selfconsistent HF”) scheme was employed to substantiate the RG procedure. As mentioned above, singleelectron wave functions within the self-consistent HF 0 0 ˆq approach obey ψq = ψq + G0 UHF ψq , where ψq and ˆq the Green’s function operator G0 describe noninteracting electrons. Let us write the correction to the transmission amplitude of second order in αL as δt(ǫ) = −Ct0 R0 (αL)2 . The self-consistent HF approximation gives C = T0 +1/2, different from C = (3T0 − 1)/2 following from the RG expansion (3). In Ref. 32, only UHF is renormalized while 0 ψq remain unperturbed, i.e., ψq → ψq in the right-hand side of the above HF equation. This yields C = 1, which accidentally coincides with the RG result for a weak impurity. However, for arbitrary T0 this approach can be seen to fail as well, already at order O (α2 L2 ). It follows that the RG (1) in fact cannot be obtained from the above HF schemes, i.e., the RG is not a method of solving the HF equations. There are other scattering processes, not captured by the HF approximation, that at higher orders in α almost compensate the HF result: a comparison of Eqs. (10),(11) says that the coefficients cn in the corresponding sums n cn (αL)n are different in the RG and HF expansions by a factor of n!. One can see that the HF approach misses important scattering processes also by noticing that Friedel oscillations in a Luttinger liquid decay in the limit of large |x| ≫ vF /Dr as |x|−1+α , more slowly40,41 (for the repulsive interaction) than in a Fermi liquid. The Bragg reflection by a potential created by the Friedel oscillations can then be easily seen to yield a power-law behavior of the transmission coefficient only if α is put equal to zero in the exponent of the Friedel oscillations. For any α > 0 the Bragg reflection leads to an exponential decay of the wave functions, ln |ψ| ∝ −|x|α , instead of a power law required by self-consistency and by Eq. (3). We can summarize the above observations by stating that, although it is certainly appealing to think of the RG (1) as being associated with scattering off Friedel oscillations, the mechanism of the RG is more complicated. In fact, in a closely related calculation of various correlation functions of a clean Luttinger liquid without impurities the inapplicability of the HF theory was realized long ago:42,43,44 it is the interaction in a Cooper (particle-particle) channel that interferes with the HF interaction. As a result, instead of the summation of a 6 HF ladder one has to use a much more involved parquet technique.42,43,45 In the present problem, the parquet summation is equivalent to the RG (1). A rigorous microscopic justification of the RG (1) can be done by using Ward identities in a diagrammatic approach and will be given elsewhere.46 Our purpose in Sec. III is to extend the RG (1) to the case of two impurities. Green’s function is characterized by bare scattering amplitudes t0 (ǫ), rL0,R0 (ǫ). To first order in α, transforming to the (x, ǫ)-representation, which is most convenient in the present context, we have a correction to Gµν (xf , xi ; ǫ) (summation over branch indices assumed): ∞ δGµν (xf , xi ; ǫ) = dx Gµµ′ (x, xi ; ǫ) −∞ III. × Σµ′ ν ′ (x)Gν ′ ν (xf , x; ǫ) , DOUBLE BARRIER (12) where Consider now two potential barriers located at x = 0 and x = x0 and let the distance x0 be much larger than the width of each of them. Clearly, the spatial structure itself yields an energy dependence of the total (describing scattering on both impurities) amplitudes t(ǫ) and rL,R (ǫ), even if such a dependence may be neglected for each of the impurities, as was assumed in the derivation of Eqs. (1). Specifically, without interaction the energy ∆ = πvF /x0 gives a period of oscillations in the total scattering amplitudes with changing Fermi level. If the impurities are strong, ∆ is the level spacing inside a quantum dot formed by the barriers and the period of resonant tunneling oscillations. It follows that an RG description of scattering off a double barrier requires a generalization of the RG29,30 to the case when the bare amplitudes t0 (ǫ) and rL0,R0 (ǫ) are energy dependent. A question, however, arises if it is at all possible to construct the RG theory for a compound scatterer in terms of only total S-matrix, as in Eqs. (1). Put another way, the question is if total scattering amplitudes generated by RG transformations are expressed in terms of themselves only. As we will see below, the answer depends on the parameter ∆/D0 . We recall that D0 in our problem is the smallest of two energy scales given by ǫF and vF /d, where d is the radius of interaction. If ∆ ≪ D0 , so that there are many resonances within the band (−D0 , D0 ), the RG transformations generate more terms than are encoded in the total S-matrix. This is a generic case we are interested in. On the other hand, if ∆ ≫ D0 (e.g., when d ≫ x0 ), one would have from the very beginning a model of a single impurity with no spatial structure but with possibly energy dependent scattering amplitudes. In that case, it is sufficient to deal with total amplitudes only (see Secs. III D,III E). For the case of a single resonance this is a model studied, e.g., in Ref. 47 (for an exactly solvable case of the Luttinger liquid parameter g = 1/2) and Ref. 33 (for a weak interaction, 1 − g ≪ 1). A. Perturbative expansion Building on our knowledge of the single impurity case, we start the derivation of a double barrier RG with a calculation of perturbative in α corrections to a timeordered single-particle Green’s function Gµν (x, x′ ) = † −i T ψν (x)ψµ (x′ ) , where x = (x, t) and µ, ν = ± label two branches of right (+) and left (−) movers. The bare D0 dǫ Gµν (x, x; ǫ) . Σµν (x) = iα vF (13) −D0 In the Luttinger liquid model, only forward scattering due to interaction is present: we explicitly assume this in Eqs. (12),(13) by writing the self-energy that depends on two branch indices only [backward scattering can be straightforwardly incorporated for spinless electrons, see Eq. (2)]. As explained in Sec. II A, we also assume in Eq. (12) that the interaction is effectively shortranged and write the self-energy as a spatially local quantity. Indeed, the perturbative logarithmic correction to Gµν (xf , xi ; ǫ) comes from energies |ǫ| vF /d, i.e., from spatial scales |x| larger than d. We start with the case d ≪ x0 (we will return to the simpler case d ≫ x0 in Sec. III D). Moreover, in the following, we neglect the forward scattering of electrons belonging to the same branch: to one-loop order, such processes can be seen to yield only a non-singular renormalization of the bare parameters. The forward-scattering interaction that generates RG transformations similar to Eqs. (1) is that of electrons from different branches.48 We thus formulate an impurity problem with only non-diagonal couplings Σ+− (x) and Σ−+ (x). The scattering amplitudes are related to Gµν (xf , xi ; ǫ) for ǫ > 0 (we measure ǫ from the Fermi level upwards) as rL (ǫ) = ivF G+− (xf , xi ; ǫ)e−iǫ(xf +xi )/vF |xf ,xi →−∞ , rR (ǫ) = ivF G−+ (xf , xi ; ǫ)eiǫ(xf −xi )/vF |xf ,xi →∞ , t(ǫ) = ivF G++ (xf , xi ; ǫ)e −iǫ(xf −xi )/vF (14) |xf →∞,xi →−∞ , and similarly for ǫ < 0 by changing ± → ∓ in branch indices of Gµν . We count the phases of the reflection amplitudes from x = 0. Since the integral over x in Eq. (12) involves integration over the interval of x inside the dot, 0 < x < x0 , corrections δt(ǫ), δrL,R (ǫ) cannot be expressed solely in terms of the bare amplitudes t0 (ǫ) and rL0,R0 (ǫ). A closed set of equations can be written by introducing amplitudes to stay inside the dot Aµ,−µ (ǫ) = ivF Gµ,−µ (x, x; ǫ)e2iµǫx (15) (we need only nondiagonal amplitudes Aµ,−µ ), amplitudes to escape from the dot, to the left or to the right, d± (ǫ) = ivF Gµ,± (xf , x; ǫ)e−iǫ(xf −µx) |xf →±∞ , µ (16) 7 and, similarly, amplitudes to get into the dot from outside b± (ǫ) = ivF G±,µ (x, xi ; ǫ)eiǫ(xi ∓x) |xi →∓∞ . µ (17) The amplitudes (14)–(17) are constrained by unitarity and, moreover, bν (ǫ) = dν (ǫ) by time reversal symmeµ −µ try which we assume throughout the paper. In Eqs. (15)– (17), x lies within the dot. Without interaction, the amplitudes (15)–(17) do not depend on x. This property is preserved in the leading-log approximation. The first-order corrections to t(ǫ) and rL (ǫ) read δt(ǫ) = − D0 α 2 −D0 dǫ′ L+ (ǫ, ǫ′ ) ǫ − ǫ′ (18) ∗ ∗ +θ(−ǫ′ )t(ǫ)[rR (ǫ)rR (ǫ′ )χǫ−ǫ′ + rL (ǫ)rL (ǫ′ )] , δrL (ǫ) = − α 2 D0 −D0 dǫ′ L− (ǫ, ǫ′ ) + θ(ǫ′ )rL (ǫ′ ) ǫ − ǫ′ (19) 2 ∗ ∗ +θ(−ǫ′ )[t2 (ǫ)rR (ǫ′ )χǫ−ǫ′ + rL (ǫ)rL (ǫ′ )] , and similar equations can be written for corrections to rR (ǫ) and the amplitudes defined in Eqs. (15)–(17). Here θ(ǫ) is the step function and χǫ = exp(2πiǫ/∆) . (20) The terms Lµ (ǫ, ǫ′ ), which correspond to the integration over x in (12) inside the dot, are given by Lµ (ǫ, ǫ′ ) = b− (ǫ)A+− (ǫ′ )dµ (ǫ)(χǫ−ǫ′ − 1) + − + b− (ǫ)A−+ (ǫ′ )dµ (ǫ)(1 − χǫ′ −ǫ ) . − + (21) The amplitudes to stay inside the dot satisfy the relation Aµ,−µ (ǫ) = −A∗ (−ǫ) and it is useful to decompose −µ,µ them as Aµ,−µ (ǫ) = θ(ǫ)Bµ,−µ (ǫ) + θ(−ǫ)Cµ,−µ (ǫ) . (22) We thus arrive at a closed system of perturbative equations written in terms of quantities describing the dot as a whole, without directly referring to parameters characterizing two barriers. B. Renormalization group for a double barrier Generically (singular exceptions are discussed below), the above first-order corrections diverge logarithmically as ǫ → 0, which implies that higher orders of the perturbation theory are important. Naively, one could try to treat Eqs. (18),(19) and other perturbative equations self-consistently, i.e., not as expressions for small corrections δt(ǫ), etc., but as equations to be solved for t(ǫ) = t0 (ǫ) + δt(ǫ), etc., where δt(ǫ) is not necessarily small. However, this would not correctly describe the case of strong impurities, either the case of a dot formed by two barriers or even that of a single strong barrier. To see this in the simpler case of a single impurity, take the limit x0 → 0. The terms Lµ (ǫ, ǫ′ ) and factors χǫ−ǫ′ then drop out in Eqs. (18),(19) and the self-consistent equations acquire the form t(ǫ) = t0 (ǫ) (23) −|ǫ| ′ dǫ ∗ ∗ t(ǫ)[ rR (ǫ)rR (ǫ′ ) + rL (ǫ)rL (ǫ′ ) ] , ′ −D0 ǫ α D0 dǫ′ rL (ǫ) = rL0 (ǫ) + rL (ǫ′ ) (24) 2 |ǫ| ǫ′ + α 2 + α 2 −|ǫ| dǫ′ 2 ∗ 2 ∗ [ t (ǫ)rR (ǫ′ ) + rL (ǫ)rL (ǫ′ ) ] . ′ −D0 ǫ As a comparison of Eqs. (18),(19) and (23),(24) shows, the latter do not describe two impurities since they miss the terms Lµ (ǫ, ǫ′ ) and the factors χǫ−ǫ′ . Moreover, Eqs. (23),(24) do not describe even a single strong structureless (with no dependence of the bare amplitudes on ǫ) impurity. This can be checked straightforwardly, e.g., by putting all reflection coefficients in the integrand of Eq. (23) equal to unity: in this limit the integration over ǫ′ in Eq. (23) gives a geometric progression of logarithms, instead of reproducing Eq. (1) for t(ǫ). Also, plugging in Eq. (3) into the integrand of Eq. (24) can be easily seen to give a logarithmic divergence at ǫ → 0, instead of a power law. The logarithmic divergencies are characteristic to HF approaches discussed in Sec. II B. Equations (23),(24), however, do describe correctly a weak structureless impurity, but only for |ǫ| ≫ Dr . We now derive nonperturbative amplitudes for a double barrier using an appropriate RG scheme. To account for the ǫ dependence of the bare amplitudes, the derivation of the RG from the perturbative results (18),(19) necessitates introduction of two energies, ǫ and D. The latter is a flow parameter in RG transformations, i.e., an ultraviolet cutoff rescaled after tracing over states with energies ǫ′ in the interval |ǫ′ | ∈ (D, D0 ). The renormalization stops at D = max{|ǫ|, T }. The essence of the RG procedure29,30 is a perturbative treatment of contributions to the renormalized amplitudes at energy ǫ from all states with energies ǫ′ in the interval |ǫ′ | ∈ (D, ΛD), starting from D = D0 /Λ, such that Λ ≫ 1 but α ln Λ ≪ 1.49 The RG equations thus differ from both the HF equations and Eqs. (23),(24) in that all the amplitudes depend on D and, moreover, the HF-type integration over projected states with energies ǫ′ only goes over the interval |ǫ′ | ∈ (D, ΛD) instead of (0, D0 ). In effect, each step of the RG transformations accounts for the scattering off the Friedel oscillations in a finite spatial region, |x|, |x − x0 | ∈ (vF /ΛD, vF /D). Moreover, the Friedel oscillations are only partly modified, through the (already performed) renormalization of the reflection amplitudes at energies larger than D. At the same time, the scattering matrix at energies smaller than D is taken at its bare value. This should be contrasted with the HF approach, where the scattering amplitudes are determined by interaction processes on all energy scales on every step of the HF iterations. 8 The system of one-loop RG equations for a double barrier reads ∂t(ǫ, D) ˆ = Iǫ′ (ǫ, D) L+ (ǫ, ǫ′ ; D) ∂LD ∗ + θ(−ǫ′ )t(ǫ, D)[ rR (ǫ, D)rR (ǫ′ , D) χǫ−ǫ′ ∗ + rL (ǫ, D)rL (ǫ′ , D) ] , (25) ∂rL (ǫ, D) ˆ = Iǫ′ (ǫ, D) L− (ǫ, ǫ′ ; D) + θ(ǫ′ )rL (ǫ′ , D) ∂LD ∗ + θ(−ǫ′ )[ t2 (ǫ, D)rR (ǫ′ , D) χǫ−ǫ′ + 2 ∗ rL (ǫ, D)rL (ǫ′ , D) ] , (26) and similar equations for other amplitudes. Here LD = ln(D0 /D) and we introduced D dependent amplitudes t(ǫ, D), etc., so that integration of Eqs. (25),(26) over D acts on all the amplitudes [which should be contrasted with Eqs. (23),(24), where the integration acts on only one amplitude in the products of three]. All amplitudes in Lµ (ǫ, ǫ′ ) (21) are now also functions of D. The integral ˆ operator Iǫ′ (ǫ, D) is defined as α ˆ Iǫ′ (ǫ, D) = − 2 ln Λ ΛD −D + D −ΛD dǫ′ {. . .} , ǫ − ǫ′ (27) where Λ ≫ 1 is restricted by the condition α ln Λ ≪ 1. The theory is renormalizable if the action of the operator ˆ Iǫ′ (ǫ, D) to leading order is independent of Λ, which is the case for the present problem within the leading-log approximation. Needless to say, in the limit x0 → 0 [i.e., L± (ǫ, ǫ′ ; D) → 0 and χǫ−ǫ′ → 1] Eqs. (25),(26) describe a single impurity. At finite temperature T , one should substitute the Fermi distribution function nF (ǫ) for the step functions in Eqs. (25),(26) and also in Eq. (22) according to θ(±ǫ) → nF (∓ǫ). The factor (ǫ − ǫ′ )−1 in Eq. (27) effectively stops the renormalization at D ∼ |ǫ|, while the factors nF (±ǫ′ ) do so at D ∼ T , otherwise the renormalization can be carried out down to D = 0. The infrared cutoff at D ∼ T establishes a characteristic spatial scale of LT = vF /T . Due to the thermal smearing, the Friedel oscillations decay exponentially on a scale of LT [which can be seen from Eq. (8) if one plugs in nF (ǫ) into the integrand and extends the integration to ǫ > 0]. The absence of renormalization at D ≪ T is closely related to the thermal cutoff of interaction-induced corrections50 to the conductance in higher dimensions. The RG equations (25),(26) should be solved with proper boundary conditions at D = D0 , where the renormalization starts from the bare values of the amplitudes (14)–(17). For the double-barrier problem, the boundary conditions for the transmission and reflection amplitudes are: t0 (ǫ) = t1 t2 , S(ǫ) rL0 (ǫ) = r1 + r2 t2 1 χǫ , S(ǫ) (28) where ′ S(ǫ) = 1 − r2 r1 χǫ , (29) ∗ and rR0 (ǫ) = −rL0 (ǫ)t0 (ǫ)/t∗ (ǫ) by unitarity. The coef0 ′ ficients r1,2 (r1,2 ) are the noninteracting reflection amplitudes from the left (right) and t1,2 are the transmission amplitudes of each of the two barriers, respectively. Similarly for other amplitudes: d+ (ǫ) = + d+ (ǫ) t0 (ǫ) − = , ′ r1 t1 d− (ǫ) t0 (ǫ) + = , r2 χǫ t2 ′ t0 (ǫ)r1 t0 (ǫ)r2 χǫ , C+− (ǫ) = − B+− (ǫ) = t1 t2 t1 t2 d− (ǫ) = − (30) ∗ . We count the phases of rL,R from x = 0 and the phases of r1,2 are defined for an impurity sitting at x = 0. C. Separate renormalization of two impurities: D ≫ ∆ We are now in a position to solve the system of RG equations (25),(26) by integrating out all states with energies |ǫ′ | max{|ǫ|, T }. We begin with the case D0 ≫ ∆, which is a typical case unless interaction is very long ranged. We proceed in two steps. Let us first integrate over D ≫ ∆. This can be done for arbitrary ǫ. Specifically, if |ǫ| ∆, this will already solve the problem by providing us with fully renormalized amplitudes. In the more interesting case of |ǫ| ∆, we will only sum up contributions to the renormalized amplitudes from states with |ǫ′ | ∆ and, as a second step, will have to proceed with renormalization for D ∆. Since the renormalization for D ≫ ∆ involves many resonant levels, the amplitudes contain slowly varying parts and parts oscillating rapidly with changing ǫ′ on a scale of ∆. Integration over ǫ′ in Eq. (27) allows us to separate the slow and fast variables: as a result, the dependence of the amplitudes on D will be slow on the scale of ∆. However, even after that the RG equations look rather cumbersome. To construct the solution to these equations, note that an important parameter D/Drmin is available, where Drmin = min{Dr1 , Dr2 } (31) and Dr1,2 are defined for each of two barriers by Eq. (4). If both barriers are initially (i.e., at D = D0 ) strong (|t1,2 | ≪ 1), then this parameter is small for all D < D0 . However, if one or both of the barriers are initially weak, there is a range of D ∈ (Drmin , D0 ) where at least one barrier still remains weak. It is useful first to examine some general properties of integrals over ǫ′ that appear in the course of renormalization. Let us return to the perturbative expansion 9 (18),(19). We see that the averaging over ǫ′ involves two types of integrals D0 I1 = |ǫ| D0 dǫ′ 1 , ǫ′ S(ǫ′ ) I2 = |ǫ| dǫ′ χǫ′ , ǫ′ S(ǫ′ ) (32) where S(ǫ) is given by Eq. (29) and I1,2 are related by ′ I2 = (I1 − L)/r1 r2 . The integrals (32) are evaluated in different ways depending on whether at least one of the ′ barriers is weak (so that |r1 r2 | ≪ 1) or both barriers are ′ strong (|r1 r2 | ≃ 1). In the former case the integrand of I1 is only slightly modulated, so that one can expand the factor S −1 (ǫ′ ) and average over harmonics χn (ǫ′ ). Then only zero harmonics give rise to singular (logarithmic) corrections and in the leading-log approximation we have I1 = L , I2 = 0 . and the scaling exponent is now half that for D ≫ Drmin . We thus have two solutions given by Eqs. (37),(38) and Eq. (39), respectively, that match onto each other at D ∼ Drmin . Due to the slow power-law dependence, for α ≪ 1 the matching is exact. Lε Lε (a) (33) In the opposite case of strong barriers, sharp resonances appear that are described by a Breit-Wigner formula for S −1 (ǫ′ ) and yield |I1 | ≃ |I2 |: ′ I2 = −L/2r1 r2 . I1 = L/2 , (34) The situation repeats itself in the RG equations (25),(26). The difference in the factor of 1/2 between the values of I1 in Eqs. (33),(34) implies that the renormalization should be carried out differently (see Fig. 2) in the regions D ≫ Drmin and ∆ ≪ D ≪ Drmin . For D ≫ Drmin , similarly to Eq. (33), after the averaging over ǫ′ only zero harmonics contribute to the renormalization. The solution at D ≫ Drmin is then simple: it has the form of Eqs. (28),(30) with the reflection and transmission amplitudes of each of two barriers renormalized separately (Fig. 2a), according to the RG (1),(3) for a single impurity: ∂t1,2 (D) = −αt1,2 (D)|r1,2 (D)|2 , ∂LD ∂r1,2 (D) = αr1,2 (D)|t1,2 (D)|2 . ∂LD (35) (36) One can check straightforwardly that Eqs. (28),(30) (describing the Fabry-Perot resonance) with the replacement (D/D0 )α t1,2 , (37) [ |r1,2 + (D/D0 )2α |t1,2 |2 ]1/2 ′ r1,2 (r1,2 ) ′ (38) r1,2 (r1,2 ) → [ |r1,2 |2 + (D/D0 )2α |t1,2 |2 ]1/2 t1,2 → |2 solve Eqs. (25),(26) averaged over harmonics for D ≫ max{Drmin , ∆}. On the other hand, if Drmin ≫ ∆, there is an interval of D ∈ (∆, Drmin ) in which each of the impurities is strongly reflecting (Fig. 2b), so that the averaged equations are again simplified by summing over resonance poles, similarly to Eq. (34). We get an independent renormalization of t1,2 (D) according to t1,2 (D)/t1,2 (Drmin ) = (D/Drmin )α/2 , (b) (39) ∆ FIG. 2: Sketch of different stages of the RG procedure for a double barrier for D ≫ ∆ ≫ T , Eqs. (37)-(39). The Friedel oscillations shown schematically around each of the barriers yield a renormalization of the scattering amplitudes and are renormalized themselves as well. At energy ǫ, the renormalization comes from spatial scales smaller than Lǫ = vF /|ǫ| (these regions are marked by the solid lines). (a) Separate renormalization of two weak barriers. Since there is no energy quantization between the barriers, the Friedel oscillations both inside and outside the dot contribute to the renormalization and the barriers “do not talk to each other”. The scaling exponent α is the same as for a single barrier, Sec. II A. (b) Separate renormalization of two strong barriers. This figure describes either initially strong barriers or those that are initially weak but both become strongly reflecting due to the renormalization. The inner part of the dot with a discrete energy spectrum does not contribute to the renormalization. The renormalization of each of the barriers is governed by the exponent α/2, twice as small as for weak barriers. We conclude that the key difference between the renormalization for D larger and smaller than Drmin is that for D ≫ Drmin the transmission amplitudes for each barrier are renormalized with the exponent α, whereas for D ≪ Drmin with the exponent α/2. In both limiting cases, two barriers are renormalized separately for D ≫ ∆ (Fig. 2). It is worth stressing that generally the independent renormalizations of two barriers cannot be derived from RG equations written in terms of only t(ǫ) and rL,R (ǫ), i.e., the terms Lµ (ǫ, ǫ′ ; D) are of crucial importance in the derivation of Eqs. (35)–(38). However, the renormalization of resonant tunneling amplitudes for 10 energies near resonances allows for another formulation which involves t(ǫ) and rL,R (ǫ) only, we will return to this issue in Sec. III D. If ǫ is close to one of resonant energies, Eqs. (25),(26) can be further simplified by expanding χǫ near the resonance: the renormalized amplitudes for strong barriers take then the form of Breit-Wigner amplitudes with D dependent widths Γ1,2 (D) = (∆/2π)|t1,2 (D)|2 ∝ Dα . Specifically, for initially strong barriers: Γ1,2 (D) = ∆ |t1,2 |2 2π α D D0 , (40) where |t1,2 |2 ≪ 1 are the bare transmission probabilities at D = D0 and resonant peaks are sharp [i.e., Γ1,2 (D) ≪ ∆] for all D < D0 . If at least one barrier is initially weak, the resonant structure develops only at D ≪ Drmin . Provided one barrier is initially weak (assume this is the right barrier and Dr2 ≫ ∆), whereas the other is strong, then ∆ DDr2 |t1 |2 2 2π D0 α D ∆ . Γ2 (D) = 2π Dr2 Γ1 (D) = If ∆ ≪ Dr2 weak, then boundary conditions in the double-barrier case should be written at D ∼ ∆, instead of D ∼ D0 . We obtain for |ǫ|, |ǫ′ |, D ≪ ∆: ∂t(ǫ, D) α ∗ = − t(ǫ, D) [ rR (ǫ, D) rR (D) ∂LD 2 ∗ +rL (ǫ, D) rL (D) ] , α ∂rL (ǫ, D) ∗ = [ rL (D) − t2 (ǫ, D) rR (D) ∂LD 2 2 ∗ −rL (ǫ, D) rL (D) ] , (44) (45) and similarly for rR (ǫ, D), where the bar over the reflection amplitudes denotes the averaging (27) over ǫ′ . α , (41) (42) Dr1 ≪ D0 , i.e., both barriers are initially Γ1 (D) = ∆ 2π DDr2 2 Dr1 α (43) and Γ2 (D) is given again by Eq. (42). To summarize this section, substituting D → |ǫ| in Eqs. (37),(38),(39) and using the Fabry-Perot equations (28) gives the fully renormalized scattering amplitudes for |ǫ| ∆ if |ǫ| ≫ T . Also, if T ≫ ∆, substituting D → T solves the problem for arbitrary ǫ. However, when both |ǫ|, T ≪ ∆, we should proceed with the renormalization in the range D ≪ ∆. D. Single resonance: D ≪ ∆ Let us now consider Eqs. (25),(26) for |ǫ|, |ǫ′ | ≪ ∆. In this limit, the terms (21) containing the amplitudes Aµ,−µ (ǫ′ ) to stay inside the dot become irrelevant in the RG sense: the phase factors χǫ−ǫ′ in Eq. (21) can be expanded about ǫ, ǫ′ = 0, which leads to the cancellation of the singular factor (ǫ − ǫ′ )−1 in Eq. (27). As a result, the terms Lµ (ǫ, ǫ′ ; D) do not contribute to the renormalization at D ≪ ∆. The factors χǫ−ǫ′ should also be omitted in the terms of Eqs. (25),(26) that are proportional to ∗ rR (ǫ′ ). Thus we are led to a coupled set of RG equations that describe also a single impurity with energy dependent scattering amplitudes: the spatial structure of the double barrier system is of no importance for the renormalization at D ≪ ∆ (see Fig. 3). However, the Lε FIG. 3: Illustration of the renormalization of a double barrier for D ≪ ∆, Eqs. (44)-(52). The Friedel oscillations are shown schematically within the range Lǫ = vF /|ǫ| outside the dot, where they contribute to the renormalization of the scattering amplitudes at energy ǫ (provided |ǫ| ≫ T ). The double barrier for small D may be considered as a single barrier with an energy dependent scattering matrix, which describes the resonance (sketched by the dotted line) on a single level. The scattering matrix inside the renormalized resonance is discussed in Sec. III E. We integrate now Eqs. (44),(45) assuming that each of two barriers is characterized by Dr1,2 ≫ ∆. This condition means that either the barriers are strong initially at D = D0 or get strong in the course of renormalization (35),(36) before D equals ∆. We will analyze the case of both or one of Dr1,2 being smaller than ∆ in Sec. III F. Thus, if Dr1,2 ≫ ∆, we have narrow peaks of resonant transmission in a neighborhood of the Fermi level, even if the bare reflection coefficients are small. Consider first the case of a resonance energy ǫ0 lying exactly on the Fermi level, ǫ0 = 0. Let Γ(D) be a renormalized width of the resonance peak at the Fermi energy (to be found below). If D ≫ Γ(D), then |rL,R (D)| ≃ 1, which allows for a significant simplification of Eqs. (44),(45). It is convenient to introduce phase-shifted amplitudes rL = ˜ −iϕr′ +2πi(ǫF +ǫ0 )/∆ −iϕr1 ˜ = te−i(ϕt1 +ϕt2 ) , 2 , rR = rR e ˜ ,t rL e 11 T (ε) where ϕr1 is the phase of r1 , etc., in obvious notation. Then we get, by putting the averaged amplitudes far from ˜ the resonance rL,R (D) = 1: T1 >T2 ˜ ∂ t(ǫ, D) α ˜ = − t(ǫ, D) [ rL (ǫ, D) + rR (ǫ, D) ] , (46) ˜ ˜ ∂LD 2 ∂˜L,R (ǫ, D) r α ˜ = [ 1 − rL,R (ǫ, D) − t2 (ǫ, D) ] , (47) ˜2 ∂LD 2 with the following solutions [u2 (D) − u2 (D)]1/2 + − , u+ (D) + 2iǫ u− (D) + 2iǫ rL (ǫ, D) = ˜ , u+ (D) + 2iǫ −u− (D) + 2iǫ , rR (ǫ, D) = ˜ u+ (D) + 2iǫ ˜ t(ǫ, D) = (48) (49) (50) where u± (D) = Γ± (∆)(D/∆)α , (51) and Γ± (∆) = Γ1 (∆) ± Γ2 (∆) should be found by matching onto Eqs. (40)–(43). The width of a resonant tunneling peak Γ(D) is thus given by u+ (D). Note that the only condition we have assumed in the above derivation is D ≫ Γ(D) with D = max{|ǫ|, T }, otherwise ǫ in Eqs. (48)–(50) may be arbitrary. Thus, Eqs. (48)–(50) give the shape of the ǫ dependence of fully renormalized amplitudes for the case of temperature T ≫ Γ(T ) (with T substituted for D). In particular, Eq. (51) says that the width of the resonance behaves as T α : Γ(T ) = Γ+ (∆)(T /∆)α . (52) As follows from Eq. (48), while the resonance becomes sharper with decreasing T , the peak value of the transmission amplitude is not renormalized (see Fig. 4), since the D dependent factors cancel in Eq. (48) at ǫ = 0. The absence of renormalization stems from the vanishing of the sum rL (ǫ, D) + rR (ǫ, D) in Eq. (46) at ǫ = 0, which ˜ ˜ can be seen from Eqs. (49),(50). We recognize Eqs. (48)–(50) as Breit-Wigner solutions that take into account renormalization at D ∆. On the other hand, we have already obtained Breit-Wigner formulas in Sec. III C, where the renormalization has been carried out for D ≫ ∆, for ǫ close to a resonance energy. In particular, the results of Sec. III C apply for |ǫ| ≪ ∆ if ǫ0 = 0. The matching of the two solutions at D ∼ ∆ implies that Eqs. (44),(45) are in fact valid in a broader range of D, namely for D ≪ Drmin , provided only that one averages rR (ǫ′ , D) over ǫ′ together with the phase factor χǫ′ . It follows that in the case of strong barriers close to resonances the RG equations can be cast in the form (46),(47) containing t(ǫ, D) and rL,R (ǫ, D) only. Note also that for D0 ∆ the boundary conditions to Eqs. (46),(47) are fixed at D = D0 , which leads to the change ∆ → D0 in Eq. (51). 0 ε FIG. 4: Transmission coefficient T(ǫ) for temperatures Ds ≪ T ≪ ∆. Two curves correspond to two temperatures: T1 (solid) larger than T2 (dashed). While the peak height is not renormalized by interactions, the resonance gets narrower with decreasing T , Eq. (52). At this point, one might be concerned about a possible contribution to t(ǫ = 0, D) from other resonances. Indeed, in the derivation of Eq. (48), which gives no renormalization of t(ǫ = 0, D), we approximated |r(ǫ′ , D)| by unity for large |ǫ′ |, D ≫ Γ(D). Corrections coming from other resonances are clearly small in the parameter Γ1,2 (D)/∆ but one should check if they might contribute to the renormalization of t(ǫ = 0, D). Relaxing the above approximation by allowing for resonant “percolation” of electrons through the barriers at D ∆ does give a perturbative correction to the RG (46): ˜ ∂[δ t(ǫ, D)] πu− (D) ˜ = −αt(ǫ, D) [ rL (ǫ, D) − rR (ǫ, D) ] , ˜ ˜ ∂LD 4∆ (53) which, in contrast to Eq. (46), does not vanish at ǫ = 0 (unless the double barrier is symmetric: the correction is then always zero). However, Eq. (53) tells us that the correction is irrelevant since u− (D) itself scales to zero as Dα . We thus conclude that the “single-peak approximation” of Eqs. (46),(47) correctly describes the renormalization of the resonant amplitudes for all D ≫ Γ(D). E. Inside a peak: D ≪ Ds Now that we have integrated out all D ≫ u+ (D), let us continue with the renormalization for D inside a resonant tunneling peak. The point at which D and u+ (D) become equal to each other yields a new characteristic scale Ds : ′ Ds = Γ+ (∆)(Ds /∆)α = Γ+ (∆)[ Γ+ (∆)/∆ ]α , (54) where Γ+ (∆) is obtained from Eqs. (40)–(43) depending on the ratio of Dr1,2 and D0 . To leading order in α the exponent α′ = α/(1 − α) → α. As will be seen below, 12 the significance of Ds is that the width of the tunneling resonance saturates with decreasing D on the scale of Ds . For D ≪ Ds , the RG equations (44),(45) can be simplified since the scattering amplitudes now depend on a single variable, which is D = max{|ǫ|, T }. The averaged reflection amplitudes rL,R (D) coincide then with rL,R (D), and the RG equations can be written in precisely the same form as for a single impurity: ∂t(D) = −αt(D)R(D) , ∂LD ∂rL,R (D) = αrL,R (D)T(D) , ∂LD (55) with matching conditions at LD = ln(D0 /Ds ). The difference between the single structureless impurity and the resonance peak is that in the latter case the ultraviolet cutoff is Ds . The fact that the scattering amplitudes inside the peak are described by Eqs. (55) was used also in Ref. 33. In the preceding Secs. III C,III D, asymmetry of the double barrier, i.e., a possible difference between t1 and t2 was seen to determine the amplitude and the width of a resonance peak but otherwise did not lead to any qualitatively different consequences, as compared to the symmetric case. However, for small D ≪ Ds the renormalization of the scattering amplitudes is essentially different depending on whether the barriers are identical to each other or not. Consider first the symmetric case. Assuming, as in Sec. III D, that the resonance energy ǫ0 = 0, we get the Breit-Wigner formula with a width given by Eq. (51): ˜ t(D) = u+ (D) , u+ (D) + 2iǫ (56) which turns out to be valid down to D = 0. A remarkable consequence of Eq. (56) is that the resonance in the symmetric case is perfect, T = 1 at ǫ = 0. While this is trivial for noninteracting electrons, weak interaction in the Luttinger liquid is seen to preserve the perfect transmission, in agreement with the result obtained by a bosonic RG.3 So long as inelastic scattering is not taken into account, the perfect transmission at ǫ = 0 is not affected by finite temperature T , either, as can be seen from Eq. (56) if one puts D = T . However, the width of the resonance does depend on T . At T = 0, the width is finite and given by Ds (see Fig. 5a), which follows from Eq. (56) for D = |ǫ|. For T ≫ Ds , the width obeys Eq. (52). The shape of the perfect resonance depends on the parameter T /Ds . If T ≪ Ds , the reflection probability as a function of |ǫ| behaves near ǫ = 0 first as ǫ2 for |ǫ| ≪ T and then as |ǫ|2(1−α) for T ≪ |ǫ| ≪ Ds . For larger energies, the transmission probability falls off with increasing |ǫ| as T(ǫ) = (Ds /2|ǫ|)2(1−α) . (57) This lineshape should be contrasted with the Lorentzian which describes the transmission peak for T ≫ Ds up to |ǫ| ∼ T , at which point a crossover to Eq. (57) occurs. Let us now turn to the asymmetric case. Inspecting Eqs. (55), we see that a new characteristic scale D− emerges: D− = Ds |Γ− (∆)| Γ+ (∆) 1/α , (58) which coincides with Ds for strongly asymmetric barriers but vanishes for symmetric ones. For T D− we get the same results for T(ǫ) as in the symmetric case, only with an overall factor of λ= Γ2 (∆) − Γ2 (∆) 4Γ1 (∆)Γ2 (∆) + − . = Γ2 (∆) [ Γ1 (∆) + Γ2 (∆) ]2 + (59) However, for T ≪ D− a new feature in the behavior of T(ǫ) shows up, namely a power-law falloff with decreasing |ǫ|. The function T(D), as obtained from Eqs. (55), for D ≪ Ds reads T(D) = λ(D/Ds )2α . 1 − λ [ 1 − (D/Ds )2α ] (60) One sees that T(ǫ) behaves as (Fig. 5a) T(ǫ) = λ(|ǫ|/D− )2α (61) in the interval T ≪ |ǫ| ≪ D− and saturates at smaller energies: T(ǫ = 0) = λ(T /D− )2α . Thus, in the limit T ≪ D− , the resonant transmission probability as a function of ǫ exhibits a double-peak structure, see Fig. 5a. If, however, the barriers are only slightly asymmetric, the gap near ǫ = 0 develops in a range of ǫ which is much narrower than the width of the resonance peak. Specifically, T(ǫ) first grows up wit increasing |ǫ| for T ≪ |ǫ| ≪ D− , then there is a plateau with an energy independent transmission for D− ≪ |ǫ| ≪ Ds , and T(ǫ) starts to fall off as |ǫ| is further increased. In the above, we analyzed the behavior of T(ǫ) for the resonance energy ǫ0 = 0, i.e., when it coincides with the Fermi level. Let us now examine the case ǫ0 = 0. Again, let the barriers first be symmetric. Then, in Eq. (56), the resonant denominator changes to u+ (D) + 2i(ǫ − ǫ0 ) and D is, as before, max{|ǫ|, T }. The innocent looking shift ǫ → ǫ − ǫ0 leads at T = 0 to dramatic consequences for transmission at the Fermi energy, ǫ = 0. Namely, T(ǫ) is now seen to vanish at ǫ = 0 and zero T , whatever ǫ0 unless it is exactly zero. A new characteristic scale D1 becomes relevant at ǫ0 = 0: it is defined by u+ (D1 ) = 2|ǫ0 |, which is rewritten as D1 = Ds (2|ǫ0 |/Ds )1/α . (62) The significance of the energy D1 is that the width of the gap in the dependence of T(ǫ) around ǫ = 0 at T = 0 is given by D1 for |ǫ0 | Ds . Note that D1 ≪ |ǫ0 | for |ǫ0 | ≪ Ds . The shape of the resonant peak as a function of ǫ changes in an essential way for |ǫ0 | Ds . Specifically, if T ≫ D1 , the changes are weak; however, for T ≪ D1 13 T (ε) ε 0 =0 T (ε) ε 0 =0 sym asym ε 0 Ds gap width = 0 ε0 D_ gap width = (a) ε D1 (b) FIG. 5: Schematic summary of transmission peak structure for zero temperature and strong barriers: (a) ǫ0 = 0, a single peak of T(ǫ) for symmetric barriers with a width Ds transforms into a double peak with a reduced height and a gap width D− < Ds for asymmetric barriers, Eqs. (58),(60); (b) 0 < |ǫ0 | < Ds , a single peak for symmetric barriers, centered at ǫ = ǫ0 , shows a dip at the Fermi energy, ǫ = 0, with a gap width D1 < |ǫ0 |, Eq. (62). For ǫ0 = 0 and asymmetric barriers, (a) describes the case of sufficiently small ǫ0 , namely D− ≫ D1 , whereas (b) with a properly rescaled height describes the opposite case. The only effect of finite T Ds is to fill the gaps at |ǫ| T . a range of ǫ arises, T ≪ |ǫ| ≪ D1 , within which T(ǫ) behaves as (Fig. 5b) T(ǫ) = (|ǫ|/D1 )2α . rL (ǫ, D) = (r1 − r2 χǫ )(D0 /D)α Weak barriers In Secs. III C–III E, we have assumed that Drmin ≫ ∆, which means that even if the impurities are initially (at D = D0 ) weak, they get strong in the process of renormalization before D becomes equal to ∆. Under this assumption, we have sharp resonant peaks close to the Fermi level and the bare scattering amplitudes only (64) with D = max{|ǫ|, T }. It is worth noting that, due to the oscillating factor χǫ (20), it is not possible to derive Eq. (64) from Eqs. (23),(24) even in the simplest case of a weak double barrier. Suppose first that the barrier is symmetric and ǫ0 = 0. Then Eq. (64) simplifies to R(ǫ, D) = 2[ 1 − cos(2πǫ/∆) ] (Dr /D)2α . (65) One sees that reflection is enhanced by interaction, but the reflection coefficient is always small, R ≪ 1 for any ǫ, provided Dr ≪ ∆. No sharp features in the ǫ dependence of the scattering amplitudes emerge around the Fermi energy (see Fig. 6a). T (ε) T (ε) δ− δ1 sym |ε 0 | > 0 asym (63) The power-law falloff (63) saturates close to the Fermi level at T(ǫ = 0) = (T /D1 )2α . We thus see that the width of the resonance in the transmission through a symmetric barrier exactly at the Fermi energy T(ǫ = 0) as a function of ǫ0 vanishes as T → 0. On the other hand, the width of the resonance in the transmission at ǫ0 = 0 as a function of ǫ is finite even at T = 0 and is given by Ds . This peculiar feature is in sharp contrast to the resonant tunneling of noninteracting electrons, for which the two widths are the same. For asymmetric barriers, T(ǫ) does not change substantially with increasing |ǫ0 | as long as D1 ≪ D− and is given by the formulas for symmetric barriers with an overall reduction of the transmission probability by a factor of λ (59) otherwise (see Fig. 5). F. rescale [according to Eqs. (40)–(43)] parameters in otherwise general formulas for the resonant tunneling. Let us now examine the resonant transmission in the case of at least one barrier being initially so weak that the renormalization does not make it strong at D ∼ ∆. We begin with the case of both barriers characterized by Dr1,2 ≪ ∆. The total reflection coefficient rL (ǫ, D) as obtained from Eqs. (25),(26) in the limit |rL | ≪ 1 is given by ∆/2 ε 0 0 ε0 (a) ε (b) FIG. 6: Schematic behavior of the transmission coefficient for zero temperature and weak barriers, Eq. (71): (a) for ǫ0 = 0, the modulation of T(ǫ) for both symmetric (solid) and asymmetric (dashed) barriers is strongly increased. However, for symmetric barriers T(ǫ) remains close to unity for all ǫ, while there appears a dip of width δ− < Dr , Eqs. (67),(68), around ǫ = 0 for asymmetric barriers; (b) for ǫ0 = 0 and symmetric barriers, the bare transmission (dash-dotted) is strongly renormalized (solid) due to interactions and exhibits a gap of width δ1 < Dr , Eq. (70), around the Fermi level. For ǫ0 = 0 and asymmetric barriers, (a) describes the case of sufficiently small ǫ0 , namely δ− ≫ δ1 , whereas (b) with a properly rescaled height describes the opposite case. Similarly to Fig. 5, finite T fills the gaps at |ǫ| T . It is now instructive to introduce a weak asymmetry R− = |R2 − R1 |, such that R− ≪ R ≃ R1,2 . Given that the asymmetry is weak, it can manifests itself only at small energies. Expanding Eq. (64) about ǫ = 0, we get for |ǫ| ≪ ∆: R(ǫ, D) = R− 2R 2 + 2πǫ ∆ 2 Dr D 2α . (66) 14 As can be seen from Eq. (66), asymmetry sets two new characteristic scales of energy: (R− /R)∆ and a smaller scale δ− = Dr (R− /2R)1/α . (67) Provided that temperature T ≪ (R− /R)∆, the reflection coefficient starts to grow with approaching the Fermi level at |ǫ| ∼ (R− /R)∆. The enhancement of reflection is cut off by temperature before R becomes of order unity if T is not too low, specifically if δ− ≪ T . However, if T ≪ δ− , then reflection gets strong at |ǫ| ∼ δ− . To describe the scattering probabilities at |ǫ| δ− , one should solve Eqs. (55), derived in the same way it was done in Sec. III E, now with matching onto the perturbative (in R1,2 ) solution (64) anywhere in the region δ− ≪ |ǫ| ≪ ∆. For D ≪ (R− /R)∆ the solution reads: T(D) = 1/ [ 1 + (δ− /D)2α ] . (68) Thus, the weak double barrier remains slightly reflecting after the renormalization provided that T ≫ δ− . However, for both T, |ǫ| ≪ δ− the transmission probability is small: within the range T ≪ |ǫ| ≪ δ− , T(ǫ) behaves as (Fig. 6a) T(ǫ) = (|ǫ|/δ− )2α , (69) and saturates for smaller |ǫ| at T(ǫ = 0) = (T /δ− )2α . Comparing Eqs. (69),(61) with each other, we see that the energy δ− is a counterpart of D− for the case of a weak barrier. Generalizing to ǫ0 = 0, we have for |ǫ0 | ≪ ∆ a shift ǫ → ǫ − ǫ0 in Eq. (66), while D = max{|ǫ|, T }. A new energy scale δ1 ≪ ǫ0 appears, at which the reflection coefficient becomes of order unity in the symmetric case: δ1 = Dr (2π|ǫ0 |/∆)1/α , (70) analogous to D1 in Eq. (62) for tunneling barriers. The energy δ1 gives the width of the gap in the transmission probability at the Fermi level at T = 0. For T ≪ |ǫ| ≪ δ1 , we get a power-law vanishing of T(ǫ) = (|ǫ|/δ1 )2α with decreasing |ǫ| (see Fig. 6b) and a saturation for smaller |ǫ| at (T /δ1 )2α . A general expression for the scattering probabilities, valid for arbitrary Dr1,2 ≪ ∆ and |ǫ|, D ≪ ∆, can be obtained from Eqs. (55): R(ǫ, D) = T(ǫ, D) 1/2 R1 1/2 − R2 + (R1 R2 )1/2 2π ∆ 2 (71) 2 (ǫ − ǫ0 )2 D0 D 2α . Equation (71) reproduces Eqs. (66)–(70) in the corresponding limits. We conclude that if the barriers are symmetric but ǫ0 is nonzero, or if the barriers are asymmetric, the transmission probability vanishes (Fig. 6) at the Fermi level in the limit T → 0. We will see in Sec. IV that these features lead to the emergence of a sharp peak in the low-temperature conductance as a function of ǫ0 even for two weak impurities, provided only that they are slightly asymmetric. Finally, when two strongly asymmetric barriers are located nearby, so that at D ∼ ∆ one barrier is strongly reflecting whereas the other is still weak, the effect of the latter on the transmission probability remains small for any D. Let us take the example Dr1 ≫ ∆ and Dr2 ≪ ∆. Then we get for Dr1 ≫ D ≫ ∆ T(ǫ, D) = (D/Dr1 )2α [ 1 + 2(Dr2 /D)α cos θ ] , (72) where θ = 2π(ǫ − ǫ0 )/∆. One sees that the presence of the weak impurity only leads to a weak modulation with changing ǫ. For D ∆, the independent renormalization of the weaker impurity is suppressed by reflection from the strong barrier and T(D) behaves as (D/Dr1 )2α down to D = 0. IV. CONDUCTANCE PEAK The solution to the problem of transmission through a double barrier given in the preceding sections allows us to examine the linear conductance of the system G(ǫ0 , T ) as a function of temperature T and the energy distance between the Fermi level and a resonance level ǫ0 . Recall that we have studied the elastic transmission of interacting electrons, i.e., the energy ǫ of an incident electron before and after the transmission is the same, while in the process of scattering off the barrier ǫ is not conserved due to interaction with other electrons both in the particle-hole and Cooper channels. At finite T , there are also inelastic processes, characterized by the inelastic scattering length Lin . Neglecting the inelastic scattering is legitimate if Lin ≫ LT ∼ vF /T , which is satisfied in the present problem for weak interaction α ≪ 1. On the other hand, it is worthwhile to note that the very formulation of the scattering problem in the interacting case even for elastic scattering is a delicate issue if one keeps scaling exponents of higher order than linear in α.46 Also, interaction-induced current-vertex corrections in the Kubo formula in the presence of impurities and interaction are governed by exponents of higher order in α. Here, we avoid these complications by treating the scattering problem within the one-loop approximation, i.e., keeping only first order terms in the exponents. Under these conditions, one can use the Landauer-B¨ ttiker u formalism relating the conductance and the transmission probability. The conductance G(ǫ0 , T ) in units of e2 /h reads G(ǫ0 , T ) = dǫ T(ǫ) (−∂nF /∂ǫ) . (73) We are interested in the low-temperature regime with T ≪ ∆, otherwise we intend to keep T arbitrary, i.e., T may be as small as zero. 15 Below we analyze various limiting cases. As we have seen in Sec. III, the strength of interaction, the strength of the barriers, and the degree of asymmetry set a number of characteristic energy scales which yield a variety of different regimes in the temperature and energy dependence of the transmission probability T(ǫ, T ). By means of Eq. (73), these regimes manifest themselves in the behavior of the conductance G(ǫ0 , T ), which is a directly measurable quantity. Before turning to the calculation of G(ǫ0 , T ), let us briefly discuss the restrictions and possible implementations of our model. Even though we are dealing with the case of weak interaction, α ≪ 1, the product α ln(D0 /∆) may be large in small quantum dots, so that the renormalization of scattering amplitudes on scales D ≫ ∆ is necessary, as has been done in Sec. III C. Experimentally, as discussed in Sec. I, the parameter α is typically not small in carbon nanotubes [in Refs. 6,7, the value of α extracted according to Eq. (5) from the measured αe ∼ 0.6 − 1.0 is α ∼ 1.3 − 3.0)]. On the other hand, in single-mode semiconductor quantum wires the strength of interaction is typically smaller (in Ref. 12, αe ∼ 0.2 − 0.5 corresponds [Eq. (5)] to α ∼ 0.3 − 0.8) and can be more easily made small through screening by nearby metallic gates. In weakly interacting wires, our results would be applicable directly. At the same time, our theory captures much of the essential physics of strongly [with α defined in Eq. (2) of order unity] correlated wires too, as follows from the comparison with the known results obtained by different methods.3,16,19 As for the impurity strength, both transport and tunneling experiments on quantum wires with imperfections have been so far focused on the case of initially strong inhomogeneities (impurities, artificially created tunneling barriers, or non-adiabatic contacts). Sections III C-III E describe this situation in detail. Moreover, as shown in Secs. III C,III F, weak inhomogeneities, which are potentially realizable in a controllable way both in semiconductor quantum wires and carbon nanotubes, are of special interest, since the Coulomb interaction transforms two weak impurities into a quantum dot with a pronounced resonance structure. Finally, a relatively strong asymmetry of the quantum dot appears to be inevitably present in the experimental setups of Refs. 12,13. A. Strong barriers Consider first the case of strong barriers (more precisely, the bare transmission through the barriers may be high, T1,2 ≃ 1, but we assume that the barriers get strong before the RG flow parameter D equals the single-particle energy spacing inside the dot, ∆), i.e., Drmin = D0 (min{R1 , R2 })1/2α ≫ ∆. Then we have a sharp peak of the transmission probability centered at ǫ = ǫ0 whose width is max{Ds , Γ(T )} ≪ ∆, where Ds and Γ(T) are defined in Eqs. (52),(54). In other words, the width of the peak in T(ǫ, T ) is Γ(T ) = Ds (T /Ds )α for T ≫ Ds , whereas for smaller T ≪ Ds the width is of order Ds and does not depend on T . 1. T ≫ Ds , Sequential tunneling For T ≫ Ds , the shape of the conductance peak is given by (see Fig. 7) G(ǫ0 , T ) = ζ Gp , cosh2 (ǫ0 /2T ) (74) where the peak value of the conductance Gp = πλΓ(T )/8T , (75) with λ defined in Eq. (59), and ζ = (max{|ǫ0 |, T }/T )α. The width of the conductance peak w is of order T , as for noninteracting electrons; however, the power-law behavior of Gp (T ) is seen to be modified by interaction, in accordance with the results derived in Refs. 16,19. Note that the scaling of Gp ∝ T α−1 is governed by the singleparticle density of states ρe (T ) for tunneling into the end of a Luttinger liquid, namely Gp ∝ ρe (T )/T .28 We recognize Eqs. (74),(75) as the conventional sequential tunneling formulas, but with a T dependent resonance width Γ(T ). Far in the wings of the resonance the exponential falloff (74), G(ǫ0 , T ) ∼ λT −1 Γ(|ǫ0 |) exp(−|ǫ0 |/T ), crosses over onto the cotunneling (determined by the processes of fourth order in tunneling amplitudes) power law G(ǫ0 , T ) = λΓ2 (T )/4ǫ2 , 0 as usual. The crossover between the sequential tunneling and cotunneling regimes occurs at |ǫ0 | ≃ T ln [ T /Γ(T ) ]. In fact, this formula is valid19 for an arbitrary strength of interaction with Γ(T ) ∝ ρe (T ) ∝ T αe , where αe (equal to α for a weak interaction) is the end-tunneling exponent (5). It is worthwhile to note that for strong enough interaction (namely for αe > 1) the sequential mechanism of tunneling is effective for the resonance peak for all T , down to T = 0.19 Moreover, for αe > 1 the crossover to the cotunneling regime shifts towards larger |ǫ0 | with increasing strength of interaction. As demonstrated in Ref. 19, Eqs. (74),(75) can be obtained from a classical kinetic equation51 for occupation numbers characterizing the state of the quantum dot. We get the same results from the fermionic RG equations. It is worth mentioning that, although Eqs. (74),(75) follow from a classical kinetic equation51 which involves diagonal elements of the density matrix only, the transmission in our derivation is fully coherent. The fact that in the high-temperature limit it can also be described in terms of the classical kinetic equation merely means that quantum corrections to the kinetic equation may be neglected for small Γ(T )/T [whereas the quantum suppression of the tunneling density of states ρe (T ) may be treated in this equation in a phenomenological way]. 16 G G 1 1 (a) T1