Electron transport in disordered Luttinger liquid 1 arXiv:cond-mat/0608590v2 [cond-mat.dis-nn] 3 Jan 2007 2 I.V. Gornyi1,∗, A.D. Mirlin1,2,† , and D.G. Polyakov1,∗ 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 (Dated: February 6, 2008) We study the transport properties of interacting electrons in a disordered quantum wire within the framework of the Luttinger liquid model. We demonstrate that the notion of weak localization is applicable to the strongly correlated one-dimensional electron system. Two alternative approaches to the problem are developed, both combining fermionic and bosonic treatment of the underlying physics. We calculate the relevant dephasing rate, which for spinless electrons is governed by the interplay of electron-electron interaction and disorder, thus vanishing in the clean limit. Our approach provides a framework for a systematic study of mesoscopic effects in strongly correlated electron systems. PACS numbers: 71.10.Pm, 73.21.-b, 73.63.-b, 73.20.Jc I. INTRODUCTION Mesoscopics of strongly correlated electron systems has emerged as an area of great interest to both experimental and theoretical communities working in the field of nanoscale physics. Recently, progress in manufacturing of nanodevices has paved the way for systematic transport measurements on narrow quantum wires with a few or single conducting channels. Most prominent examples of these are: single-wall carbon nanotubes,1 cleaved-edge,2 V -groove,3 and crystallized in a matrix4 semiconductor quantum wires, quantum Hall edges running in opposite directions and interconnected by means of tunneling,5,6 polymer nanofibers,7 and metallic nanowires.8,9 Much experimental attention has been focused on the electrical properties of single-wall nanotubes (Refs.10,11, 12,13,14,15,16,17,18,19,20,21,22,23,24,25 and references therein). In particular, evidence has emerged pointing towards the existence of Luttinger liquid in metallic singlewall carbon nanotubes, as expected for strongly interacting electrons in one dimension (1d). 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. In the past few years, technological advances have made possible the fabrication of contacts between nanotubes and metallic leads with a very low contact resistance, which allows one to observe mesoscopic effects.19,20,21,22 Further, techniques to grow very long (of the mm scale) nanotubes have been developed and the corresponding transport measurements performed.26 On the theoretical side, the challenge is to expand the ideas that have been developed for mesoscopic disordered systems on one side and for strongly correlated clean systems on the other. The situation is particularly interesting in 1d, where both disorder and electronelectron (e-e) interaction, even if they are weak, modify dramatically the large-scale, low-energy physics of the problem. In a clean wire, e-e interaction leads to the formation of a Luttinger-liquid ground state. The prop- erties of the Luttinger liquid without impurities and in the presence of a single barrier are known in great detail, see Refs. 27,28,29,30,31,32 for review. Much less is known about the Luttinger liquid in the presence of many impurities (disordered Luttinger liquids). The influential works by Apel and Rice33 and by Giamarchi and Schulz34 have defined the state of the art in this area for two decades. Recent years have seen a revival of interest in disordered Luttinger liquids,35,36,37,38,39,40,41,42,43 largely motivated by the technological advances mentioned above. However, the very applicability of such key notions of the mesoscopic physics as weak localization (WL) and dephasing to a disordered 1d system, not to mention a detailed analysis of these effects, remained an open problem. This problem is a subject of the present paper. Throughout the paper we study spinless (spinpolarized) electrons. Spin-related effects will be considered in Ref. 44. The structure of the paper is as follows. In Sec. II we summarize the earlier achievements in the mesoscopics of higher-dimensional systems on one side and in the theory of strongly correlated 1d systems on the other. Section III contains an exact formulation of the problem, as well as a discussion of our strategy in solving it. In Sec. IV we present the bosonization method and the renormalization-group technique that we use to account for the Luttinger-liquid renormalizations. Having analyzed in Sec. V the e-e inelastic scattering rate in 1d, we turn to the calculation of a weak-localization correction to the conductivity and the associated dephasing rate in Sec. VI. We also compare this dephasing rate with the one that governs the damping of AharonovBohm oscillations. Section VII is devoted to the analysis of the transport properties of a disorder Luttinger liquid by using an alternative approach—the “functional bosonization”—which allows one to treat the renormalization and the inelastic scattering on an equal footing. Our results are summarized in Sec. VIII. Technical details of the calculations are presented in several Appendices. Some of the results of this paper were published in a brief form in Letter Ref. 45. 2 II. BACKGROUND In this introductory section, we briefly summarize the known results for higher-dimensional mesoscopic systems (Sec. II A) and for strongly correlated 1d systems (Sec. II B). These will serve as a starting point for our work. A. ∆σAA ≃ −α σD Moderately low temperatures: Weak-localization regime The quantum localization is the most prominent manifestation of the quantum interference in disordered systems. At not too low temperatures, the localization effects are cut off by inelastic processes, resulting in a WL correction52 ∆σwl to the Drude conductivity σD , ∆σwl ≃ − σD ∼ − l−1 L−1 φ (dq) πν0 Dq 2   ln(τφ /τ ) , 1 ν0 D  (Dτ )1/2 , φ 2d , (2.1) quasi − 1d , where D is the diffusion constant, ν0 the density of states, l the mean free path, τ the transport mean free time, τφ and Lφ = (Dτφ )1/2 are the interaction-induced dephasing time and length, respectively. We use notation (dq) = dd q/(2π)d . Throughout the paper = 1. Another manifestation of the quantum coherence are mesoscopic fluctuations of the conductance and of other observables. As far as e-e interaction in the WL regime is concerned, it is responsible for two distinctly different main effects. First, it renormalizes elastic scattering on impurities through the creation of virtual electron-hole excitations (screening) with a high energy transfer ω larger than the temperature T , T |ω| ǫF . Here ǫF is the Fermi energy. At sufficiently high T (in the ballistic regime T τ ≫ 1), i.e., when the relevant spatial scales are smaller than l, the screening can be described in terms of a dressing of impurities by Friedel oscillations53,54 and yields a l−1 L−1 T (dq) πν0 Dq 2  ln(T τ ) , α  ∼ ν0 D  −(D/T )1/2 , Mesoscopics of higher-dimensional systems During the last three decades, mesoscopic physics of low-dimensional diffusive systems has been attracting a great deal of attention.46,47,48,49,50,51 It has been recognized that the effects of the quantum interference and e-e interaction become strongly enhanced in disordered systems of reduced dimensionality d, affecting in an essential way the low-temperature transport properties of the systems. These effects lead to a set of remarkable phenomena, including the WL, mesoscopic conductance fluctuations, and the Altshuler-Aronov interaction-induced quantum correction to the conductivity. At still lower temperatures, these effects may drive the system towards strong localization. We now briefly remind the reader the basic facts in this field. 1. T -dependent Drude conductivity.54 At lower T (in the diffusive regime T τ ≪ 1), effects of this type generate the Altshuler-Aronov correction to the conductivity,46 2d , (2.2) quasi − 1d , where LT ∼ (D/T )1/2 is the thermal length. This correction is perturbative both in the effective interaction constant α and in the disorder strength. The Coulomb interaction corresponds to α ∼ 1. The T dependence of ∆σAA is determined by the thermal smearing of the distribution function, which introduces an infrared cutoff on spatial scales of the order of LT . Using the renormalization-group (RG) approach within the replicated σ-model, Finkel’stein extended this theory to include the interaction nonperturbatively.55 Effects related to the interaction-induced renormalizations and screening are also responsible for a zero-bias anomaly46,55 in the tunneling density of states as a function of electron energy, ν(ǫ). It is worth noting that, in the case of Coulomb interaction, ν(ǫ) may become strongly suppressed,55,56,57 ν(ǫ) 1 ln2 Dκ2 /|ǫ| ∼ exp − 2 ν0 8π ν0 D − ln2 (Dκ2 τ ) 1, 2d , (2.3) where κ−1 is the screening radius, while the conductivity can still remain close to its Drude value (∆σAA /σD ≪ 1). The second effect of interaction is inelastic e-e scattering which breaks the phase coherence.46 A key concept in the localization theory of a disordered Fermi liquid −1 is that of the dephasing rate τφ . The phase-breaking processes are of crucial importance in the problem of transport since without dephasing the conductivity of low-dimensional systems would be zero at any T , see Sec. II A 2. Two qualitatively different sources of dephasing are possible: (i) scattering of electrons by external excitations (in practice, phonons) and (ii) e-e scattering (at low T the dephasing is mostly due to e-e interactions). In either case, at sufficiently high T , the dephasing rate −1 τφ is high, so that the localization effects are reduced to the WL correction to the Drude conductivity. A characteristic energy transfer in the real inelastic processes that determine the phase relaxation is restricted by temperature: |ω| T , so that the dephasing rate vanishes at zero temperature. The infrared divergency characteristic of a diffusive system makes it necessary to introduce a self-consistent cutoff in the equation for the dephasing rate:46 −1 τφ 2 ∼ α T L−1 T L−1 φ (dq) πν0 Dq 2 3 ∼  2  α T ln    ν0 D     α2 T ν0 D ν0 D α2 , 2d , (2.4) 2/3 D 1/3 , quasi − 1d . A rigorous calculation necessitates employing the pathintegral technique.58 The different energy scales relevant to the disorder renormalization due to virtual processes with T < |ω| < ǫF and to the dephasing due to real inelastic scattering with |ω| T allow for a straightforward separation of these two effects of e-e interaction. At sufficiently high T , the conductivity σ(T ) ≃ σD + ∆σwl + ∆σAA (2.5) is close to the Drude value σD . It is worth mentioning that for weak interaction (α ≪ 1) the WL correction [Eq. (2.1) with τφ given by Eq. (2.4)] is much stronger than the Altshuler-Aronov correction [Eq. (2.2)]. The WL correction grows with lowering T and eventually becomes strong (|∆σwl |/σD ∼ 1) when Lφ reaches ξ or, −1 equivalently, when τφ becomes of order ∆ξ . Here ξ is the localization length and ∆ξ is the corresponding energy level spacing. At this temperature, |∆σAA |/σD ≪ 1 for α ≪ 1. This means that in a weakly interacting system the strong localization occurs due to the growth of the interference-induced WL correction. In the next subsection, we briefly overview the strong localization regime −1 which emerges at low temperatures, when τφ ∆ξ . 2. Low temperatures: Fate of dephasing in the strongly localized regime In a pathbreaking paper59 Anderson demonstrated that a quantum particle may become localized by a random potential. In particular, in noninteracting systems of one- or two-dimensional geometry even weak disorder localizes all electronic states,52,60 thus leading to the exactly zero conductivity, σ(T ) = 0, whatever temperature T . A nonzero σ(T ) in such systems may only occur due to inelastic scattering processes leading to the dephasing of electrons. As discussed in Sec. II A 1, the WL correction (2.1) to the Drude conductivity of a diffusive system behaves (2−d)/2 as τφ (ln τφ for d = 2) and diverges with lowering temperature for d ≤ 2. This singular behavior of the perturbative corrections is a manifestation of strong Anderson localization. It prompts a question as to how the disordered interacting low-dimensional systems conduct at low temperatures, when the localization effects are no longer weak. It is commonly believed that transport in the localized regime is of the hopping nature. Mott61 has developed the picture of variable-range-hopping (VRH) as the transport mechanism in the insulating state. The VRH mechanism of Ref. 61 is well justified in the presence of delocalized phonons which provide inelastic scattering between single-particle states with different energies (phonon-assisted hopping). The finite-temperature conductivity is then proportional to the strength of the electron-phonon coupling and governed by Mott’s VRH:61 σ(T ) ∝ exp[−(T0 /T )µ ] with µ = 1/(d + 1) depending on the spatial dimensionality d. In the presence of a long-range Coulomb interaction, the Coulomb gap in the tunneling density of states modifies62 the VRH exponent, µ = 1 . 2 But what if the coupling to phonons is negligibly weak? It was proposed in Ref. 63 that the e-e interaction by itself is sufficient to induce VRH at low T . Although the idea63 of a phononless VRH was widely exploited in the community, it is, however, in conflict with the argument by Fleishman and Anderson64 that elementary hops in the low-T limit are forbidden for a short-range interaction (including the case of 1/r Coulomb interaction in d < 3), because energy conservation cannot be respected when an electron attempts a real transition by exciting an electron-hole pair. Since there is a mismatch in the positions of the energy levels of individual single-particle localized states, inelastic scattering is necessary to broaden these levels to allow for hopping between them. It was argued in Ref. 64 that at low T a short-range e-e interaction alone is not sufficient to broaden the localized states and to induce a nonzero conductivity. The situation is particularly interesting in 1d and 2d, where no mobility edge exists, activation to which otherwise might give σ(T ) = 0. Very recently, the problem was reconsidered in Refs. 65,66. It was found there that the Anderson localization in Fock space crucially affects the conductivity at low T . As a result, the phononless VRH should not exist; instead, the system undergoes a transition at certain temperature Tc below which the conductivity vanishes, in agreement with Ref. 64. The mechanism of low-T transport in low-dimensional disordered systems is thus completely determined by inelastic processes which govern the lifetime of localized states. This lifetime may be viewed as a low-temperature reincarnation of the dephasing time that controls the cutoff of quantum-interference effects at higher T . Therefore, an analysis of the WL effects turns out to be important also for understanding of the establishment of the strong localization with lowering T . B. Strongly correlated 1d electron systems This paper is concerned with transport in a singlechannel quantum wire, where both disorder and interaction more strongly modify the transport properties as compared to higher dimensions. In the absence of disorder, the e-e correlations drive the system into the nonFermi liquid state known as Luttinger liquid. Yet another peculiarity of the single-channel 1d system is that the bal- 4 listic motion on short scales crosses over in the absence of interaction directly to the localization regime, with no diffusive dynamics on intermediate scales. We are now going to expand on these known features of 1d electronic systems. 1. Noninteracting case: Anderson Localization We first briefly recall the peculiarities of 1d noninteracting disordered systems. In the absence of interaction, transport in a disordered single-channel 1d gas has been studied in great detail.67,68,69,70,71,72,73,74,75 It is known that arbitrarily weak disorder leads to the localization of all electronic states in this system, with the localization length ξ being the mean free path. The dc conductivity is then zero, σ(T ) ≡ 0 for any T , since the temperature merely determines the distribution function over the localized states. The characteristic spatial and temporal scales of the problem are probed in the absence of interactions by the ac response at external frequency Ω. The appropriate technique for the calculation of the response, which exploits the separation of fast and slow degrees of freedom in the case of weak disorder, ǫF τ ≫ 1, was put forward in a pioneering paper by Berezinskii;69 later, alternative approaches were developed by Abrikosov and Ryzhkin71 and by Berezinskii and Gor’kov.73 In the limit of low frequency, Ωτ ≪ 1, the leading contribution to the ac conductivity is given by the Berezinskii-Mott formula:69,70 Re σ(Ω)/σD = 4(Ωτ )2 ln2 (Ωτ ) . Here ν0 is the density of states in the absence of interaction, Λ is the ultraviolet cutoff (determined by the band width, the Fermi energy, or the range of interaction). The exponent β is determined by the interaction strength and depends on the system geometry (tunneling into the bulk of a wire differs from tunneling into the end of a wire). The suppression of the tunneling density of states in Luttinger liquid is similar to the interaction-induced zero-bias anomaly in higher dimensions, Eq. (2.3). According to a common wisdom, electronic excitations do not exist in Luttinger liquid and a proper language to describe the strongly correlated electrons is the language of bosonic excitations. This notion is based on the fact that the bosonic density-fields in a clean Luttinger liquid do not decay in the linearized-spectrum approximation, when a finite curvature of the electron spectrum is neglected. 3. Renormalization of an impurity in Luttinger liquid Interactions between oppositely moving electrons generate charge- and spin-density wave correlations that lead to striking transport properties of Luttinger liquid in the presence of impurities. The densitywave correlations yield a strong renormalization of an impurity:76,77,78,79,80,81 if the bare strength of the impurity is small, the renormalized backscattering amplitude U (2kF , ǫ) seen by an electron with energy ǫ scales as a U (2kF , ǫ) ∝ (Λ/|ǫ|) , (2.6) In the opposite (ballistic) limit of high frequency, Ωτ ≫ 1, the conductivity σ(Ω) is approximately equal to the Drude formula, σD (Ω) = σD /(1 − iΩτ ), with a quantum interference (WL) correction arising at fourth order in the expansion in (Ωτ )−1 ≪ 1. At Ωτ ∼ 1, the ballistic regime crosses over directly to the strongly localized regime, Eq. (2.6), so that the diffusive regime is absent in the single-channel quantum wire without e-e interaction. where a is determined by the interaction strength. As a result, a single impurity in the infinite Luttinger liquid with a repulsive interaction effectively decouples the wire into two independent pieces, thus leading to a vanishing linear conductance at zero temperature. This conclusion remains valid no matter how weak the interaction is. 4. 2. Clean quantum wires with interaction: Luttinger liquid The interest in the role of e-e interaction in quantum wires is largely inspired by the fact that in 1d the interaction breaks down the Landau’s Fermi liquid approach. Owing to the particular geometry of the Fermi surface, systems of dimensionality one are unique in that the e-e correlations change the noninteracting picture completely and lead to the formation of a strongly correlated state. A remarkable example of the correlated 1d electron phase is the Luttinger-liquid model (for a review see, e.g., Refs. 27,28,29,30,31,32). The Luttinger-liquid correlations show up in a power-law singularity of the tunneling density of states at low energy ǫ counted from the Fermi surface, β ν(ǫ) ∼ ν0 (|ǫ|/Λ) . (2.7) (2.8) Many impurities: Disordered Luttinger liquid The conductivity of a disordered Luttinger liquid was discussed for the first time more than 30 years ago in Refs. 76,77,78 in terms of the renormalization of individual impurities (see also Refs. 82,83 for early attempts to construct the phase diagram of a disordered Luttinger liquid). In the beginning of the 80’s, Apel and Rice33 combined the scaling arguments developed for noninteracting systems60,72,84 with the results of the theory of a clean Luttinger liquid. Later, the problem of transport in a disordered Luttinger liquid was studied by perturbative in disorder methods based on bosonic field theories in Refs. 34,85,86. A small correction to the quantized conductance of a short quantum wire due to a Gaussian disorder was derived in Ref. 86 (for a multichannel wire see Ref. 87). A detailed exposition of these issues can be found in the recent lecture course Ref. 88. 5 The RG equations for a disordered Luttinger liquid were derived by Giamarchi and Schulz.34 Their approach bears a similarity with the Finkel’stein RG55 developed earlier for higher-dimensional diffusive systems (both RG schemes treat disorder perturbatively and e-e interaction exactly). The flow equations of Ref. 34 describe, in addition to the renormalization of impurities by interaction (Sec. II B 3), also a renormalization of the interaction by disorder. A sufficiently strong attractive interaction was shown to yield a metal-insulator transition34 at zero T with changing interaction strength. A repulsive interaction was predicted to enhance the disorder-induced localization. In the latter case, the RG equations show that at a certain temperature (such that LT ∼ ξ) the renormalized disorder becomes strong. This means that the disordered interacting 1d system enters a “strongcoupling regime”, where the Drude-like approach breaks down. At higher temperatures, these equations describe the temperature dependence of an effective scattering time renormalized by interaction (which can be viewed as screening of impurities by Friedel oscillations), thus yielding a temperature-dependent Drude conductivity, 2a σD (T ) ∝ (T /Λ) . (2.9) Recently, a crossover between Eqs. (2.9) and (2.2) with increasing number of channels was studied in Ref. 43. It is important to emphasize that the RG approach formalizes the renormalization of the Drude conductivity due to the T -dependent screening but does not capture the essential physics of the Anderson localization. In particular, the RG equations of Ref. 34 miss the interference effects (coherent scattering on several impurities) that lead to the WL correction to the conductivity. As we demonstrate below, it is the WL correction that governs the temperature dependence of the conductivity at moderately high temperatures for the case of weak interaction and eventually drives the quantum wire into the strong-localization regime with decreasing temperature. Another related deficiency of the RG scheme of Ref. 34 is the treatment of the scalings with length and u/T as interchangeable. While this approach is justified for the “elastic renormalization” of disorder (i.e., the screening effects), it does not properly account for the dephasing (since the dephasing length Lφ is in general different from LT , see Sec. III B 1 and Sec. IV C). 5. Dephasing and localization in disordered Luttinger liquid As already mentioned in Sec. II B 4, a key ingredient of transport theory as regards the WL and the interactioninduced dephasing in a strongly correlated 1d system is missing. Some papers (e.g., Refs. 34,81) have suggested that the dephasing length that controls localization effects in a disordered Luttinger liquid is LT = u/T , where u is the plasmon velocity. According to this approach, the interference effects get strong with lowering T at LT ∼ ξ. An alternative approach33 is predicated on the assumption that the dephasing rate is determined by the singleparticle properties of a clean Luttinger liquid. On top of that, one might think that since in the case of linear dispersion the clean interacting electron system can be equivalently represented in terms of noninteracting bosons, the interaction should not induce any dephasing at all. The conductivity would then be exactly zero at any T . As we demonstrate below, none of the approaches captures the essential physics of dephasing in the conductivity of a disordered 1d system. As for the strongly localized regime, recently Ref. 37 used bosonization to study the problem and concluded that transport is of variable-range-hopping character. Subsequently, Ref. 38 argued that this result requires a nonzero coupling to a thermal bath, in agreement with Ref. 64, see Sec. II A 2. C. Summary of known results: 1d vs 2d We summarize the basic facts discussed in Secs. II A and II B in Table I: for simplicity, we compare here 2d and single-channel 1d systems. It turns out that the difference between the disordered Luttinger liquid and the higher-dimensional (quasi-1d and 2d) systems is not so strong as one might think judging by the proverbial nonFermi-liquid character of the Luttinger liquid. III. DISORDERED LUTTINGER LIQUID A. Formulation of the problem Let us now introduce the model we are going to study. We consider a single-channel infinite wire at sufficiently low temperatures, much lower than the bandwidth and the Fermi energy. The effective low-energy theory can then be described by the Luttinger model,27,28,29,30,31,32 with a linear dispersion of the electronic spectrum ǫk = kvF , where vF is the Fermi velocity. We use the notion of right- and left-movers corresponding to two Fermi points k ∼ ±kF . The fermionic operators ψµ are then labeled by µ = ± for the right-/left-movers. We consider a finite-range (screened) pairwise e-e interaction potential V (x−x′ ). We need only its Fourier transforms at q = 0 and q = ±2kF : Vf and Vb , respectively, which give the amplitudes of forward and backward scattering. In terms of the g-ology,27,28,32 the interaction in a homogeneous system is characterized by three coupling constants g1 = Vb (e-e backscattering), g2 = Vf (forward scattering between right- and left-movers), and g4 = Vf (forward scattering of right-/left-movers on right-/leftmovers). In several places below, it will be convenient to treat g2 and g4 as independent parameters. Next, we introduce weak (ǫF τ0 ≫ 1) white-noise dis- 6 † † + ψµ,σ ψ−µ,σ g1 ψ−µ,σ′ ψµ,σ′ TABLE I: The 2d-1d vocabulary. Hdis = 2d 1d disorder, no interaction: σ(T ) = 0 disorder, no interaction: σ(T ) = 0 interaction, common tool: fermionic diagrams interaction, common tool: bosonization field theory: σ-model Q-matrices field theory: bosonic density fields zero-bias anomaly in tunneling DoS, Eq. (2.3) Luttinger (“non-Fermi”) liquid, power-law DoS, Eq. (2.7) T-dependent screening (Friedel oscillations) renormalization of impurity (Friedel oscillations), Eq. (2.8) Altshuler-Aronov correction to the conductivity, Eq. (2.2) renormalization of disorder −→ σD (T ), Eq. (2.9) Finkel’stein RG: disorder (perturbative) & interaction (exact) Giamarchi-Schulz RG: disorder (perturbative) & interaction (exact) LT ∼ ξ : strong Altshuler-Aronov corrections = Anderson localization LT ∼ ξ : strong coupling in Giamarchi-Schulz RG = Anderson localization Lφ ∼ ξ, ∆σwl ∼ σD : strong Anderson localization Lφ ∼ ξ, ∆σwl ∼ σD : strong Anderson localization weak localization correction to the conductivity, Eq. (2.1) this work, Secs. VI,VII dephasing time, Eq. (2.4) this work, Secs. VI,VII σ (3.1) where τ0 is the elastic scattering time and ν0 = 1/πvF is the density of states per spin in the absence of interaction. The Hamiltonian for spinful electrons takes the form: H = Hkin + Hee + Hdis , Hkin = k,µ=±,σ Hee = 1 2 µ,σ,σ′ † vF (µk − kF )ψµσ (k)ψµσ (k) , σ (3.4) † † ∗ dx Ub ψ+,σ ψ−,σ + Ub ψ−,σ ψ+,σ † † ∗ dx Uf ψ+,σ ψ+,σ + Uf ψ−,σ ψ−,σ . (3.5) Here † nµ,σ (x) = ψµ,σ (x)ψµ,σ (x) (3.6) is the fermionic density, Ub and Uf are backscattering and forward-scattering amplitudes due to disorder. It can be shown32,34 that the forward-scattering amplitude can be gauged out in the calculation of the conductivity. Therefore, from now on we set Uf ≡ 0. The correlation functions for Ub (x) read Ub (x)Ub (x′ ) = 0 , ∗ Ub (x)Ub (x′ ) = U (x)U (x′ ) . (3.7) (3.8) Two points are worth mentioning here. First, all the parameters of the above Hamiltonian include, in general, high-energy renormalization effects (similar to the Fermi-liquid renormalization in higher dimensionalities) and should be considered as effective couplings of the low-energy theory. Second, the low-energy theory is only then well-defined when supplemented by an ultraviolet energy cutoff Λ. The latter is determined by the Fermi energy, the bandwidth, or the spatial range of interaction in the original microscopic theory, whichever gives the smallest cutoff. Let us now concentrate on the spinless model. In this case, two essential simplifications occur. First, the interaction-induced backward scattering and forward scattering relate to each other as direct and exchange processes. Therefore, the backscattering Vb only appears in the combination Vf − Vb and thus merely redefines the parameters of the Luttinger model, formulated27,28 in terms of forward-scattering amplitudes only. In other words, the term with g1 in Eq. (3.4) has the same structure as the g2 -term, so that we can absorb g1 into g2 , g2 ≡ Vf − Vb = g2 − g1 , ˜ order, described by the correlation function U (x)U (x′ ) = δ(x − x′ )/2πν0 τ0 , + , and start our consideration with the following form of the Hamiltonian (3.2): Hkin = k,µ=± Hee = (3.2) (3.3) dx {nµ,σ g2 n−µ,σ′ + nµ,σ g4 nµ,σ′ Hdis = † vF (µk − kF )ψµ (k)ψµ (k) , 1 2 µ=± (3.9) dx {nµ g2 n−µ + nµ g4 nµ } , (3.10) ˜ † † ∗ . dx Ub (x) ψ+ ψ− + Ub (x) ψ− ψ+ (3.11) In what follows we omit the tilde over the shifted coupling g2 . The second simplification is related to the g4 term in Eq. (3.10). Naively, one could think it is simply 7 zero, since it describes a local interaction between identical fermions. In fact, the situation is slightly more tricky. While yielding no genuine interaction, this term generates a shift of the velocity due to a quantum anomaly, ∗ vF → vF , where ∗ vF = vF + g4 . 2π (3.12) This is also straightforwardly seen in the bosonization approach, see Sec. IV. Therefore, the complete set of parameters defining the ∗ spinless problem includes vF , g2 , τ0 , and Λ, with the interaction characterized by a single coupling g2 . It is customary to define the corresponding dimensionless parameter in the form ∗ 1 − g2 /2πvF ∗ 1 + g2 /2πvF K= 1/2 1. . 1 . (1 + g2 /πvF )1/2 (3.14) For the case of a weak interaction (considered in the present paper), K ≃ 1 − α, where α= g2 g2 ≪1 ∗ ≃ 2πv 2πvF F Step 1: Luttinger-liquid renormalization (3.13) Note that in a 1d system with g2 = g4 Eq. (3.13) is identical to another conventional representation of K in terms of the bare velocity vF , K= that the processes with high and low energy transfers are treated on the same footing. This method yields results identical to the two-step approach and is presented in Sec. VII. Let us describe now the ideas underlying our two-step approach. The method is based on the separation of two types of processes: (i) “elastic renormalization” of disorder, which is associated with virtual transitions with energy transfers |ω| T and (ii) inelastic processes with |ω| T which lead to dephasing in quantum-interference effects and thus to a destruction of the Anderson localization. This separation of energy scales is very much similar to that in physics of mesoscopic phenomena in higher-dimensional case, see Sec. II A. (3.15) is the dimensionless strength of the interaction. B. Qualitative discussion: Luttinger-liquid renormalization vs Fermi-liquid dephasing We begin by considering the Drude conductivity under the condition that the dephasing time τφ is much shorter than the transport time of elastic scattering off disorder τ . The source of the strong dephasing may be external (say, interaction with phonons, or Coulomb interaction with “environment”) or intrinsic, that is the inelastic e-e interaction within the quantum wire. In the latter case, the limit τφ ≪ τ is achieved at sufficiently high T , as shown below in Sec. VI. For τφ ≪ τ , the Andersonlocalization effects are strongly suppressed by dephasing. To leading order in τφ /τ ≪ 1, the conductivity is given by the Drude formula 2 σD = e2 ρvF τ , where ρ is the compressibility. The Drude conductivity (3.16) depends on T through a T -dependent renormalization of the static disorder:34,77,78,79,80 τ (T ) = τ0 (T /Λ)2a . Having formulated the model in Sec. III A, we turn now to a discussion of our strategy in its analysis. In principle, one could proceed in a number of alternative ways: (i) First of all, one can bosonize the action and try to study the transport properties of interacting electrons in the bosonic language, Sec. IV. While this approach is very powerful for the problem of 1d interacting fermions, it turns out to be poorly suited for the analysis of transport problems in the presence of disorder. For this reason, we develop an alternative “two-step procedure”. (ii) The two-step procedure combines the language of bosonization for high energies with a fermionic treatment of processes with low-energy transfers. This method is discussed on the qualitative level in the remainder of this subsection and is implemented in Secs. IV,VI below. (iii) Finally, one more alternative is the “functional bosonization” approach.89,90,91,92,93 This method also combines the languages of fermionic (electrons) and bosonic (plasmons) excitations, but in contrast to the two-step procedure this is done in a uniform fashion, so (3.16) (3.17) For spinless electrons, the exponent in Eq. (3.17) reads a=1−K ≃α . (3.18) In principle, the interplay of disorder and interaction leads to the renormalization of both disorder, Eq. (3.17), and interaction.34 However, the value of α in Eq. (3.18) is given by the bare interaction constant (the one in a clean system), see Sec. IV for details. The Luttinger-liquid renormalization (3.17) is similar to the Altshuler-Aronov corrections46 [Eq. (2.2)] in higher dimensionalities. The underlying physics of the elastic renormalization of disorder can be described in terms of the T -dependent screening of individual impurities and scattering by slowly decaying in real space Friedel oscillations, similarly to higher-dimensional systems.53,54 At this level, the only peculiarity of Luttinger liquid as compared to higher dimensionalities is that the renormalization of τ is more singular and necessitates going beyond the Hartree-Fock (HF) approach even for the weak interaction.94 8 Technically, Eq. (3.17) can be derived by a variety of methods: either fermionic27,79 or bosonic.34,80 In particular, a conceptually important framework is based on the bosonic RG procedure developed by Giamarchi and Schulz.34 This approach allows one to integrate out all energy scales between Λ and T , which are responsible for the elastic Luttinger-liquid renormalization due to virtual e-e scattering processes. We will discuss the features of the bosonic RG approach34 relevant to our analysis in more detail in Sec. IV. The renormalization of τ stops with decreasing T at T τ (T ) ∼ 1. Physically, this is due to the fact that the long-range Friedel oscillations are cut off even at zero T on the spatial scale of the order of the disorder-induced mean free path. The condition T τ (T ) ∼ 1 (3.19) The physics of these processes is in essence equivalent to the Fermi-liquid physics of dephasing in higher dimensionalities. Having integrated out the virtual transition, we have to choose a controllable method of dealing with the dephasing. The bosonization approach (whose advantage is that it can be straightforwardly formulated for the arbitrary strength of interaction) is not particularly beneficial here. Although, in principle, it contains all effects of the Anderson localization in 1d, the machinery of bosonization is poorly suited to deal with such ingredients of the localization theory as the interference and dephasing. Even for noninteracting electrons, it is by far not straightforward to derive the Anderson localization [for instance, to obtain the Berezinskii-Mott formula (2.6) for the case of weak disorder] in the language of bosonization. For this reason, we refermionize the theory at this step, Secs. V,VI. gives the zero-T localization length 1/(1+2a) ξ ∝ τ0 . It is important to stress, however, that Eq. (3.19) does not correctly predict the onset of localization, in contrast to the argument made in Refs. 34,81. This can be seen, in particular, by noting that the temperature T ∼ τ −1 does not depend on the strength of interaction for small α, whereas it is evident that for noninteracting electrons there is no interaction-induced dephasing (the dephasing length is infinite, Lφ = ∞) and hence σ(T ) = 0 for any T due to the Anderson localization, see Sec. II B 1. The onset of strong localization occurs at Lφ ∼ ξ, as in the case of higher dimensionality, see the discussion in Sec. II A. The temperature T1 below which the Anderson-localization effects become strong is determined in 1d by the condition τ (T1 ) ∼ τφ (T1 ) . (3.21) At this temperature, the quantum-interference effects leading to the WL correction ∆σwl to the conductivity are no longer weak, |∆σwl (T1 )| ∼ σD (T1 ) . 2. IV. (3.20) (3.22) A. Bosonized action As discussed in Sec. III B, the first part of our program, namely the renormalization of the model due to highenergy processes, is done most efficiently in the framework of the bosonization technique. In this section, we present a calculation of τ (T ), using the RG scheme developed by Giamarchi and Schulz.34 After a brief description of the derivation of the RG equations, we will discuss an important subtlety of the scheme,34 namely a mixing of disorder and interaction. For simplicity, we consider here the spinless case (in notation of Ref. 34, we set g1⊥ = g2⊥ = 0 and hence uρ = uσ , Kρ = Kσ ). As discussed in Sec. III A, for spinless electrons the backscattering amplitude g1 can ∗ be absorbed into g2 , and g4 into vF , so that only one interaction coupling constant remains (g2 ), yielding the dimensionless Luttinger-liquid parameter K, Eq. (3.13). Using the boson representation of the fermionic operators,32 the Hamiltonian can be expressed in terms of the boson operators φ and Π that satisfy the commutation relation [Π(x), φ(x′ )] = −iδ(x − x′ ). In a given realization of disorder the Hamiltonian reads 1 u dx uK(πΠ)2 + (∂x φ)2 2π K 1 ∗ dx {Ub (x) exp(2iφ) + H.c.} + 2πλ Step 2: Fermi-liquid dephasing It is worth stressing that the Luttinger-liquid physics in the present problem is fully accounted for by Eq. (3.17), i.e., all the Luttinger-liquid power-law singularities are now incorporated in the renormalization of τ (T ). At this point, we are left with the system of interacting electrons with a new effective “Fermi energy” T . These electrons are scattered by disorder with renormalized τ0 → τ (T ), which is still weak for T τ (T ) ≫ 1. What remains after the renormalization is the e-e scattering with energy transfers |ω| T . The interaction is thus still important and leads to real inelastic processes. BOSONIZATION APPROACH H = (4.1) with ∗ 2 u = (vF )2 − g2 /4π 2 1/2 . (4.2) The ultraviolet length scale λ is related to the energy cutoff Λ as λ = u/πΛ . (4.3) 9 We will concentrate on the case of e-e repulsion, implying that K < 1. The disorder-induced forward scattering amplitude Uf (x) is absorbed34 in the quadratic part of H as a shift of φ(x). In order to perform the disorder averaging, one introduces replicas, φn . In the resulting action the interaction (Luttinger-liquid) term SLL is quadratic in fields φn , while disorder generates a term SD of cos 2φ-type which breaks the Gaussian character of the theory: S = SLL [φn ] + n SLL = SD [φn , φm ] , (4.4) B. n,m 1 uK SD = −Db dxdτ [∂τ φn (x, τ )]2 + u2 [∂x φn (x, τ )]2 , 2π (4.5) ′ dx dτ dτ cos{2[φn (x, τ ) − φm (x, τ ′ )]} , (2πλ)2 (4.6) ∗ where Db = vF /2τ0 . The integration over the imaginary times is performed within the interval 0 ≤ τ ≤ 1/T. The conductivity is expressed in terms of the currentcurrent correlation function at zero momentum via the Kubo formula: σ(Ω) = − e2 vF − iΩ π 1/T 0 dτ eiΩn τ K(τ ) (4.7) iΩn →Ω+i0 with K(τ ) = dx Tτ j(x, τ )j(0, 0) , i ∂τ φ(x, τ ) . π Giamarchi-Schulz RG Giamarchi and Schulz34 performed the renormalization of the coupling constants K, u, and Db in the action (4.4) upon rescaling the ultraviolet cutoff λ → L = λ exp(ℓ). Their analysis is done to first order in the dimensionless strength of disorder D = 2Db λ/πu2 . (4.10) To this order, one can omit replica indices in Eq. (4.4). Before presenting the RG equations, we discuss a subtle point of the approach developed in Ref. 34. Specifically, ˜ the coupling constant K in terms of which the RG equations are obtained in a natural way, is not simply K but rather contains an admixture of disorder. We now briefly reproduce the corresponding argument from Ref. 34. In the fermionic representation, the part of the action related to the disorder-induced backscattering acquires, after the disorder averaging, the form (4.8) where in the bosonic language the currents are expressed via the field φ(x, τ ) as j(x, τ ) = can be represented as a certain approximation in the framework of the memory-function formalism, see, e.g., Refs. 78,97). In view of technical difficulties, in this paper we do not attempt to evaluate the conductivity, including the localization effects, by using the Kubo formula in the bosonic language either. Having used the bosonization for the analysis of the high-energy renormalizations, we refermionize the theory and calculate the conductivity in the fermionic language, Sec. VI. (4.9) The first (diamagnetic) term in Eq. (4.7) is canceled by a contribution of the singularity at τ = 0 in the second term. It is worth noting that, strictly speaking, the correlation function Tτ ∂τ φ(x, τ )∂τ φ(0, 0) is a reducible (with respect to the interaction) correlator, when it is calculated with the use of SLL as given by Eq. (4.5). On the contrary, the conductivity is given by the irreducible current-current correlator. However, at zero external momentum Q, the two correlators are equal to each other, since the polarization operator vanishes at Q = 0 for arbitrary nonzero frequency. This should be contrasted with a calculation of the conductance of a finite interacting wire: the latter is related to the response at finite Q and the Kubo formula should be supplemented with an analysis of the effect of renormalization of the electric field by e-e interaction.95,96 Note that the authors of Ref. 34 did not calculate the conductivity using the Kubo formula (4.7), (4.8), (4.9). Instead, they found the renormalized value of 1/τ and substituted it into the Drude formula, Eq. (3.16) (which SD = −Db † dx dτ dτ ′ ψ+ (x, τ )ψ− (x, τ ) † × ψ− (x, τ ′ )ψ+ (x, τ ′ ) . (4.11) In order to derive the RG equations, Giamarchi and Schulz singled out the ultraviolet contribution of close times u|τ − τ ′ | ≤ λ, splitting the time integration into two parts: SD = −Db dx dτ dτ ′ u|τ −τ ′ |>λ † † × ψ+ (x, τ )ψ− (x, τ )ψ− (x, τ ′ )ψ+ (x, τ ′ ) (4.12) 2Db λ − dx dτ u † † × ψ+ (x, τ )ψ− (x, τ )ψ− (x, τ )ψ+ (x, τ ) . (4.13) The second term in Eq. (4.13) is local in time and thus equivalent to an interaction term generated by g1 processes. For this reason, the authors of Ref. 34 redefined the interaction constant, g1 ≡ g1 −2Db λ/u. For the ˜ case of spinless electrons considered here, this is equivalent to a modification of the constant g2 : g2 = g2 + 2Db λ/u . ˜ (4.14) 10 In terms of the modified interaction constants the RG equations34 (in the spinless case only three equations out of original six remain) are: ˜ ˜ dK/dℓ = −K 2 D/2 , (4.15) d˜/dℓ = −˜KD/2 , u u˜ (4.16) ˜ dD/dℓ = (3 − 2K)D , (4.17) ˜ where u and K are related to the modified interaction ˜ constant g2 (ℓ) = g2 (ℓ) + πuD(ℓ) ˜ (4.18) according to Eqs. (4.2) and (3.13), respectively. The modification of the plasmon velocity u by disorder, as well as its renormalization, can be neglected in Eq. (4.18), since this generates higher powers of D. Then Eq. (4.16), which describes the renormalization of u, turns out to have no influence on the renormalization of disorder and interaction, described by the remaining ˜ u two equations. [In fact, Eqs. (4.15),(4.16) show that K/˜ is the invariant of the flow.] Therefore, we will concentrate on the coupled set of the flow equations (4.15), ˜ (4.17). It follows from Eq. (4.17) that for K < 3/2 the strength of disorder D(ℓ) increases upon renormal˜ ization. At the same time, the value of K decreases. The system flows then to the strong-coupling regime where D becomes of order one and the first-order treatment of disorder is no longer valid. ˜ The decrease of K(ℓ) upon renormalization suggests that the initially weak interaction (K close to 1) becomes strong upon approaching the strong coupling regime. However, this is not so, as we demonstrate below: the ˜ decrease of K occurs solely due to the admixture of increasing disorder in this quantity. It is also worth noticing that Eq. (4.15) seems to generate finite interaction due to the scattering off disorder in the noninteracting case. However, as verified in Ref. 34, ˜ this is not the case. The point is that the condition K = 1 does not mean that the system is noninteracting, because of the mixing of interaction and disorder in Eq. (4.18). Therefore, it is desirable to rewrite Eqs. (4.15), (4.17) in terms of the true interaction constant K instead of ˜ the one modified by the admixture of disorder, K (see Fig. 9.6 in Ref. 32). Let us first consider the case of weak interaction, 1 − K ≃ α ≪ 1, retaining only the terms of first order in α in the RG equations. From Eqs. (3.13),(4.2),(4.10), and (4.14) we get, to first order in α, ˜ K = 1 − α − D/2 + αD/2 . (4.19) Substituting this formula in Eq. (4.17), we express the flow equation for the disorder strength in terms of the renormalized true interaction constant α(ℓ): dD/dℓ = D + 2αD + D2 . (4.20) In this equation, the first term in the right-hand side describes Ohm’s law, while the second term (∝ α) yields the Luttinger-liquid renormalization of τ , Eq. (3.17). The last (quadratic-in-disorder) term in Eq. (4.20) is beyond the accuracy of the derivation of the RG equations and should be neglected. We note, however, that this spurious term is hidden in the RG equations of Ref. 34. Accidentally, this spurious quadratic term qualitatively mimics the effect typical for the WL. Clearly, this purely accidental similarity does not hold on the quantitative level: the true WL—when included into consideration—would only appear in the r.h.s. of Eq. (4.20) at D3 order. Now let us turn to the flow equation (4.15) for the interaction strength. We rewrite it using Eqs. (4.19) and (4.20) in the form dα/dℓ = −3αD/2 − D2 . (4.21) Again, the last (quadratic-in-disorder) term here is beyond the accuracy of the calculation and has to be neglected (this unphysical term generates interaction from disorder). It is important that the linear-in-disorder term in Eq. (4.21) vanishes for α = 0. In other words, disorder does not generate interaction, which was not trivial to see from Eq. (4.15). To summarize, for a weak interaction, α ≪ 1, the set of two scaling equations for α and D reads dα/dℓ = −3αD/2 , dD/dℓ = (1 + 2α)D . (4.22) (4.23) One sees that Eq. (4.22) yields decreasing α(ℓ): while the disorder strength grows with increasing L, the interaction gets weaker (“disorder kills interaction”—Ref. 32; also Ref. 98), in contrast to a naive expectation based on Eq. (4.15). This resembles the behavior of the renormalized interaction in 2d diffusive systems. Specifically, according to the Finkel’stein RG equations,55 the combination of the singlet and triplet amplitudes (an analog of α for the spinful case) which determines the interactioninduced correction to the conductivity ∆σAA , decreases for weak interaction (α ≪ 1) due to the disorder-induced renormalization of interaction. For completeness, let us also consider the RG equations ˜ for arbitrary strength of interaction. Expanding K in D and using Eqs. (4.2) and (4.14), we obtain ˜ K ≃ K − (1 + K 2 )D/4 . (4.24) Substituting Eq. (4.24) in Eqs. (4.15) and (4.17), we arrive at dK (1 + K 2 )(3 − 2K) 1 D , (4.25) = − K2 − dℓ 2 2 dD = (3 − 2K)D . dℓ (4.26) The solution to Eqs. (4.25),(4.26) is shown in Fig. 1; for comparison, we show also the solution to 11 0.3 D 0.4 0.3 D 0.4 0.2 0.1 0.1 0 0.2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 0.25 0.5 0.75 K 1.25 1.5 1.75 1.25 1.5 1.75 K 0.08 0.06 0.06 DL 0.08 DL 1 0.04 0.02 0.02 0 0.04 0 0.25 0.5 0.75 1 1.25 1.5 1.75 K FIG. 1: Upper panel: the RG flows given by Eqs. (4.25),(4.26) in the plane (K, D), where K is the true interaction constant and D is the strength of disorder (the dimensionless resistance). Dashed line: the separatrix touches the quantum critical point at K = 3/2, D = 0. Long-dashed line: the line of K = 1 separates the repulsion (K < 1) and attraction (K > 1) regions. Lower panel: the RG flows from Eqs. (4.25),(4.26) in the coordinates K and D/L, the latter corresponds to the resistivity. The renormalization for each curve is stopped at D = 1 (the strong coupling regime). The dashed line separates the repulsion and attraction regions: an attractive (repulsive) interaction cannot become repulsive (attractive) in the course of the renormalization. Initial couplings in the upper panel from left to right: D0 = 0.001, K0 = 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 and K0 = 1.9, D0 = 0.3, 0.25, 0.2, 0.15, 0.05. Separatrix: K0 = 1.9; D0 ≃ 0.079. Initial couplings in the lower panel: K < 1 (from left to right): D0 = 0.001, K0 = 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9; K > 1 (from top to bottom): K0 = 1.9, D0 = 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.10, 0.05. Eqs. (4.15),(4.17) in Fig. 2. The flow equation (4.25) for K tells us that the repulsive interaction, K < 1, gets weaker in the course of renormalization, dK/dℓ > 0, whatever the interaction strength is, see Fig. 1. Equations (4.25),(4.26) reduce to the set of Eqs. (4.15),(4.17) around K = 3/2, where the Giamarchi-Schulz RG predicts the metal-insulator transition. Note that for K > ˜ 3/2 the flows in terms of K and K are qualitatively similar to each other, as is evident from a comparison of 0.25 0.5 0.75 1 K FIG. 2: Upper panel: the RG flows given by Eqs. (4.15),(4.17) in the plane of the modified interaction con˜ stant K, introduced in Ref. 34, and the strength of disorder (the dimensionless resistance) D. Dashed line: the separa˜ trix touches the quantum critical point at K = 3/2, D = 0. Lower panel: the RG flows from Eqs. (4.15),(4.17) in ˜ the coordinates K and D/L (corresponding to the resistivity). The renormalization for each curve is stopped at D = 1 (the strong coupling regime). Initial couplings in ˜ the upper panel from left to right: D0 = 0.001, K0 = 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 ˜ and K0 = 1.9, D0 = 0.3, 0.25, 0.2, 0.15, 0.05. Separatrix: ˜ K0 = 1.9; D0 ≃ 0.104. Initial couplings in the lower panel: ˜ D0 = 0.001, K0 = 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and ˜ K0 = 1.9, D0 = 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.10, 0.05. Figs. 1 and 2. Let us return to the case of weak interaction and analyze the RG flow generated by Eqs. (4.22),(4.23), see Fig. 3. An approximate solution of these equations in the range of weak disorder, D(ℓ) ≪ 1 (which is where they are valid), is easily found by an iterative procedure. The result is: 1+2α 0 D(ℓ) ≃ D0 (L/λ) , α(ℓ) ≃ α0 [ 1 − 3D(ℓ)/2 ] , (4.27) (4.28) where D0 and α0 are the initial (ultraviolet) values of the coupling constants at L ∼ λ. It is worth emphasizing that whereas the correction to α0 is of the order of α0 itself when D(ℓ) ∼ 1, the renor- 12 1 0.9 K 0.8 0.7 0.6 0.5 0 0.05 0.1 0.15 0.2 0.15 0.2 ΛL 1 D0 L DΛ 0.8 0.6 0.4 0.2 0 0 0.05 0.1 ΛL FIG. 3: Upper panel: the renormalization of the interaction constant K due to disorder for various initial values of K in the ultraviolet limit λ/L = 1. The strength of interaction decreases upon renormalization, K → 1. Lower panel: the renormalization of the “conductivity” L/D in units of the bare “conductivity” λ/D0 (solid line). The dashed lines describe the dependence of L/D given by Eq. (4.27) with the exponent α0 = 1 − K0 , where K0 is the ultraviolet value of the Luttinger constant. Each flow is stopped at D = 1. Initial couplings from top to bottom, both panels: D0 = 0.001, K0 = 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5. malization of disorder is governed by the initial value of the interaction constant and not by the running coupling constant α(ℓ). The renormalization of the interaction constant is irrelevant in the present case, and hence the renormalization of disorder reduces to the renormalization of an individual impurity. As a consequence, the exponent a in Eq. (3.17) for the renormalized scattering time τ (T ) is given by the bare interaction constant (the one in a clean system), a = a0 ≃ α0 . Thus, in the Drude formula (3.16) the exponent of the Luttingerliquid power-law function (T /Λ)a is T -independent. longer valid. Since in the absence of interaction the strong-coupling fixed point D/L → ∞ corresponds to the complete Anderson localization, Giamarchi and Schulz identified the regime D ≫ 1 as the localized phase. It should be stressed, however, that the notion of Anderson localization does not follow from the RG equations of Ref. 34. Indeed, Eq. (4.23) does not account for the renormalization of disorder D/L in the absence of interaction: the effects of Anderson localization (arising due to interference terms which involve higher powers of D) are not included in this RG scheme. In the weak-coupling regime, D ≪ 1, the Anderson localization would show up in an additional α-independent term, fwl (D) ∼ D3 on the right-hand side of Eq. (4.23), which is beyond the accuracy of the derivation of these equations. It is also worth noting that in the presence of interaction all the RG trajectories for D/L tend to saturate at finite values of D/L, as can be seen from Fig. 1, in contrast to what one expects for the Anderson-localization behavior. The strong-coupling fixed point D → ∞ is therefore a manifestation of Ohm’s law, rather than of the Anderson localization. In order to study the temperature behavior of σ(T ), the renormalization procedure in Ref. 34 was stopped at L = LT ∼ u/T , with the argument that the coherence effects disappear on length scales greater than LT due to inelastic processes. Although this is perfectly correct for the interaction-induced renormalization of disorder which is cut off by the thermal smearing of the electron distribution function [similarly to the Altshuler-Aronov correction to the conductivity in higher-d systems, see Eq. (2.2)], the coherence effects relevant to the Anderson localization are governed by the dephasing length, Lφ . The condition LT ∼ ξ yielding the temperature ∼ u/ξ for the onset of the Anderson localization is in general not correct. Indeed, the additional term fwl (D) responsible for the WL (quantum-interference) effects is cut off not by LT but rather by the dephasing length Lφ , which is parametrically different from LT . In particular, in the absence of interaction Lφ diverges, while LT remains finite, being independent of α. For weak interaction Lφ ≫ LT and hence, there exists an intermediate range of length scales, LT L Lφ , in which only the WL term fwl (D) contributes to the renormalization of D. Thus, it turns out that the scalings with T and L are not interchangeable, when σ(T ) is considered. For α ≪ 1, the onset of strong localization is determined by the condition Lφ ∼ ξ , (4.29) which in 1d is equivalent to C. Bosonization vs Localization As discussed in Sec. IV B, the solution of the RG equations flows to the strong-coupling regime, where D 1 and the perturbative-in-disorder RG equations are no τφ ∼ τ . (4.30) After the RG transformation, the bosonization action retains its original form (4.4)–(4.6), but with a renormalized ultraviolet cutoff, Λ → T , and a renormalized scat−1 tering rate, τ0 → τ −1 (T ). Performing a transformation 13 from the bosons back to fermions, we thus return to the original fermionic theory with a corresponding renormalization of its parameters. This theory is analyzed in the following sections. V. GOLDEN RULE IN LUTTINGER LIQUID In order to analyze the effects of dephasing, we will follow the route suggested by earlier works on higherdimensional (in particular, diffusive) systems.46 Since the physics of dephasing is governed by electronic inelastic scattering processes, a natural first step is to cal−1 culate the e-e scattering rate τee . Indeed, for higherdimensional systems in the high-temperature (ballistic) −1 regime, T τ ≫ 1, the dephasing rate τφ to leading order −1 is given by τee , see Ref. 99. For quasi-1d and 2d systems in the diffusive regime the scattering kernel acquires an −1 infrared singularity leading to a divergent τee . However, even in this case the calculation of the e-e collision rate turns out to be instructive: a parametrically correct result for the dephasing rate can be obtained from the −1 expression for τee supplemented with an appropriate in−1 frared cutoff. It is thus useful to begin by analyzing τee . We will show that this quantity is finite and meaningful in the interacting 1d system (contrary to what one might expect based on the famous non-Fermi-liquid character of the Luttinger liquid). On the other hand, we will demonstrate that, while governing the dephasing of Aharonov−1 Bohm oscillations, τee does not give the dephasing rate for the WL of spin-polarized electrons. The Golden Rule expression for the e-e collision rate following from the Boltzmann kinetic equation reads 1 = τee (ǫ) × dω H H Here K++ and K+− are related to scattering of two electrons from the same or different chiral spectral branches, H respectively, KF is the exchange counterpart of K++ , ηs is the spin degeneracy (ηs = 1 for the spinless case and ηs = 2 for the spinful case), and D± (ω, q) are the two-particle propagators, given by Eq. (A.4) with iΩn → ω + i0. Substituting Eqs. (5.2)–(5.5) in Eq. (5.1), we obtain the lowest-order result for the e-e scattering rate at the Fermi level (ǫ = 0) in terms of the corresponding contributions to the retarded electronic self-energy ΣR defined + by GR (ǫ, p) = [ ǫ − vF p − ΣR (ǫ, p) ]−1 , where GR is the + + + retarded Green’s function for right-movers: 1/2τee = −Im ΣR (0, 0) , + ΣR (0, 0) + + + ΣH +− (5.1) (5.8) a) g2 − g2 + g4 g4 + b) c) where K(ω) is the kernel of the e-e collision integral St{fǫ } = dω dǫ′ K(ω) h h h h −fǫ fǫ−ω fǫ′ fǫ′ +ω + fǫ fǫ−ω fǫ′ fǫ′ +ω fǫ h + dω dǫ′ K(ω)fǫ−ω fǫ′ fǫ′ +ω . = − τee (ǫ) × h Here fǫ is the Fermi distribution function, fǫ = 1 − fǫ . To leading (second) order in interaction, the scattering kernel K(ω) in a clean Luttinger liquid has the form H H K(ω) = ηs [ K++ (ω) + K+− (ω) ] + K F (ω) , (5.2) where 1 π3 ρ 2 1 = g2 3 π ρ H 2 K++ = g4 H K+− H K F = −K++ dq 2 [ ReD+ (ω, q) ] , (5.3) 2π dq ReD+ (ω, q) ReD− (ω, q) , 2π (5.4) . (5.5) (5.7) We refer to this cancellation as the “Hartree-Fock cancellation”. g4 , . ΣF = −ΣH . ++ + h fǫ−ω fǫ′ fǫ′ +ω +Σ (5.6) F The diagrams for ΣH , ΣH , and ΣF are shown in +− ++ Figs. 4a, 4b, and 4c, respectively. The contribution of the diagram in Fig. 4b (Hartree term) is canceled by the diagram in Fig. 4c (Fock term) due to the Pauli principle, dǫ′ K(ω) h h fǫ−ω fǫ′ fǫ′ +ω = ΣH ++ + g4 FIG. 4: Second-order (“Golden Rule”) diagrams for the selfenergy of a right-mover. Wavy lines: interaction of the rightmover with left- (g2 ) or right- (g4 ) movers. (a) ΣH , the +− Hartree contribution due to the interaction with a left mover; (b) ΣH , the Hartree contribution due to the interaction with ++ a right mover; (c) ΣF , the Fock (exchange) contribution. Let us now calculate the imaginary parts of the perturbative self-energies. Using Eq. (A.4) to get the retarded propagators D± (ω, q) by analytically continuing iΩn → ω + i0 and putting g2 = g4 = 2πvF α we find: 3 ImΣH = −2π 2 α2 vF ++ × × dω ω ω ω coth − tanh 2π 2T 2T dq 2 [ ReD+ (ω, q) ] ≃ −πα2 vF T 2π dω |ω| T dq δ(ω − vF q)δ(ω − vF q) , (5.9) 14 dω ω ω ω coth − tanh 2π 2T 2T One sees that the contribution of ΣH is diverging. For ++ spin-polarized electrons (i.e., for ηs = 1) it is, however, exactly canceled by the exchange interaction, Eq. (5.8). Indeed, as we have discussed in Sec. III A, the g4 interaction drops out of the problem in this case, inducing only a shift of the velocity, Eq. (3.12). The remaining term ΣH is determined by ω, q → 0 and is given by +− property. The Luttinger-liquid physics is in fact encoded in the singular real part of the perturbative self-energy, Eq. (5.12). For α2 ln(Λ/T ) ≪ 1, the Green’s function can be written as GR (ǫ, p) ≃ Z/(ǫ − vF p + iπα2 T /2), + where the single-particle weight Z ≃ 1 − (α2 /2) ln(Λ/T ) is suppressed by interaction. It is in fact the product α2 ln(Λ/T ), coming from the real part of the self-energy, that after resummation of all orders gets exponentiated and leads to the Luttinger-liquid power-law singularities. Those singularities have been accounted for by the renormalization at step 1 in the present approach. It is instructive to compare the perturbative collision rate in a clean Luttinger liquid, Eq. (5.13), with the damping100 of the exact retarded single-particle Green’s function in the coordinate-time (x, t) representation, ImΣH = −πα2 T /2 . +− R g± (x, t) = 2iΘ(t) Im g± (x, t) . ¯ 3 ImΣH = −2π 2 α2 vF +− × × dq ReD+ (ω, q) ReD− (ω, q) ≃ −πα2 vF T 2π dω |ω| T dq δ(ω − vF q)δ(ω + vF q) . (5.10) (5.11) Already the perturbative expansion demonstrates a qualitative difference between two cases of spinless and spinful electrons—the spinful case will be considered in Ref. 44. The characteristic frequencies in Eqs. (5.9),(5.10) satisfy |ω| T , because of the factor coth(ω/2T ) − tanh(ω/2T ), even if we would perform this calculation in the original theory with the ultraviolet cutoff Λ. For the sake of comparison, we present also the corresponding contribution to the real part of the self-energy: Re Σ+ (ǫ, p) = × 2 −2π 2 α2 vF dω 2π coth , which for T ≫ |ǫ ± vF p| gives Λ 1 Re Σ+ (ǫ, p) ≃ − α2 (ǫ − vF p) ln . 2 T (5.12) In contrast to the imaginary part of the self-energy, the real part is determined by energy transfers |ω| T , which is characteristic of the elastic virtual processes. Furthermore, ReΣ+ is logarithmically divergent in the ultraviolet and is cut off by Λ: this behavior is specific to 1d. It is worth stressing that in the considered case of weak interaction, α ≪ 1, the lowest-order contribution to the inelastic scattering rate obtained from Eqs. (5.6),(5.11), −1 τee = πα2 T , (5.13) is much smaller than temperature: T τee ≫ 1 . Considering right-movers for definiteness, we have g+ (x, t) = (Λ/2πu)A1+αb /2 (x/u − t + i/Λ) ¯ × Aαb /2 (x/u + t − i/Λ) , (5.16) A(z) = πT /Λ sinh(πT z) (5.17) where and αb = (K + K −1 − 2)/2, which becomes αb ≃ α2 /2 for α ≪ 1. The Green’s function is peaked at the “classical trajectory”x = ut. In the limit of large t → ∞ we have an exponential decay100 dq 2π q ω δ(ω + vF q) 2T ǫ + ω − vF (p + q) ω+ǫ q + tanh δ [ ǫ + ω − vF (p + q) ] 2T ω + vF q (5.15) (5.14) In higher dimensionalities, this fact is commonly referred to as one of the conditions of the existence of Fermi liquid: the inverse lifetime of a quasiparticle is smaller than its characteristic energy. In this respect, the weakly interacting Luttinger liquid, while being a canonical example of a non-Fermi liquid, reveals the typical Fermi-liquid g+ (x = ut, t) ≃ ¯ −iΛ 2πu 2πT iΛ αb /2 exp(−παb T t) , (5.18) which should be contrasted with a power-law tdependence at zero temperature, g+ (ut, t) = ¯ −iΛ 1 . 2πu (2iΛt)αb /2 (5.19) The large factor Λ in the numerator of Eq. (5.18) is reminiscent of the pole in the noninteracting Green’s function. It is worth stressing that, since |A(i/Λ)| → 1, the nonanalytical singularity of the factor sinh−αb /2 [ πT (x/u − t + i/Λ) ] in the main peak at x = ut is exactly canceled by the Luttinger-liquid renormalization factor (T /Λ)αb /2 which arises due to the resummation of the real part of the self-energy. Thus the exponential decay in Eq. (5.18) comes solely from the factor [A(2t − i/Λ)]αb /2 which describes the tail of the “left-moving peak” centered at x = −ut. The decay exp(−παb T t) of the “residue” R [ x − u(t − i/Λ) ]g+ (x, t)|x→ut ∝ sinh−αb /2 (2πT t) for t → ∞ agrees with the Golden Rule expression (5.13) to order α2 : παb T ≃ 1/2τee . Therefore, the temporal decay of the Green’s function is determined by the processes of inelastic scattering of right-movers on left-movers. 15 We see that the perturbative result for the scattering rate yields the asymptotic behavior of the exact Green’s function. This means that the Golden Rule calculation of the inelastic scattering time is meaningful in Luttinger liquid. However, the Golden Rule time τee does not cut off the WL correction in the spinless case, as we demonstrate below in Sec. VI. B. We begin by identifying the leading contribution to the WL correction to the conductivity in the regime of wl strong dephasing, τφ /τ ≪ 1. This regime takes place at sufficiently high temperatures—the precise condition wl is found below in Sec. IIIc, where we evaluate τφ . The diagrams that give the main contribution to ∆σwl are shown in Fig. 5. VI. WEAK LOCALIZATION AND DEPHASING: FERMIONIC PATH–INTEGRAL APPROACH A. Weak localization in 1d a) WL dephasing vs Golden Rule The notion of dephasing associated with the behavior of the single-particle Green’s function (5.20) makes sense in a clean Luttinger liquid in the ring geometry, see Sec. VI E. However, as we are going to show in this wl section, the WL dephasing time τφ that governs the cutoff of the WL correction ∆σwl is parametrically different from the Golden Rule time τee , Eq. (5.13). This difference can in fact be anticipated based on the following observation. The integral (5.10) for the Golden Rule rate is determined by the energy and momentum transfer q, ω = 0. On the other hand, in the spirit of wl Ref. 58, soft inelastic scattering with qvF , ω ≪ 1/τφ is not expected to contribute to the WL dephasing. This wl suggests that the dephasing rate 1/τφ requires a selfwl consistent cutoff in Eq. (5.1) at qvF , ω ∼ 1/τφ , similarly to Eq. (2.4) for higher-dimensional systems, and so is parametrically different from the one in Eq. (5.13). In particular, if one introduces, similarly to higher dimensionalities, a sharp low-frequency cutoff in the right-hand side of Eq. (5.10), one immediately gets zero instead of Eq. (5.13). On the other hand, introducing disorder in D+ and Π− in Eq. (5.10), one effectively broadens the delta functions by 1/τ , and the self-consistent equation can be solved. Such a self-consistent equation should be treated with the utmost caution, in view of the singular character of the ballistic propagators. In particular, when transformed to the space-time representation, the integral in Eq. (5.10) is determined by a single point, x, t = 0, so that one could be drawn to the erroneous conclusion that it is high—rather than low—frequencies that matter. In any case, this equation is at most qualitative. However, the message that we learn from this consideration is correct: (i) the WL dephasing rate differs from the Golden Rule result, and (ii) to evaluate it, we have to include disorder in the interaction propagators. To perform a controllable quantitative analysis, we employ the pathintegral technique, which is a natural tool in the situation when the diagrammatic calculation suffers from the infrared problems. b) c) FIG. 5: Diagrams describing the leading WL correction to wl the conductivity of Luttinger liquid for τφ ≪ τ . The dashed lines represent backscattering off impurities. The current vertices are dressed by impurity ladders. The solid lines are the Green’s functions with disorder-induced self-energies. The diagrams are understood as fully dressed by e-e interactions. The “three-impurity Cooperon” in Fig. 5a describes the propagation of two electron waves along the path connecting three impurities (“minimal loop”) in timereversed directions, Fig. 6a. Indeed, one can easily check that the interference paths involving only two backscattering impurities are impossible merely due to geometrical reasons. This means that the Cooperon with two backscattering impurity lines101 yields zero contribution to the conductivity to leading order in (ǫF τ )−1 . If one retains the forward scattering and does not gauge it out, the two-impurity Cooperon diagram (Fig. 7a) is canceled by the contribution of two other non-Drude diagrams with two impurity lines (Fig. 7b,c). The diagrams represented in Fig. 5b,c (the corresponding trajectories are shown in Fig. 6b) are of the same order as the Cooperon diagram. In the absence of interaction, all higher-order diagrams (including non-Cooperon diagrams) which describe quantum-interference processes involving a larger number of impurities (4, 5, . . .) are equally important and together with the diagrams in Fig. 5 sum up to exactly cancel σD , which spells a complete localization.69 In the WL regime, they only yield subleading corrections through a systematic expansion in powers of the small wl parameter τφ /τ . Thus, the WL in 1d is governed by the “shortest possible Cooperon”, namely, the Cooperon with three impurity lines. In essence, this is due to the wl fact that the WL condition 1/τφ ≫ ∆ξ reduces in 1d to wl the condition of the “ballistic dephasing”, τφ /τ ≪ 1. In turn, this means that the contribution to ∆σwl of typical 16 xc i f forward path xb xa f xc i backward path xb a) xa xc i f xb b) ± U ± (x, t) = U0 (x ∓ ut) . forward path xa f xc i xb backward path xa FIG. 6: (a) Interference paths described by the Cooperon diagram, Fig. 5a. Black dots: the backscattering off impurities. The one-dimensional graphs are stretched vertically for ease of visualization. The two waves start at point i and interfere at point f after traversing the closed loop in opposite directions. (b) Interfering paths for the diagram in Fig. 5c. The dashed circle denotes the absence of backscattering off the impurity at point xc . The paths corresponding to the diagram in Fig. 5b are obtained by interchanging the upper and lower plots in Fig. 6b. Note that similar paths describe the WL in the ballistic regime in higher dimensionalities.102 + a) =0 + b) configurations of three impurities separated by a distance of order vF τ is strongly suppressed by dephasing. In the compact configurations that do contribute the three impurities are anomalously close to each other, with a characteristic distance between them being much smaller than the mean free path. WL dephasing in 1d: Qualitative discussion of the leading-order cancellation Before turning to the path-integral evaluation of the WL dephasing rate in a spinless 1d system, we present a simple argument demonstrating the cancellation of the wl leading contribution 1/τee to 1/τφ . For the purpose of (6.1) We now demonstrate that the phases acquired by the time-reversed waves propagating along a closed three± impurity loop are equal for arbitrary U0 (x) and hence cancel each other in the interference contribution even ± before the averaging over fluctuations of U0 (x). As we show in Appendix F, the velocity of electrons moving along the “Cooperon path” is renormalized by the interaction and coincides with u. Referring to the paths shown in Fig. 6a, we first follow the “forward” trajectory xc = 0 → xa → xb → 0 for a particle which starts moving to the right at t = 0 and is backscattered at these points at times ta , tb , and tc , respectively, see Fig. 8. It is important to recall that the scattering amplitude g4 has been eliminated through the renormalization of the Fermi velocity. Therefore, on the right-moving parts of the forward trajectory (segments 0 → xa and xb → 0) the phase changes by φF only due to the scattering off U − (x, t), + while on the left-moving parts (segment xa → xb ) only the scattering off U + (x, t) matters and yields the phase shift φF . Similarly, there are phase shifts for the back− ward path φB due to the scattering off U ∓ (x, t). ± The phase φF acquired on the forward path between − points xa and xb is given by c) FIG. 7: Two-impurity non-Drude diagrams. C. a qualitative discussion, we can consider inelastic scattering within the picture of a classical thermal noise ± with characteristic frequencies |ω| ≪ T . Let U0 (x) be a given realization of the fluctuating scalar potential U ± (x, t = 0) created by right-/left-movers at time t = 0. The peculiarity of the 1d geometry is that in the clean case the initial density profiles of right-/left-movers, whose dispersion relation is linear, remain unchanged: they move with the plasmon velocity either to the left or to the right as a whole, and so does the fluctuating potential: φF = − − 1 u 0 −utc + U0 (x) dx . (6.2) Here we use tb − ta = tc /2. Electrons moving on the backward path interact with the same right-moving potential U + (x, t) on the segments 0 → xb and xa → 0, see Fig. 9. The phase φB is written as − φB = − − 1 u 0 −2u(tc −tb +ta ) + U0 (x) dx = φF . − (6.3) Similarly, φF = φB . This is clearly seen in Figs. 8d and + + 9d, where the regions of the potential profile that contribute to the phase shifts are identical at time tc for the forward and backward paths. As a result, the interactioninduced phase shifts cancel out in the total Cooperon phase (φF + φF ) − (φB + φB ) for an arbitrary profile + − + − U0 (x). This means, in particular, that in a calculation of the conductance through a short piece of Luttinger liquid to third order in the impurity strength, which is 17 backward path forward path a) + U0(x) a) + U0(x) b) + U (x,ta ) b) + U (x,tc−tb ) c) + U (x,tb ) c) + U (x,tc−ta ) d) + U (x,tc ) d) + U (x,tc ) xb x c = 0 xb x c = 0 xa FIG. 8: Illustration of the quasiclassical electron dynamics in the presence of three impurities (whose positions are denoted by the dashed lines) and a right-moving component of the fluctuating scalar potential U + (x, t). Top: the forward path corresponds to that in Fig. 6a. The position (black dots) and the direction of the velocity (arrows) of the particle, as well as the profile of the fluctuating potential, are shown at different times: (a) t = 0; (b) t = ta + 0; (c) t = tb + 0; (d) t = tc + 0. The phase φF of the electronic wave is accumulated when − the particle moves to the left, that is between the snapshots (b) and (c). The segment of the fluctuating potential that contributes to the phase shift is marked in bold. the order at which the interference term arises, the latter would remain unaffected by inelastic e-e collisions. In the calculation of the WL correction to the conductivity of an infinite quantum wire, we have, indeed, to include disorder in the interaction propagator, despite the ballistic character of dephasing in the WL regime and in accordance with the argument made in Sec. VI A. This is done systematically within the path-integral approach in the next subsection. D. WL dephasing in 1d: Path-integral calculation The dephasing-induced action S(tc , ta ) acquired by the Cooperon is accumulated on the classical (saddle-point) path, whose geometry for three impurities if fixed by two length scales, the total length of the path vF tc and the distance between two rightmost impurities vF ta ≤ vF tc /2 (Fig. 10). As shown in Appendix B, the WL correction can then be represented as ∞ ∆σwl = −2σD ∞ dtc 0 dta P2 (tc , ta ) exp [−S(tc , ta )] , 0 (6.4) xa FIG. 9: Same as in Fig. 8 but for the backward part, Fig. 6a. (a) t = 0; (b) t = tc − tb + 0; (c) t = tc − ta + 0; (d) t = tc + 0. The phase φB of the electronic wave is accumulated when the − particle moves between the snapshots (a) and (b) and between the snapshots (c) and (d). where P2 (tc , ta ) = (1/8τ 2 ) exp(−tc /2τ )Θ(tc − 2ta ) (6.5) is the probability density of return to point x = 0 after two reflections at points x = vF ta and x = −vF (tc /2−ta ) (in this subsection, we ignore the difference between vF and u). The contribution Sij to the dephasing action associated with inelastic interaction between electrons propagating along the paths xi (t) and xj (t) is obtained similarly to higher dimensionalities:46,58 tc dω 1 dq tc dt2 Im Vµν (ω, q) dt1 2π 2π 0 ω 0 × exp{iq [ xi (t1 ) − xj (t2 ) ] − iω(t1 − t2 )} , (6.6) Sij = −T where Vµν (ω, q) is the dynamically screened retarded interaction. Because of the HF cancellation of the bare interaction between electrons from the same chiral branch (g4 processes), the screened interaction V (ω, q) acquires the indices µ, ν denoting the direction of motion of the interacting electrons, µ = sgn xi , ν = sgn xj . More specif˙ ˙ ically, the g4 processes have been incorporated in the renormalization of the bare Fermi velocity. It is worth stressing that if we would keep both g2 and g4 processes in V (ω, q), the dephasing action in 1d could not be written in the form of Eq. (6.6). Indeed, in that case the dephasing action would generate the contribution of the diagrams of type (b) in Fig. 4, which is actually absent due to the HF cancellation (5.8). The main steps in the derivation of Eq. (6.6) are as follows. First, the interaction is treated within the randomphase approximation (RPA), neglecting closed electronic 18 xf,b(t ) xa and b stand for “forward” and “backward” time-reversed paths (Fig. 10). Let us first calculate S for the case when the disorder in interaction propagators (6.9),(6.10) is neglected, τ −1 = 0. We get straightforwardly F−− backward + 0 tc ta F−+ xb Sff = Sfb = πα2 T tc /2 . t forward FIG. 10: Illustration of electron dynamics governing the WL and dephasing: Time-reversed paths xf (t) (solid) and xb (t) = xf (tc − t) (dashed) on which the interaction-induced action S that yields dephasing of the Cooperon is accumulated. Dotted lines: the propagation of dynamically screened interaction. The interaction may change the direction of propagation upon scattering off disorder (as marked by a cross). Each interaction line gives a contribution to S proportional to (Nf − Nb )2 , where Nf,b is the number of its intersections with the forward (f ) and backward (b) paths. One sees that Nf = Nb only due to impurity scattering in the interaction propagator. Interaction and electron lines lying on top of each other do not yield dephasing because of the HF cancellation. bubbles with more than two interaction legs, which is justified for the weak interaction, α ≪ 1. Second, we perform an independent averaging of the RPA bubble and the electronic trajectories over the disorder. This is wl justified if the characteristic energy transfer ω ∼ 1/τφ is much larger 1/τ . Further, we treat the thermal electromagnetic fluctuations through which electrons interact with each other as a classical field, which is a valid approximation provided that ω is much smaller than T . As will be seen from the results of our calculation, the charwl acteristic energy transfer ω ∼ 1/τφ does satisfy both these requirements for the considered case of weak interaction, α ≪ 1. Finally, we will neglect the influence of the interaction on the velocity (u vs vF ), which is again justified to the leading order in the interaction strength. Expanding Vµν to second order in α, we have ImVµν (ω, q) = −πα2 vF ωFµν (ω, q) , (6.7) where the functions Fµν Fourier-transformed to (x, τ ) space are given by Eqs. (C.2),(C.3) with γ = 1/τ . The action Sij can then be written in a simple form: tc tc Sij = πα2 vF T dt1 0 0 dt2 Fµν [ xi (t1 )−xj (t2 ), t1 −t2 ] , (6.8) where, to first order in τ −1 , Fµν (x, t) read F++ (x, t) ≃ δ(x + vF t) (1 − |t|/2τ ) , 2 F+− (x, t) ≃ Θ(vF t2 − x2 )/4vF τ , (6.9) (6.10) and F−− (x, t) = F++ (−x, t), F−+ (x, t) = F+− (x, t). The total action is given by S = 2(Sff − Sfb ), where f (6.11) One sees that Sff reproduces the Golden Rule result, Eqs. (5.13),(5.20). As we explain in Sec. VI E, this rate is relevant for the AB dephasing in the multiple-connected (ring) geometry. However, in the present case of dephasing in a wire the self-energy processes (Sff + Sbb ) are exactly canceled in the action by the vertex corrections (Sfb + Sbf ), yielding S = 0. This confirms the conclusion of Sec. VI C: for clean-system interaction propagators, the WL dephasing is absent. Therefore, the dephasing in Eq. (6.4) is only due to the dressing of the dynamically screened interaction by impurities. A calculation of the dephasing action S to order O(τ −1 ) is presented in Appendix C, with the result S(tc , ta ) = 2πα2 T ta (tc − 2ta ) /τ . (6.12) The dephasing vanishes for ta = 0, tc /2, since in these cases the two paths f and b are identical. wl Substituting Eq. (6.12) in Eq. (6.4) we find for τφ ≪ τ : 1 ∆σwl = − σD 4 tc /2 × 0 ∞ dtc 0 1 exp(−tc /2τ ) τ2 dta exp[ −4πα2 T ta (tc /2 − ta )/τ ] 1 = − σD 4 wl τφ τ 2 ln 1 τ ∝ 2 ln(α2 T ) , wl α T τφ (6.13) where 1 wl τφ =α πT τ 1/2 , T ≫ T1 = 1 . α2 τ (6.14) wl In agreement with the above findings, 1/τφ vanishes in 103 the clean limit, in contrast to the total e-e scattering rate, Eq. (5.13). The logarithmic factor in Eq. (6.13) is related to the contribution of rare configurations of disorder in which two of the three impurities are anomalously close to each other (ta → 0 or ta → tc /2): in these configurations the dephasing is strongly suppressed. As a result, the characteristic t in the integral (6.13) are wl spread over the whole range between τφ and τ . The scale T1 defined in Eq. (6.14) marks the temperature below which the localization effects become strong. It is worth mentioning that the T dependence of σ(T ) = σD (T ) + ∆σwl (T ) starts to be dominated, with decreasing T , by the WL term rather than by σD (T ) at T ∼ T1 /α, i.e., well above T1 for α ≪ 1. These results are illustrated in Fig. 11. Let us briefly mention what happens below T1 . In the strongly localized phase, the only peculiarity of Luttinger liquid is the T dependence of the scattering rate 19 σ(Τ) AB i.e., τφ = τee . The difference from the WL correction is in the absence of the vertex parts of the dephasing action. Specifically, S12 = S21 = 0 in the AB setup [where Sij is given by Eq. (6.6) with i, j equal to 1 or 2 for trajectories in arms 1 or 2, correspondingly], because of the absence of interaction between the arms, see Fig. 12. On Drude WL strong localization Τ Τ1 =1/α 2 τ 1/τ Τ1 /α φ (t ) εF π FIG. 11: Schematic behavior of σ(T ) on the log-log scale. Dotted line: the T -dependent Drude conductivity. Below T1 /α the WL correction, Eq. (6.13), dominates d ln σ/d ln T . Below T1 the localization becomes strong. F−− E. Weak localization vs Aharonov-Bohm dephasing As shown in Sec. VI D, the dephasing rate that cuts off the WL correction is much smaller than the Golden Rule result from Sec. V, which governs the damping of the single-particle Green’s function, Eq. (5.20). One may ask if the Golden Rule rate 1/τee is observable at all. The answer is yes: this type of dephasing makes sense in a clean Luttinger liquid in the ring geometry, where it AB governs the decay rate (τφ )−1 of Aharonov-Bohm (AB) 45,100,106 oscillations. Indeed, the amplitude GAB of the first harmonic of the AB conductance is proportional to the product of the Green’s functions in two arms of the interferometer: GAB ∝ 1 Ωn 1/T dτ eiΩn τ 0 × g+ (L1 , τ )g+ (−L2 , −τ )}Ωn →+0 , (6.15) where L1,2 are the lengths of the arms of the ring. In the case of L1 = L2 ≡ L, the damping is solely due to the dephasing and is given by AB GAB ∝ exp(−L/uτee) ≡ exp(−L/uτφ ) , (6.16) + 0 τ −1 (T ) and the related T dependence of the localization length ξ(T ), persisting down to T τ (T ) ∼ 1 (see Sec. III B 1); otherwise, the picture is similar to that in many-channel quantum wires or 2d systems. If α ≪ 1, there exists a wide range of T where the localization is strong but the conductivity is a power-law function of T . We call this transport regime in a weakly disordered system, associated with interaction-induced diffusion over localized states, Power-Law Hopping (PLH); see Refs. 65,70,72,104,105 for details. An important point is that the lifetime of localized states due to inelastic e-e scattering can be calculated in the PLH regime at the Golden Rule level. At still lower T , the lifetime becomes infinite, so that the e-e interaction can no longer support hopping via localized states and the conductivity vanishes at a certain critical Tc , see Refs. 65,66 for details. Arm 1 t L/u F−+ −π Arm 2 FIG. 12: Illustration of electron dynamics governing the dephasing of the first harmonics GAB of the AB oscillations in the ring geometry with two arms of equal length L. The position of a particle on the ring is parametrized by the angle φ(t) : φ = 0 corresponds to the point at which the ring is connected to one lead, φ = ±π—to the point at which it is connected to the other lead. The paths φ1 (t) (solid line) and φ2 (t) (dashed) traveling along arm 1 and arm 2, respectively, interfere at the second contact at time t = L/u. Dotted lines: the propagation of the dynamically screened interaction. The interaction may change the direction of propagation upon scattering off the contact (as marked by a cross). Similarly to Fig. 11, each interaction line gives a contribution to the AB-dephasing action proportional to (N1 − N2 )2 , where N1,2 is the number of its intersections with the trajectory in arm 1 or arm 2. One sees that for any dotted line the number of possible intersections in the time domain 0 < t < L/u is strictly one, yielding only the “self-energy” contribution to the action and no “vertex part”. Interaction and electron lines lying on top of each other do not contribute to the dephasing because of the HF cancellation. the other hand, the self-energy part S11 = S22 (analog of Sff and Sbb in the WL problem) is given by Eq. (6.11) AB with tc = L/u. An analogous T −1 dependence of τφ was also obtained in Ref. 107 for a quantum wire coupled capacitively to metallic gates. The preexponential factor in Eq. (6.15) depends on details of how the AB ring is connected to the leads. A particular AB setup— two Luttinger liquids weakly coupled to each other at two points by tunneling—was thoroughly studied in Ref. 106. To summarize, the WL and AB dephasing rates in a Luttinger liquid are parametrically different in the spinless AB wl case studied in this paper; specifically, τφ ≪ τφ . In diffusive many-channel wires, an analogous difference was shown in Ref. 108. 20 VII. FUNCTIONAL BOSONIZATION: ALL-IN-ONE APPROACH Equations (6.13),(6.14) have been derived in two steps, emphasizing two key ingredients of the theory: the Luttinger-liquid renormalization of disorder and the Fermi-liquid-like dephasing of electrons scattered by the renormalized disorder. Although both effects are generated by e-e interactions, they are of essentially different origin: the renormalization of disorder is due to virtual processes with energy transfers in the range between T and Λ, whereas the dephasing stems from real processes with energy transfers smaller than T . Conceptually, an advantage of the two-step approach is that it reveals the transparent physical picture of transport in the disordered Luttinger liquid. Alternatively, one can develop a unified approach where the virtual and real transitions are treated on an equal footing, which is what we do below. A fully bosonized approach, as well as a purely fermionic one, turns out to be not particularly advantageous for this purpose. Instead, in this paper we employ the method of so-called “functional bosonization”, adjusted to deal with the particulars of e-e scattering in one spatial dimension. The method was introduced for a clean Luttinger liquid by Fogedby89 and by Lee and Chen.90 A transparent exposition and further development of the method in 1d can be found in recent reviews by Yurkevich92 and Lerner and Yurkevich.93 For recent applications to single-impurity problems see Refs. 109,110. In essence, the functional bosonization is a technique based on the conventional Hubbard-Stratonovich transformation that decouples the density-density interaction term in the fermionic action. Bosonic fields appear in this approach as auxiliary fields of the HubbardStratonovich transformation, whose fluctuations mediate the dynamically screened interaction between electrons. The peculiarity of a clean Luttinger liquid is that the RPA approximation for the dynamically screened interaction is exact.111 This is a drastic simplification, so that the ease in which an arbitrary strong interaction in the clean case can be incorporated via the fully bosonized method is also the property of the functionalbosonization approach. In the disordered case, the functional-bosonization method has a technical advantage, especially useful for weak interactions, in that the conductivity is represented (in contrast to the fully bosonized approach) as a fermionic loop, albeit dressed by interactions in all possible ways. A. Functional bosonization: Background Our purpose here is to develop the functionalbosonization scheme in the presence of disorder and to apply it for the calculation of the weak-localization term (6.13) in the conductivity. The quantity to begin with is the fermionic Green’s function. In the absence of disor- der, we write it in the Matsubara representation for an electron moving to the right (+) or to the left (–) as (0) gµ (x, τ ) = gµ (x, τ ) e−Bµµ (x,τ ) , µ=±, (7.1) where (0) g± (x, τ ) = ∓ iT 1 2vF sinh[πT (x/vF ± iτ )] (7.2) is the free Green’s function and the factor exp(−Bµµ ) accounts for interactions between electrons. The function Bµµ can be found by using the fact that, in the absence of backscattering due to e-e interaction, the Hubbard-Stratonovich transformation characterized by a decoupling field ϕ(x, τ ) results in an identical representation of gµ (x, τ ) as the average over ϕ(x, τ ) of a function gµ (x, τ ),90,92,93 ˜ gµ (x, τ ) = gµ (x, τ ) , ˜ (7.3) where gµ (x, τ ) describes the propagation of a particle of ˜ the same chirality in the field ϕ(x, τ ): [ ∂τ − iµvF ∂x − iϕ(x, τ ) ] gµ (x, τ ) = δ(x)δ(τ ) . ˜ (7.4) The angular brackets in Eq. (7.3) and throughout Sec. VII below denote the functional averaging over the field ϕ(x, τ ). The representation (7.3) is equivalent to a local gauge transformation for the fermionic field of the right-/left-mover93 ψµ (x, τ ) → ψµ (x, τ ) exp[ iθµ (x, τ ) ] , (7.5) where the bosonic field θµ (x, τ ) is related to the decoupling field ϕ(x, τ ) by (∂τ − iµvF ∂x )θµ (x, τ ) = ϕ(x, τ ) . (7.6) The Gaussian character of the Luttinger-liquid theory in the clean case means that the averaging in Eq. (7.3) is performed with an action in which all terms containing powers of ϕ(x, τ ) higher than second vanish to zero. The correlator ϕ(x, τ )ϕ(0, 0) = V (x, τ ) then gives the dynamically screened RPA interaction. As a result, the functions Bµµ (x, τ ) are represented as propagators of the fields θµ (x, τ ): Bµν (x, τ ) = [ θµ (0, 0) − θµ (x, τ ) ] θν (0, 0) , (7.7) where for later use we also introduce the correlator of fields θµ of different chirality, µ, ν = ±. It is convenient to put the bare coupling constant g4 = 0, shift the Fermi velocity accordingly (as discussed in Sec.III A) and deal with two different interaction propagators V++ = V+− (see Appendix A). Then the correlators Bµν (x, τ ) read B+± (x, τ ) = T n dq iqx−iΩn τ e −1 2π V+± (iΩn , q) , × (iΩn − qvF )(iΩn ∓ qvF ) (7.8) 21 where Ωn = 2πnT is the bosonic frequency, B−− (x, τ ) = B++ (−x, τ ), and B−+ (x, τ ) = B+− (x, τ ). The integration with Vµν given by Eq. (A.5) [or, equivalently, by Eqs. (A.7),(A.8) with a renormalized Fermi velocity, see discussion in Sec. III A] yields (see Appendix D): Bµµ (x, τ ) = − ln ηµ (x, τ ) − Bµ,−µ (x, τ ) = − αr L(x, τ ) , 2 αb L(x, τ ) , 2 (7.9) (7.10) where αb = (1 − K)2 /2K , αr = (1 − K 2 )/2K , (7.11) and L(x, τ ) = ln (πT /Λ)2 . sinh[ πT (x/u + iτ ) ] sinh[ πT (x/u − iτ ) ] (7.12) e-iθ+(1) B. + 1 2 1 (7.13) (0) g± (x, τ ) 2 FIG. 13: (a) The Green’s function of a right-mover propagating between points 1 = (x1 , τ1 ) and 2 = (x2 , τ2 ) in a given field ϕ(x, τ ) acquires two phase factors exp[iθ+ (2)] and exp[−iθ+ (1)] (denoted as wavy lines) at the end points. Solid (0) line: the free Green’s function g+ (x2 − x1 , τ2 − τ1 ). (b) The wavy line connecting the points 1 and 2 denotes averaging of the phase factors over fluctuations of θ+ (1) and θ+ (2). e-iθ+(1) i[θ+(2)-θ-(2)] 1 eiθ-(3) e + vF sinh[πT (x/vF ± iτ )] η± (x, τ ) = u sinh[πT (x/u ± iτ )] ,τ2-τ1) e-B++(x2-x1 + The functions replace vF → u in the free Green’s functions Eq. (7.1). e+iθ+(2) -2 3 in FIG. 14: The Green’s function with a backscattering vertex (marked by a cross) between the end points. The vertex is dressed by a local fluctuating factor exp{i[θ+ (2) − θ− (2)]}. Functional bosonization: Diagrammatic technique The exactness of the RPA approximation in 1d suggests the following diagrammatic technique. The Green’s function (7.1) for a right-mover propagating from point 1 = (x1 , τ1 ) to point 2 = (x2 , τ2 ) can be represented by the diagram in Fig. 13, where the solid line is (0) g+ (x2 − x1 , τ2 − τ1 ) and the wavy line connecting the end points denotes the pairing of two fluctuating factors exp[iθ+ (2)] and exp[−iθ+ (1)] with the Gaussian fields θ+ (2) and θ+ (1): exp{ i[ θ+ (2) − θ+ (1) ] } = exp [ −B++ (x2 − x1 , τ2 − τ1 ) ] . (7.14) 1/T (0) dτ2 0 (0) τ1 )g− (x3 − exp{ i[ θ+ (2) − θ− (1) ] } = exp [ −B+− (x2 − x1 , τ2 − τ1 ) − M0 ] , (7.16) where The fields θ+ (2) and θ+ (1) enter the exponent of Eq. (7.14) with different signs, according to the direction of the arrow on the solid line. Consider now a backscattering vertex which describes reflection of the right-mover from an impurity (Figs. 14 and 15, before and after averaging over the fluctuations of θµ , respectively). The Green’s function propagating from point 1 to point 3 upon backscattering on the static potential Ub (x) at point 2 is given by δg(x3 , x1 , τ3 − τ1 ) = A direct consequence of Eq. (7.5) is that the vertex at point 2 is dressed by the local fluctuating factor exp{i[θ+ (2) − θ− (2)]}. The correlations of the fields θµ at different points in x space in the last factor of Eq. (7.15) account for e-e interaction. Apart from the averages (7.14) of the fields of the same chirality, there appear averages of the type dx2 Ub (x2 ) × g+ (x2 − x1 , τ2 − x2 , τ3 − τ2 ) × exp{ i[ θ+ (2) − θ+ (1) + θ− (3) − θ− (2) ] } . (7.15) 2 M0 = θ+ (0, 0) − θ+ (0, 0)θ− (0, 0) . (7.17) Note that the factors exp(±M0 ) cancel out in any closed fermionic loop, so that in the diagrammatic technique for observables one can omit them from the very beginning. The Gaussian character of the correlations means that the exponent of the last factor in Eq. (7.15) is represented as a sum of the correlators Bµν : exp{ i[ θ+ (2) − θ+ (1) + θ− (3) − θ− (2) ] } = exp [ −B++ (x2 − x1 , τ2 − τ1 ) +B+− (x2 − x1 , τ2 − τ1 ) − B−− (x3 − x2 , τ3 − τ2 ) +B+− (x3 − x2 , τ3 − τ2 ) − B+− (x3 − x1 , τ3 − τ1 ) ] , (7.18) as illustrated in Fig. 15 [we used in Eq. (7.18) the fact that B+− (2, 2) = 0]. 22 B+-(3,1) B++(2,1) 1 + B--(3,2) 2 B+-(2,1) -- 3 B+-(3,2) 2 2 g2 q 2 vF − iqvF Ωn + |Ωn |γ/2 , 2 u2 + Ω2 + |Ω |(γ + δγ) 2πvF q n n (7.20) 2 2 2 q v + Ωn + |Ωn |(γ + δγ/2) V+− (iΩn , q) = g2 2F 2 , q u + Ω2 + |Ωn |(γ + δγ) n (7.21) V++ (iΩn , q) = − B+-(2,2) FIG. 15: Illustration of the diagrammatic technique with the propagators Bµν : the correction δg(x3 , x1 , τ3 −τ1 ) [Eq. (7.15)] to the Green’s function due to the backscattering at point 2. The wavy lines denote averaging of the fluctuating factors exp(±θµ ) shown in Fig. 14. The pairing of all the fields θµ with each other according to Eq. (7.7) yields the corresponding contributions Bµν to the factor (7.18). For ease of visualization, the ends of the free Green’s functions are split off from the backscattering vertex at point 2. The appearance of the combination θ+ (2) − θ− (2) for backscattering at point 2 in Eq. (7.15) and the factorization of the correlations (7.18) in the simplest case of a single backscattering vertex suggests a way to proceed to two or more vertices. In effect, the e-e interaction can be completely gauged out to the impurity vertices, as emphasized in Ref. 93. Specifically, one can, in principle, calculate observables (given by closed fermionic loops) perturbatively to any order in the strength of disorder and exactly in the strength of e-e interaction by attaching a factor exp { ±i [ θ+(N ) − θ− (N ) ] } than two external interaction lines connect to each of the fermionic loops that constitute the interaction propagator and the averaging of the interaction propagator over disorder is performed in such a way that no impurity lines connect different fermionic loops. As we show in Appendix E, in the language of the functional bosonization, the RPA approximation in the presence of disorder translates into a disorder-induced renormalization of the correlators Bµν (x, τ ). Specifically, the renormalization consists in adding a disorder-induced damping to the propagators V++ and V+− in Eq. (7.8). The renormalized interaction propagators are derived in Appendix A [see Eqs. (A.24)–(A.25)] and are given in the Matsubara representation by (7.19) to each backscattering vertex, where N labels the vertex, and pairing all the fields θµ (N ) with each other. The averaging over the fluctuations of θµ (N ) is then performed with the correlators (7.9)–(7.10) calculated for a homogeneous system. The sign ± in Eq. (7.19) should be chosen to correspond to the chirality with which an electron is incident on the impurity. The WL correction to the conductivity is given by the fermionic loop with 6 backscattering vertices (Fig. 5), each of which should be dressed with the factors (7.19). However, taking disorder into account at this level would not be sufficient. As we showed in Sec. VI, in order to calculate the effect of dephasing for spinless electrons, disorder should also be included, to first order in the strength of disorder γ, in the loops that make up the RPA series for the effective interaction. We recall that in the RPA approximation, substantiated in Sec. VI, no more where γ denotes the transport scattering rate, Eq. (A.13), and δγ is given by Eq. (A.26) (at variance with the preceding part of the paper, in this section we do not use the notation τ for the transport scattering time in order to avoid a possible confusion with the Matsubara time). Thus, our strategy in deriving Eq. (6.13) via the functional bosonization is this: find the disorder-renormalized 2 correlators Bµν (x, τ ) to order O(γg2 ) and use them to average over fluctuations of the fields θµ “attached” to 6 impurity vertices of the 3-impurity Cooperon (Fig. 16). The diagrams (b) and (c) in Fig. 5 are calculated in the same way. C. Functional bosonization with a disorder-renormalized RPA We now turn to the calculation of the effective (disorder-renormalized) propagators Bµν (x, τ ) to order 2 O(γg2 ) by using Eq. (7.8) and the interaction propagators V++ and V+− given by Eqs. (7.20),(7.21). At second order in interaction, B++ (x, τ ) reads B++ (x, τ ) ≃ − 2 g2 T 2πvF n dq iqx−iΩn τ e −1 2π 2 q 2 vF − iqvF Ωn + |Ωn |γ/2 . × 2 (−iΩn + qvF )2 (q 2 vF + Ω2 + |Ωn |γ) n (7.22) Let us represent B++ as a sum B++ = B0 + B> + B< , where B0 (x) is the contribution of Ωn = 0, B> (x, τ ) and B< (x, τ ) = B> (−x, −τ ) are the contributions of Ωn > 0 and Ωn < 0, respectively. In turn, B> (x, τ ) is a sum of two terms: dp sp B> = B> + B> , (7.23) 23 The sum (7.27) is defined for Re z > 0. At small |z| ≪ 1, it behaves as F (z) ≃ z ln |z|, which cancels the second term in Eq. (7.26) at the ultraviolet cutoff τ → 0, |x| → 1/vF Λ. Below, we will need the analytical continuation of Bµν (x, τ ) in the complex plane of τ . In particular, an important contribution to the dephasing will come from the analytical continuation of F (z) to large negative Re z < 0. The latter can be done, e.g., by means of the integral representation 2 1 3 z F (z) = 0 3 1 B++ = B++ |γ=0 + b , dp where B> is the contribution of the double pole at qvF = sp iΩn and B> is the contribution of the single poles at qvF = ±i(Ω2 + Ωn γ)1/2 . n The term coming from zero frequency does not depend on γ: dq eiqx − 1 πT |x| , (7.24) = α2 2 2π q 2 vF vF where α = g2 /2πvF , as before. The contribution of the double pole does not depend on γ either, b1 (x, τ ) ≃ α2 where ∞ exp(−nz) − 1 . n2 n=1 (7.27) 2πT x vF 2 + 2πT |x| vF 1 , 4 sinh[ πT (x/vF + iτ ) ] sinh[ πT (x/vF − iτ ) ] |x| γ F 2πT + iτ b2 (x, τ ) ≃ α2 16πT vF |x| + F 2πT − iτ . (7.30) vF Note that the second term in Eq. (7.30) can be viewed as an x dependent renormalization of α induced by disorder, δα = −αγ|x|/4vF , as follows from a comparison with Eq. (7.9). This is consistent with the RG flow of the interaction constant (4.28). However, Eqs. (7.30),(7.30) contain more information and allow us to study the dephasing missed in the RG equations. Consider now the disorder-renormalized propagator B+− (x, τ ). To second order in g2 , πT x Θ(x) vF 1 × , (7.25) exp [ 2πT (x/vF + iτ ) ] − 1 and is an expansion of ln η+ (7.13) in powers of α. Expanding the single-pole contribution to first order in γ, we have 2πT /Λ α2 sp ln B> (x, τ ) ≃ − 4 1 − exp[−2πT (|x|/vF + iτ )] 1 γ|x| ln − 2vF 1 − exp[−2πT (|x|/vF + iτ )] γ |x| − + iτ , (7.26) F 2πT 4πT vF γ 16πT × ln B+− (x, τ ) ≃ −T dp B> (x, τ ) = α2 F (z) = (7.29) where B++ |γ=0 is given by Eq. (7.9), b = b1 + b2 , and FIG. 16: The three-impurity Cooperon diagram with interaction effects encoded in the fluctuating factors exp(±θµ ) (denoted by the wavy lines) attached to the backscattering vertices. Each impurity vertex N = (xN , τN ) is characterized by two fluctuating fields θ± (N ), and similarly for ¯ N = (xN , τN ). Pairing of all the fields with each other (simi¯ larly to Figs. 14,15) describes both the elastic renormalization of the three impurities and the dephasing of the Cooperon. 2 g2 T 2πvF (7.28) Piecing all the terms together, B++ (x, τ ) with a 2 disorder-induced correction of order O(γg2 ) is written as 2 B0 (x) = − dz ′ ln[ 1 − exp(−z ′ ) ] . × n dq eiqx−iΩn τ − 1 2 2π Ω2 + q 2 vF n g2 |Ωn |γ g2 + 2 2 2 4πvF q vF + Ω2 + |Ωn |γ n . (7.31) Expanding Eq. (7.31) to linear order in γ, we readily obtain a correction to B+− (x, τ ) [Eq. (7.10)] which is again given by Eqs. (7.30),(7.30) but with the opposite sign, so that finally B+± = B+± |γ=0 ± b . D. (7.32) Cooperon in a dynamic environment Now that we have found the propagators Bµν (x, τ ), let us turn to the calculation of the WL correction to the conductivity. The interaction induces the factor exp(−SC ) = exp[ i(θf − θb ) ] (7.33) 24 to the Cooperon loop, where θf and θb are the phases accumulated by an electron propagating along the “forward” and “backward” paths (Fig. 17 represents the paths from Fig. 6a in a way which is more convenient here) and the averaging is performed with the disorderrenormalized correlators (7.32). Again, SC accounts for both the dephasing and the elastic renormalization of impurities. The factor (7.33) is the same for all 3 diagrams for ∆σwl in Fig. 5. 3 2 _ 3 ¯ and similarly for the pairings between N and N . The constant M0 given by Eq. (7.17) cancels out in SC . Note 2 2 that on the closed contour θf = θb , which is accounted for by the factor of 2 in Eq. (7.36). E. forward 1 _ 2 + B−− (xN − xN ′ , τN − τN ′ ) − 2B+− (xN − xN ′ , τN − τN ′ ) , (7.39) Diffuson ladder and renormalization of disorder Before calculating the three-impurity Cooperon, it is instructive to look at the diffuson diagrams. More specifically, let us consider first the ladder diagrams without including self-energy corrections. _ 1 1. One impurity line backward FIG. 17: Two time-reversed paths whose interference constitutes the three-impurity Cooperon. There are three backscatterings off impurities on each of the paths (the positions of the impurities are marked by the dashed lines). The onedimensional paths are stretched vertically for ease of visualization, with the “time axis” directed upwards for the forward path and downwards for the backward path. Note that this representation of the paths is identical to the graphs in the Berezinskii technique.69 The phases θf,b are given by 3 (−1)N +1 [ θ+ (N ) − θ− (N ) ] , θf = N =1 3 ¯ ¯ ¯ (−1)N +1 [ θ+ (N ) − θ− (N ) ] , θb = (7.34) (7.35) ¯ N =1 ¯ where N, N are the impurity-vertex numbers, for the forward and backward paths, respectively; as shown in Fig. 6. The exponent in Eq. (7.33) is represented as SC = −2(Mf f − Mf b ) , where Mf f = − = Mf b 1 2 (7.36) d1 = vF γ0 2 1/T 1/T dτ1 0 d¯1 τ 0 (0) (0) × g+ (x1 , τ1 )g− (x − x1 , τ − τ1 ) exp[−M (1, ¯ 1)] (0) (0) × g+ (−x1 , −¯1 )g− (x1 − x, τ1 − τ ) . τ ¯ ′ (−1)N +N M (N, N ′ ) − M0 , (7.37) 1 = − θf θb 2 1 ¯ ¯ = (−1)N +N M (N, N ) − M0 , 2 ¯ (7.38) NN and all the pairings between the phases associated with impurities N and N ′ are given by the combination M (N, N ′ ) = B++ (xN − xN ′ , τN − τN ′ ) (7.40) The dashed line gives the bare strength of backscattering vF γ0 /2. In Eq. (7.40) we only integrate over internal time variables, keeping other coordinates in (x, τ ) space fixed. The factor exp[−M (1, ¯ which is taken at the same 1)], spatial point, depends on τ1 − τ1 only: ¯ Λ πT exp [−M (1, ¯ ] = 1) × sin πT 2 sin πT τ1 − τ1 − ¯ i Λ τ1 − τ1 + ¯ i Λ αe , (7.41) where we used the ultraviolet cutoff from Eq. (D.7) and αe = αr − αb = 1 − K . 1 2 θ 2 f NN′ The simplest diagram for the density-density correlator of order O(γ) is shown in Fig. 18 and written as (7.42) Note that the interaction-induced phase factors cancel out at the external vertices which do not change chirality. In particular, this means that in Eq. (7.40) there is no renormalization of the velocity by the factors (7.13). The latter is a general property of fermionic loops with a larger number of impurities: the Green’s functions connecting to the external vertices with a small momentum transfer are characterized by the bare Fermi velocity vF . In the absence of e-e interaction, d1 is equal to (0) d1 = vF γ0 2 I , 2 (7.43) 25 τ1 1 ix1 /vF = + 0 -- FIG. 19: Analytical structure of the integrand in Eq. (7.44), with a pole at τ1 = ix1 /vF , for x1 > 0 and x < x1 . Closing the contour of integration upwards, the integral along the real axis of the Matsubara time is represented as a sum of two integrals along the imaginary axis at τ1 = +0 and τ1 = 1/T − 0. Using the periodicity with respect to the shift τ1 → τ1 ± 1/T , the sum gives a residue at the pole. 1 FIG. 18: Density-density correlator of first order in the impurity strength. The backscattering vertices are marked by the crosses. Pairing of the fluctuating factors exp(±θµ ) (denoted by the wavy lines) with each other yields the factor exp[−M (1, ¯ in Eq. (7.40), which describes the renormaliza1)] tion of the impurity strength by the interaction. by the contributions of the pole and the branch cut. The contribution of the latter is proportional to αe and can be omitted in the limit of weak interaction, after which the remaining integral over τ1 reads d1 ≃ where × 1/T I= 0 (0) (0) = − (7.44) What is important to us here is that I is given by the pole on the classical trajectory t1 = x1 /vF of a rightmoving electron (for x1 > 0) or a left-moving hole (for x1 < 0), where t1 = −iτ1 (Fig. 19). If we proceed to the Ω representation, the diagram for nonzero x and x1 is written as 1/T γ0 2 T 2vF dτ1 C dτ1 g+ (x1 , τ1 )g− (x − x1 , τ − τ1 ) 1 T 2 sinh{πT [ (2x − x)/v + iτ ]} 2vF 1 F × [ Θ(x1 )Θ(x1 − x) − Θ(−x1 )Θ(x − x1 ) ] . = 1/T γ0 −|Ωn (2x1 −x)| 3 |Ωn |e 4πvF 0 × [ Θ(x1 )Θ(x1 − x)Θ(Ωn ) + Θ(−x1 )Θ(x − x1 )Θ(−Ωn ) ] (0) dτ eiΩn τ d1 = and again is given by the (double) pole on the classical trajectory t = (2x1 − x)/vF , where t = −iτ and |2x1 − x| is the total length of the trajectory from point x = 0 to point x with backscattering at point x1 . This simple consideration suggests that the two-particle correlators in the presence of a weak e-e interaction should still be determined by a close vicinity of the classical trajectory. To evaluate the integrals (7.40) at αe = 0, we assume for definiteness that x1 > 0 and x < x1 . The singularities of the integrand of Eq. (7.40) in the upper half-plane of τ1 ¯ for −0 < Re τ1 < 1/T − 0 are then a pole at τ1 = ix1 /vF ¯ ¯ and a branch cut which comes from the factor (7.41). The branch cut starts at τ1 = τ1 + i/Λ and is sent upwards, ¯ as shown in Fig. 20a. The contour of integration over τ1 ¯ can be closed upwards, so that the integral over τ1 along ¯ the real axis from τ1 = 0 to τ1 = 1/T is given exactly ¯ ¯ 3 (Λ/πT )2αe sinh{πT [ (2x1 − x)/vF + iτ ]} sin−1+2αe [ πT (τ1 − ix1 /vF ) ] . sin{πT [ τ1 − τ + i(x1 − x)/vF ]} (7.45) Here the contour of integration C in the complex plane of τ1 runs anticlockwise around the branch cut along the imaginary axis of τ1 , Fig. 20b. For |αe | ≪ 1 the singularity in Eq. (7.45) is almost a pole on the classical trajectory t1 = x1 /vF with t1 = −iτ1 , so that, indeed, the system dynamics is determined by a close vicinity of the classical trajectory. We observe, however, that taking the interaction-dependent factor in the integrand straight at t1 → x1 /vF yields an ultraviolet singularity that exactly cancels the factor (Λ/πT )2αe in front of the integral in Eq. (7.45), which returns us to the noninteracting result. It is thus the integration in a close vicinity t1 − x1 /vF 1/T of the classical trajectory along the branch cut that gives the renormalization of disorder by interaction. If |αe | ≪ 1, Eq. (7.45) reduces to (0) 2αe d1 ≃ d1 (Λ/πT ) . (7.46) The difference between Eq. (7.46) and the noninteracting result consists solely in the last factor which describes the renormalization of the disorder strength γ0 → γ, where γ = γ0 (Λ/πT )2αe . 2. (7.47) “Two-impurity diffuson” Before generalizing to the Cooperon diagram, it is useful to consider a diagram of order O(γ 2 ) with two parallel impurity lines, Fig. 21. What is new in this “twoimpurity diffuson” as compared to Eq. (7.40) is that now 26 _ τ1 τ1 ix1 /vF ix1 /vF τ1 0 1/T 0 1/T a) b) FIG. 20: Analytical structure of the integrand in Eq. (7.40). (a) The contour of integration over τ1 is closed upwards where ¯ the singularities of the integrand for x1 > 0 and x < x1 are a pole at τ1 = ix1 /vF and a branch cut which starts at ¯ τ1 = τ1 + i/Λ and runs upwards. (b) The integral over τ1 of ¯ the contribution of the pole in Fig. 20a is given exactly by the branch cut starting at τ1 = ix1 /vF and running upwards along the imaginary axis of τ1 . our purposes here to analyze the case of weak interaction. The analytical structure of the integral over four time variables, τ1 , τ2 , τ1 , and τ2 can be simplified by selecting ¯ ¯ the singularities that are strongest in the limit |αe | ≪ 1. Let us assume for definiteness that x1 > 0 and x2 < x1 . In the plane of τ1 we have for Im τ1 ≥ 0: a pole at τ1 = ix1 /vF and three branch cuts starting at τ1 = τ1 + i/Λ, ¯ τ2 +i(x1 −x2 )/u, and τ2 +i(x1 −x2 )/u, as shown in Fig. 22. ¯ For weak interaction, these branch cuts are “weak” in the sense that they are characterized by a small exponent αe . The approximation we make consists in neglecting all weak cuts and keeping only “strong” cuts characterized by exponents close to 1. Taking into account the weak cuts yields corrections small in αe ≪ 1. (7.48) 2 + + 1 _ 0 τ2 τ2 _ i(x1−x2 )/u _ τ 1 1/T 0 τ2 τ2 a) Each of the cross-terms (the last 4 terms above), when taken on the classical trajectory, may give a large contribution to the exponent of Eq. (7.48). An important observation, however, is that on the classical trajectory they cancel each other exactly. This can be seen immediately since on the classical trajectory τ1 = τ1 and at this point ¯ M (1, 2) = M (¯ 2) and M (¯ ¯ = M (1, ¯ The cancella1, 1, 2) 2). tion can be verified for diffuson diagrams of higher order in γ and is a manifestation of the mere fact that for the density-density correlator there is no dephasing in the diffuson channel. No dephasing is generated both for the bare correlators Bµν (x, τ ) and for those renormalized by disorder at the level of RPA. 1 τ1 i(x1−x2 )/u ix1 /vF ix1 /vF there are factors exp(±M ) which connect different impurities. Specifically, the interaction-induced factor is exp [ −M (1, ¯ − M (2, ¯ 1) 2) ¯ ¯ + M (1, ¯ + M (¯ 2) ] . 1, 2) −M (1, 2) − M (1, 2) _ τ1 FIG. 22: Integration over the internal times in the twoimpurity diffuson shown in Fig. 21. (a) The analytical structure of the integral in the complex plane of τ1 : the contour of integration runs around a pole and three vertical branch cuts. (b) The contour of integration in the complex plane of τ1 runs around three vertical branch cuts. ¯ Picking up τ1 at the pole τ1 = ix1 /vF yields the following singularities in the plane of τ1 for Im τ1 ≥ 0: a strong ¯ ¯ branch cut at τ1 = ix1 /vF and two weak branch cuts at ¯ τ1 = τ2 + i(x1 − x2 )/u and τ2 + i(x1 − x2 )/u. Neglect¯ ¯ ing the weak cuts and integrating along the strong one similarly to Eq. (7.45) gives the renormalization of the impurity at point x1 . During the integration along the strong cut, we pick up other factors in the integrand at the classical trajectory, i.e., at the starting point of the strong cut, which yields a cancellation of the cross-terms connecting two impurities. The remaining double integral over τ2 and τ2 with the factor exp[−M (2, ¯ has the ¯ 2)] same structure as in the case of a single impurity and gives the renormalization of the impurity at point x2 . The net result is that the two-impurity diffuson d2 for |αe | ≪ 1 simply acquires two identical factors (Λ/πT )2αe which renormalize the impurities: (0) 2 FIG. 21: “Two-impurity” diffuson with the fluctuating factors exp[±θµ (N ) ] (denoted by the wavy lines) attached to the backscattering vertices (marked by the crosses). As far as the integration around the classical trajectory of the two-impurity diffuson is concerned, it suffices for 1/T b) 4αe d2 ≃ d2 (Λ/πT ) , (7.49) where (0) d2 2 (T /2vF )2 sinh2 {πT [ (x + 2x1 − 2x2 )/vF + iτ ]} × [Θ(x1 )Θ(x1 − x2 )Θ(x − x2 ) + Θ(−x1 )Θ(x2 − x1 )Θ(x2 − x)] . (7.50) = γ0 2vF 27 Self-energy corrections are incorporated in the same way leading to an additional factor d2 → d2 exp(−γ|x + 2x1 − 2x2 |/vF ), where γ is given by Eq. (7.47). This procedure is straightforwardly extended to the diffuson ladder with three or more impurity legs, so that the difference of the full diffuson ladder in the presence of e-e interaction from that in the absence of interaction is only in the change γ0 → γ [Eq. (7.47)]. F. Dephasing action and weak-localization correction Let us now return to the three-impurity Cooperon whose interaction-induced factor is given by Eqs. (7.33),(7.36)–(7.39). As the analysis of the two-impurity diffuson demonstrated, the analytical structure of the diagrams with increasing number of factors exp(±M ) rapidly gets quite complicated. The same example suggested, however, a systematic way to deal with the limit of weak interaction by ignoring in the first approximation all weak branch cuts. The integration around strong cuts then leads to the renormalization of impurities, while picking up nonsingular factors at the classical trajectory (the starting points of the strong cuts) generates the dephasing action. At this level, the calculation of the Cooperon is similar to that of the diffuson, with one essential difference: in contrast to the diffuson, the Cooperon factor exp(−SC ), Eq. (7.33), does not vanish on the classical trajectory and hence leads to dephasing. Another important issue is related to the definition of the classical trajectory for the Cooperon; more specifically, to the question of what is the velocity that characterizes this trajectory. To answer this question, one has to carefully treat the factors η± (7.13) in the integration over the internal times of the Cooperon. As shown in Appendix F, the classical Cooperon trajectory is characterized by a single velocity and this is u. Having established the fact that there is the unique velocity on the whole classical trajectory of the Cooperon, we can, in the leading approximation, ignore below the difference between u and vF in the functions M (N, N ′ ). Neglecting first the disorder-induced corrections b(x, τ ) in Eq. (7.32) and summing up according to Eqs. (7.37),(7.38) all 12 terms each of which is given by Eq. (7.39), we observe that the difference Mf f − Mf b in this approximation “miraculously” cancels out to zero on the classical trajectory. This is in agreement with Eq. (6.14) saying that the dephasing rate for the Cooperon vanishes in the limit γ → 0. Combining Eqs. (7.32) and (7.39), the disorderrenormalized correlators M are represented as M = M |γ=0 + 4b , (7.51) where b = b1 + b2 is given by Eqs. (7.30),(7.30). In order to evaluate the dephasing action on the classical trajectory, we analytically continue b1,2 (x, τ ) in the complex plane of τ . The analytical continuation of b2 (x, τ ) is effected via Eq. (7.28). On the imaginary axis of τ , the functions b1,2 at order O(γα2 ) read π α2 T 2 γ|x| [ (|x| − vF t) Θ(−|x| + vF t) 4 vF + (|x| + vF t) Θ(−|x| − vF t) ] , (7.52) 2 πα T 2 b2 (x, it) ≃ − 2 γ (|x| − vF t) Θ(−|x| + vF t) 8 vF b1 (x, it) ≃ + (|x| + vF t)2 Θ(−|x| − vF t) (7.53) for both |x|/vF and |t| ≫ 1/T . Taking this limit is justified for the calculation of the dephasing rate since upon averaging over the positions of impurities in Eq. (6.4) wl the characteristic distance between impurities is vF τφ or wl larger [in fact it is spread over the range between vF τφ and vF τ because of the logarithmic factor in Eq.(6.13)], which in turn is much larger than vF /T according to Eq. (6.14). The sum of Eqs. (7.52),(7.53) gives π α2 T 2 2 2 2 γ (x − vF t ) 8 vF × [ Θ(−|x| + vF t) + Θ(−|x| − vF t) ] . b(x, it) ≃ (7.54) Introducing a short-hand notation 8 β(N, N ′ ) = πα2 T γ b[ xN − xN ′ , i(tN − tN ′ ) ] , (7.55) where b is given by the asymptote (7.54), we see that β(N, N ′ ) = 0 on the classical trajectory of the Cooperon for all N, N ′ except for β(1, ¯ = β(2, ¯ = −2ta (tc − 2ta ) . 2) 3) (7.56) Summing up the terms (7.56) with the signs prescribed by Eqs. (7.38),(7.51) we obtain the dephasing action: deph SC ≃ 2πα2 T γta (tc − 2ta ) . (7.57) This result coincides with Eq. (6.12) for the dephasing action derived via the path-integral calculation in Sec. VI D. The integration along the cuts around the classical trajectory yields the renormalization of disorder on short spacial scales of order vF /T around impurities, similarly to the renormalization of the diffuson ladder in Eqs. (7.46),(7.49). The three-impurity Cooperon with both the renormalization of disorder and the dephasing included is thus given by (0) 6α c3 ≃ c3 (Λ/πT ) exp[−2πα2 T γta (tc − 2ta )] , (7.58) where (0) c3 = × × + × 3 (T /2vF )2 sinh [ πT (xΣ /vF + iτ ) ] [ Θ(x1 )Θ(x1 − x2 )Θ(x3 − x2 ) Θ(x3 − x)Θ(x3 )Θ(x1 − x) Θ(−x1 )Θ(x2 − x1 )Θ(x2 − x3 ) Θ(x − x3 )Θ(−x3 )Θ(x − x1 ) ] (7.59) γ0 2vF 2 28 is the three-impurity Cooperon bubble without interaction, xΣ = 2(x1 + x3 − x2 ) − x , (7.60) |xΣ | is the total length of the classical trajectory along which the density fluctuations propagate from point x = 0 to point x. The length of the Cooperon loop vF tc and the distance between two right-most impurities vF ta are related to the positions of impurities by vF tc /2 = max{x3 , x1 } − x2 and vF ta = |x3 − x1 |. Equation (7.58) describes the Cooperon bubble with scalar vertices—the current vertices are straightforwardly incorporated below. To calculate the conductivity, one has to add also selfenergy corrections to the Green’s functions constituting the Cooperon bubble. In the presence of e-e interaction, this is done in essentially the same way as the renormalization of impurities in Sec. VII E and results in the additional factor exp(−γ|xΣ |/vF ), where γ is the renormalized rate of backscattering, which the Cooperon acquires on the length |xΣ |. The WL correction ∆σwl is then expressed in terms of c3 from Eq. (7.58) as ∆σwl = −2 × × dx1 dx2 1 2 2 × 2 e2 vF Ωn 1/T dτ dx eiΩn τ We have developed two alternative approaches for a systematic analytic treatment of the problem. One is a two-step procedure which combines the bosonic-RG treatment of high-energy renormalization processes at the first step with the subsequent analysis of low-energy real processes in the fermionic language. The other approach is based on the method of “functional bosonization” which makes it possible to treat both types of effects simultaneously. We have demonstrated that the notion of weak localization is applicable to the strongly correlated onedimensional electron system. This finding is of conceptual importance, showing that the famous non-Fermiliquid character of the Luttinger liquid does not prevent this system from exhibiting the features characteristic of conventional mesoscopic electron structures. Our approach thus provides a framework for systematically studying the mesoscopic phenomena in strongly correlated electron systems. Further directions of research include mesoscopic fluctuations and noise in a disordered Luttinger liquid, spin-related effects (in particular, the magnetoresistance), several-channel quantum wires, transport in a disordered system in the limit of strong interaction, transport in a “granular Luttinger liquid” (strong impurities). 0 dx3 c3 e−γ|xΣ|/vF , (7.61) IX. ACKNOWLEDGMENTS Ωn →+0 where the factor of 2 accounts for the contribution of the diagrams (b) and (c) in Fig. 5, which give exactly the same contribution as the Cooperon, and the factor of (1/2)2 comes from the dressing of the current-vertices by the diffuson ladders, similarly to Eq. (B.1). The current vertices in Eq. (7.61) are simply given by ±vF , since we formulate our model in terms of a linear electron spectrum from the very beginning.112 Integrating (7.61) we reproduce Eq. (6.13) which was obtained in Sec. VI D in two distinct steps by treating the renormalization of disorder and dephasing separately. Here, we have demonstrated how these two effects arise “side by side” from the technical point of view in the framework of the functional bosonization. Specifically, the analytical structure of the density-density correlator in the time domain is such that the dephasing comes from the classical trajectory, whereas the renormalization of impurities is associated with the integration along strong branch cuts around the classical trajectory. VIII. We thank A. Altland, D. Aristov, T. Giamarchi, L. Glazman, M. Grayson, K. Le Hur, D. Maslov, A. Nersesyan, A. Tsvelik, A. Yashenkin, O. Yevtushenko, and I. Yurkevich for interesting discussions. The work was supported by the Schwerpunktprogramm “Quanten-HallSysteme” and the Center for Functional Nanostructures of the Deutsche Forschungsgemeinschaft, by RFBR, by the Program “Leading Russian Scientific Schools” (Grant No. 2192.2003.2), and by the Program of RAS “Condensed Matter Physics”. The authors acknowledge the hospitality of the Condensed Matter and Statistical Physics Section of the Abdus Salam ICTP (Trieste), where part of this work was performed. The work of IVG, conducted as part of the project “Quantum Transport in Nanostructures” made under the EUROHORCS/ESF EURYI Awards scheme, was supported by funds from the Participating Organisations of EURYI and the EC Sixth Framework Programme. ADM acknowledges the hospitality of the Kavli Institute for Theoretical Physics at Santa Barbara during the completion of the manuscript and partial support by NSF under Grant No. PHY9907949. SUMMARY In this paper, we have studied the transport properties of interacting spinless electrons in a disordered quantum wire within the framework of the Luttinger-liquid model. Our main result is the weak-localization correction (6.13), governed by the dephasing rate (6.14). APPENDIX A: POLARIZATION OPERATOR AND SCREENING IN DISORDERED WIRES In this Appendix, we derive the polarization operator and the dynamically screened RPA-interaction for a 29 disordered 1d electron system. As everywhere in the paper, we consider spinless electrons. The Luttinger-liquid power-law singularities only show up in the renormalized strength of disorder γ and do not make any other difference in the derivation below. We begin by considering the clean case. According to Dzyaloshinskii-Larkin theorem,111 the RPA equations for screened interaction are then exact. The propagators of interaction between right-movers (V++ ) and between right- and left-movers (V+− ) obey V++ = g4 − g4 Π+ V++ − g2 Π− V+− , V+− = g2 − g2 Π+ V++ − g4 Π− V+− , For g4 = g2 we have u = vF 2 q 2 vF , + Ω2 n (A.3) 2 q 2 vF iΩn πρ , |Ωn | iΩn ∓ qvF V+− (A.4) dp (0) (0) G (iǫm + iΩn , p + q)G± (iǫm , p) = D± (iΩn , q) 2π ± × [ Θ(−ǫm )Θ(ǫm + Ωn ) + Θ(ǫm )Θ(−ǫm − Ωn ) ] , ρ |Ωn | − D± (iΩn , q) . 2 2π V+− 2 2 g4 + (g4 − g2 ) qvF 1 2πvF iΩn + qvF (A.10) (A.11) where Πµν (iΩn , q) = |Ωn | ρ δµν − Dµν (iΩn , q). 2 2π We now calculate the two-particle propagators Dµν in the RPA. Performing the analytical continuation to real frequencies iΩn → ω + i0, we introduce the retarded twoR particle propagator [Dµν (ω, q)](0) with only self-energy impurity lines included, R [D++ (ω, q)](0) = iπρ , ω − qvF + iγ/2 γ = 1/τ where 1/2 (A.9) (A.12) where 2 × (Ω2 + q 2 vF )/(Ω2 + q 2 u2 ) , n n 2 = g2 (Ω2 + q 2 vF )/(Ω2 + q 2 u2 ) , n n u = vF (1 + g4 /2πvF )2 − (g2 /2πvF )2 . V+− = g2 − (g2 Π++ + g4 Π−+ )V+− − (g4 Π−− + g2 Π+− )V++ , The solution to Eqs. (A.1) and (A.2) for arbitrary g2 and g4 reads27 V++ = 1/2 V++ = g4 − (g4 Π++ + g2 Π−+ )V++ − (g2 Π−− + g4 Π+− )V+− , −1 where = (iǫm ∓ vF p) are the free Green’s functions for a given chirality and the ultraviolet momentum cutoff is sent to infinity. The polarization operator and the two-particle propagator are related to each other via Π± (iΩn , q) = (A.8) The interaction between left-movers is given by V++ with q → −q. Let us now turn to the disordered case. We only take into account the backward scattering induced by disorder, since the forward scattering can be completely gauged out in the calculation of the conductivity. We average the RPA bubbles over disorder separately by using the ladder approximation, see Sec. VI D. The backscattering off disorder makes it necessary to introduce two indices µ, ν = ± in the polarization bubble Πµν , corresponding to the chirality of the vertices of the bubble. The RPA equations for the interaction propagators in the presence of disorder read which satisfies (0) G± (iǫm , p) (A.7) 2 where ρ = 1/πvF is the thermodynamic density of states, so that Π(0, q) = ρ. For later use we also introduce the two-particle propagator D± (iΩn , q) = 2 g2 qvF (iΩn − qvF ) , 2πvF Ω2 + u 2 q 2 n 2 Ω2 + q 2 vF = g2 n , Ω2 + u 2 q 2 n u = vF 1 − (g2 /2πvF ) are the chiral polarization operators for right(+)/left(-) movers, Ωn = 2πnT is the bosonic Matsubara frequency. The total polarization operator is a sum of the chiral terms: (A.6) V++ = (A.1) (A.2) 1 q Π± (iΩn , q) = 2π qvF ∓ iΩn (A.5) while for g4 = 0 the propagators V++ and V+− are different from each other: where Π(iΩn , q) = ρ 2 Ω2 + q 2 vF n , Ω2 + u 2 q 2 n g2 1+ , πvF V++ = V+− = g2 . (A.13) and τ is the transport scattering time. Note the factor of 1/2 in front of γ in Eq. (A.12), which reflects the fact that only backscattering off impurities is considered, i.e., τq = 2τ , where τq is the quantum scattering time, in contrast to the case of isotropic scattering, where τq = τ . The 30 R propagator [D−− (ω, q)](0) is given by Eq. (A.12) with the change q → −q, whereas the nondiagonal propagators R [Dµ,−µ (ω, q)](0) = 0. The equations for dressed two-particle retarded propagators read R R D−− (ω, q) = [D−− (ω, q)](0) 1 + γ DR (ω, q) , 2πρ +− (A.14) γ R R D+− (ω, q) = [D++ (ω, q)](0) DR (ω, q) , 2πρ −− R V+− = g2 = g2 1 + g2 Π+− 2 (1 + g2 Π+− )2 − g2 Π−− Π++ 2 q 2 vF − ω 2 − iω(γ + δγ/2) , 2 u2 − ω 2 − iω(γ + δγ) q (A.25) where u is given by Eq. (A.9) and δγ = −γ g2 . 2πvF (A.26) (A.15) APPENDIX B: WEAK-LOCALIZATION CORRECTION TO THE CONDUCTIVITY IN 1D and R R D++ (ω, q) = D−− (ω, −q) , R D+− (ω, q) = R D−+ (ω, q) . (A.16) (A.17) Solving Eqs. (A.14)–(A.17), we get ω − q + iγ/2 , (A.18) 2 − q 2 vF + iωγ 1 γ R , (A.19) D+− (ω, q) = −πρ 2 2 ω 2 − q 2 vF + iωγ R D−− (ω, q) = iπρ ω2 and Π−− (ω, q) = Π++ (ω, −q) 2 ρ q 2 vF − qvF ω − iωγ/2 = , 2 2 q 2 vF − ω 2 − iωγ Π+− (ω, q) = Π−+ (ω, q) iωγ/2 ρ = , 2 2 q 2 vF − ω 2 − iωγ Π(ω, q) = Πµν = ρ µν 2 q 2 vF (A.20) L−1 |Ub (q)|2 = 1/2πρτ (A.21) (L is the system size) and changes chirality at every impurity vertices. For the anisotropic scattering, we have τ = τq /2, where τ is the transport time and τq is the quantum lifetime. The retarded single-particle Green’s function expressed in terms of τ reads 2 q 2 vF . 2 − iωγ −ω (A.22) Equation (A.22) reduces to Eq. (A.3) in the clean limit γ → 0 and acquires the conventional diffusive form Πdiff (ω, q) = ρDq 2 /(Dq 2 − iω) in the limit ω ≪ γ, qvF , where the 1d diffusion constant 2 D = vF /γ. From Eqs. (A.10), (A.11), (A.20), and (A.21), we obtain the retarded interaction propagators for g2 = g4 : g2 1 + g2 Π q 2 v 2 − ω 2 − iωγ = g2 2 F q u2 − ω 2 − iωγ In this Appendix, we calculate the WL correction to the conductivity in a noninteracting 1d system, assuming that the phase coherence is broken by an external source of dephasing and the dephasing time is small; specifically, τφ ≪ τ . Under this condition, the conductivity can be expanded in powers of τφ /τ . The lowest-order diagrams are depicted in Fig. 5. The sum of the diagrams with two impurity lines, shown in Fig. 7, yields a zero contribution to order (ǫF τ )−1 . We formally consider only backward scattering, i.e., the impurity line corresponds to the correlation function of the backscattering random potential GR (ǫ, p) = (ǫ ∓ vF p + i/2τq )−1 = (ǫ ∓ vF p + i/4τ )−1 . ± The anisotropic scattering also introduces the current vertex renormalization: each current vertex is multiplied by a factor τ /τq = 1/2. The sum of the diagrams (b) and (c) in Fig. 5 with an impurity line covering the two-impurity Cooperon (two crossed impurity lines) is equal to the contribution σC3 of the diagram with the three-impurity Cooperon. The total three-impurity WL correction is thus given by ∆σwl = 2 σC3 . R R V++ = V+− = The expression for σC3 reads (A.23) with u given by Eq. (A.6). For g4 = 0 we have R 2 V++ = −g2 = − Π−− 2 (1 + g2 Π+− )2 − g2 Π−− Π++ 2 2 q 2 vF − qvF ω − iωγ/2 g2 , (A.24) 2πvF q 2 u2 − ω 2 − iω(γ + δγ) τ τq 2 ∂fǫ ∂ǫ dQ J(Q) , 2π (B.1) where fǫ is the Fermi distribution function, the overall minus sign reflects the fact that the product of the current vertices is negative in the Cooperon with three backscattering lines, and the factor of 2 is related to two possibilities of setting chiralities at the vertices: (+−) σC3 = −2 2 e2 vF dǫ 2π − 31 and (−+). The function J(Q) describes the fermionic loop: J(Q) = dp1 2π dp 2π 1 (2πρτ )3 dp2 2π × GR (p)GR (p1 )GR (p2 )GR (−p + Q) + − + − × GA (p)GA (−p2 + Q)GA (−p1 + Q)GA (−p + Q) . + − + − (B.2) depends not only on the total time tc but also on the geometry of the Cooperon path. We thus have to perform averaging over the geometry of the path at given tc . For this purpose we need the probability density of return P2 (tc , ta ) for a particle which starts moving to the right at point x = 0 under the condition that the total length is vF tc and the first reflection occurs at point x = vF ta . This function is written as ∞ P2 (tc , ta ) = vF Using the equality 0 × exp[ −(tc − tb )/2τ ] Θ(tc − tb ) δ[x(tc )] . GR (p)GA (p) = 2iτ GR (p) − GA (p) , ± ± ± ± 2 P1 (t) = ∗ J(Q) = 4τ |P (Q)| [P (Q) + P (Q)] , where P (Q) = 1 2πρτ dp R G (p)GA (−p + Q) . − 2π + Now we introduce a phenomenological dephasing time τφ through an additional decay of the Green’s function: GR (ǫ, p) ± −1 → (ǫ ∓ vF q + i/4τ + i/2τφ ) , is the probability density of being reflected at point vF t (again, note that τ is the transport scattering time and only the backscattering is taken into account). The factor exp [−(tc − tb )/2τ ] in Eq. (B.7) describes the probability of avoiding the backscattering on the last segment of the path and x(tc ) = 2vF (ta − tb ) + vF tc P2 (tc , ta ) = 1 1 , (B.3) 2τ ivF Q + 1/2τ + 1/τφ 1 1 1 1 . + J(Q) = τ 2τ τφ [ (vF Q)2 + (1/2τ + 1/τφ )2 ]2 (B.4) P (Q) = Substituting Eq. (B.4) in Eq. (B.1), we find for τφ ≪ τ : ∆σwl 2 , (B.5) where σD = e2 vF τ /π is the Drude conductivity. Equation (B.5) for the weak-localization correction can be cast in the form of an integral over the time needed for a particle to return to the starting point after two backscatterings: ∞ ∆σwl = −2σD dtc P2 (tc ) exp(−tc /τφ ) . (B.6) 0 Here P2 (t) ≃ t/16τ 2 , d 1 − e−t/2τ dt for tc > tb . We obtain which yields 1 τφ = − σD 8 τ (B.7) Here we simplify Eq. (B.2): 2 dtb P1 (ta )P1 (tb − ta ) Θ(tb − ta ) t≪τ is the return probability after two reflections (hence the subscript 2) for a particle which starts moving either to the right or to the left and the overall factor of 2 in Eq. (B.6) comes from the summation over the sign of the velocity at the starting point. Let us now turn from the phenomenological factor exp(−tc /τφ ) to the actual dephasing factor. The latter ∞ × = 0 1 −tc /2τ e 4τ 2 dtb Θ(tb − ta )Θ(tc − tb ) δ[ 2(ta − tb ) + tc ] 1 −tc /2τ e Θ(tc − 2ta ) . 8τ 2 For the dephasing action which depends on t and ta we have ∞ ∆σwl = −2σD ∞ dtc 0 dta P2 (tc , ta ) exp[−S(tc , ta )] , 0 which reduces to Eq. (B.6) for the phenomenological action S(tc , ta ) = tc /τφ . APPENDIX C: DEPHASING ACTION IN THE PATH-INTEGRAL APPROACH In this Appendix, we present details of the calculation of the dephasing action defined by Eq. (6.6) and given in the final form by Eq. (6.12). For the imaginary parts of the retarded interaction propagators (A.24),(A.25) which enter Eq. (6.6), we have, retaining only the terms of order 2 O(g2 ): R ImV+± (ω, q) ≃ − 2 g2 (qvF − ω)(qvF ∓ ω) ωγ 2 2 . 4πvF (q vF − ω 2 )2 + ω 2 γ 2 Next, we introduce functions F+± (ω, q) which obey R ImV+± (ω, q) = − 2 g2 ωF+± (ω, q) . 4πvF (C.1) 32 × F++ (t1 − t2 , t1 − t2 ) , Putting g2 = 0 in F+± and doing the Fourier transform to (x, t) space, we get F++ (x, t) = e−γ|t|/2 [ δ(x+ ) γ2 γ 2 |x− | f++ − + 2 2 x+ x− 16vF 16vF xf (t) = vF t tc IIf , L : ta < t < ta + ; xf (t) = vF (2ta − t) 2 t IIIf , R : ta + < t < tc ; xf (t) = vF (t − tc ) . 2 Similarly, for the backward (time-reversed) path we have xb (t) = xf (tc − t): tc Ib , L : 0 < t < − ta ; xb (t) = −vF t 2 tc IIb , R : − t a < t < tc − t a ; 2 xb (t) = vF (2ta − tc + t) IIIb , L : tc − ta < t < tc ; xb (t) = vF (tc − t) . The “self-energy” part Sff of the action is written as 6 Jn , Jn + 2 0 ta ta dt2 dt1 J1 = vF 0 0 dt2 ta tc dt2 dt1 2J5 = vF ta +tc /2 0 × [ F++ (t1 − t2 + tc , t1 − t2 ) + F++ (−t1 + t2 − tc , t1 − t2 ) ] , ta +tc /2 IIf − IIIf : 2J6 = vF tc dt2 dt1 ta +tc /2 ta × [ F−+ (−t1 − t2 + 2ta + tc , t1 − t2 ) + F+− (t1 + t2 − 2ta − tc , t1 − t2 ) ] , Sfb = 15 2 g2 2 T Jfb , 4πvF Jfb = Jn , (C.5) n=7 where tc /2−ta ta If − Ib : dt2 dt1 J7 = vF 0 0 × F+− (t1 + t2 , t1 − t2 ) , tc −ta ta If − IIb : dt2 dt1 J8 = vF tc /2−ta 0 × F++ (t1 − t2 − 2ta + tc , t1 − t2 ) , tc ta If − IIIb : dt2 dt1 J9 = vF tc −ta 0 × F+− (t1 + t2 − tc , t1 − t2 ) , ta +tc /2 IIf − Ib : J10 = vF tc /2−ta dt2 dt1 0 ta × F−− (−t1 + t2 + 2ta , t1 − t2 ) , ta +tc /2 IIf − IIb : J11 = vF tc −ta dt2 dt1 tc /2−ta ta × F−+ (−t1 − t2 + tc , t1 − t2 ) , (C.4) n=4 where each of the terms Jn corresponds to the integration over a certain pair of the forward-path segments (for brevity, we set vF = 1 in the arguments of Fµν below): If − If : dt1 2J4 = vF 0 < t < ta ; n=1 ta +tc /2 ta If − IIf : The “vertex” part Sfb of the action is given by where Sij are given by Eq. (6.6). We split the forward path into three segments corresponding to the right- (R) and left- (L) moving parts, where the coordinate xf (t) behaves as Jff = ta +tc /2 ta +tc /2 ta S(t, ta ) = Sff + Sbb − Sfb + Sbf = 2(Sff − Sfb ) , 2 g2 2 T Jff , 4πvF dt2 dt1 × F++ (t1 − t2 , t1 − t2 ) , If − IIIf : I0 (z) and I1 (z) are the modified Bessel functions. Since the dephasing action becomes of order unity at times wl t ∼ τφ ≪ τ, we expand F++ (x, t) and F+− (x, t) to the first order in the disorder strength γ, which yields Eqs. (6.9),(6.10). Let us now turn to the calculation of the dephasing action S(t, ta ) which can be represented as Sff = tc tc J3 = vF × [ F+− (t1 + t2 − 2ta , t1 − t2 ) + F−+ (−t1 − t2 + 2ta , t1 − t2 ) ] , √ 1 f++ (z) = √ I1 (2 z) Θ(z) , z f+− (z) = I0 (2z) Θ(z) , 3 dt2 ta × F−− (−t1 + t2 , t1 − t2 ) , . Here x± = x ± vF t, ta +tc /2 dt1 ta IIIf − IIIf : (C.3) R: J2 = vF , (C.2) γ −γ|t|/2 γ2 F+− (x, t) = e f+− − 2 x+ x− 4vF 16vF If , ta +tc /2 IIf − IIf : tc /2−ta ta +tc /2 IIf − IIIb : dt2 dt1 J12 = vF 0 ta × F−− (−t1 + t2 + 2ta , t1 − t2 ) , tc /2−ta tc IIIf − Ib : dt2 dt1 J13 = vF tc /2+ta 0 × F+− (t1 + t2 − tc , t1 − t2 ) , 33 tc −ta tc IIIf − IIb : APPENDIX D: FUNCTIONAL BOSONIZATION IN THE CLEAN CASE dt2 dt1 J14 = vF tc /2−ta tc /2+ta × F++ (t1 − t2 − 2ta , t1 − t2 ) , tc /2−ta tc IIIf − IIIb : dt2 dt1 J15 = vF tc −ta ta +tc /2 × F+− (t1 + t2 − 2tc , t1 − t2 ) . Carrying out the integration yields: J1 = J2 = J3 = J4 = J6 = J8 = J9 = J10 = ta , 2 tc , 4 1 tc − ta , 2 2 ta tc γ , 8 tc tc γ − ta , 8 2 ta ta tc −γ − ta 2 4 2 t2 γ a, 4 ta 1 tc − ta − γ 2 2 4 J11 = γ 4 J12 = ta ta −γ 2 4 γ 4 1 = 2 = J7 J13 = J14 J5 B++ (x, τ ) = −T × B< (x, τ ) = B> (−x, −τ ) . tc − ta 2 + t2 a , , 2 tc − ta 2 , B0 (x) = and Jff = , Jfb t2 c − 8ta 2 Jff − Jfb (C.6) tc − ta 2 − ta . , (C.7) (C.8) We see that the difference Jff −Jfb vanishes at γ → 0, i.e., in the absence of disorder in the interaction propagators. Finally, we arrive at the expression for the total action S(t, ta ) = 2 × = 2 g2 2 πvF tc − ta 2 Using α = g2 /2πvF , we obtain Eq. (6.12). . (D.2) 2 vF − u2 2πT |x| . 2u2 vF (D.3) In the integration over q at Ωn > 0, there are poles at q = iΩn /vF and q = ±iΩn /u. The first only contributes to the integral if Ωn vF /d, whereas the latter—only if Ωn u/d, which sets the limits of summation over n: ∞ B> (x, τ ) = 2πT Θ(x) −πT vF + sgn x u e−Ωn /Λv [e −Ωn (x/vF +iτ ) − 1 ] Ωn n=1 ∞ e−Ωn /Λ [ e−Ωn (|x|/u+iτ ) − 1 ] , Ωn n=1 (D.4) 2 g2 2 T (Jff − Jfb ) 4πvF T γta (D.1) To accurately treat the ultraviolet cutoff for an arbitrary ratio u/vF , we introduce a finite range of interaction d, so that g2 (q) depends on the transferred momentum q and vanishes for qd ≫ 1. The RPA approximation, leading to Eq. (D.1), is exact in the Luttinger-liquid model for arbitrary g2 (q). In the integrand of Eq. (D.1), g2 (q), vF (q), and u(q) are related to each other at given q in precisely the same way, Eq. (4.2), as in the case of g2 (q) = const. Everywhere below in this Appendix, except for Eq. (D.6), when writing g2 , vF , or u, we understand them as taken at q = 0. The term in Eq. (D.1) with Ωn = 0 at |x| ≫ d reads , tc , − ta 2 tc ta − ta − γ 2 4 = J15 = 0 , e −1 = B0 + B> + B< , −iΩn + qvF where B0 (x), B> (x, τ ), and B< (x, τ ) are the contributions of terms with Ωn = 0, Ωn > 0, and Ωn < 0 in the Matsubara sum over the bosonic frequency Ωn = 2πnT , respectively. The sums over Ωn > 0 and Ωn < 0 are related to each other as , tc − ta 2 t2 tc +γ c 2 8 tc γ = + 2 4 tc = 2γta 2 n 2 dq g2 qvF 2π 2πvF Ω2 + q 2 u2 n iqx−iΩn τ 2 tc − ta 2 In this Appendix, we calculate the correlators Bµν (x, τ ) in a clean Luttinger liquid for an arbitrary strength of interaction. Following the discussion in Sec. III A, we consider only the g2 interaction, while the g4 interaction is accounted for in the shift of the Fermi velocity, Eq. (3.12). Substituting the interaction propagator V++ (iΩn , q) given by Eq. (A.7) [with vF understood here and below as the renormalized velocity (3.12)] in Eq. (7.8), we have (C.9) where there appear two ultraviolet scales Λv = vF /d and Λ = u/d and we approximate the cutoffs by the exponential factors exp(−Ωn /Λv ) and exp(−Ωn /Λ) (the exact shape of the cutoffs plays no role in the infrared physics we deal with in this paper). 34 Using for N ≫ 1 and Re z ≫ 1/N the formula APPENDIX E: “DIRTY RPA” IN THE FUNCTIONAL BOSONIZATION N e−nz − 1 ≃ − ln(1 − e−z ) − ln N , n n=1 we write B> (x, τ ) for |x| ≫ d as Λ 1 − exp [−2πT (x/u + iτ )] Λv 1 − exp [−2πT (x/vF + iτ )] vF − u 2πT /Λ − ln . 2u 1 − exp [−2πT (|x|/u + iτ )] (D.5) B> (x, τ ) = Θ(x) ln Combining Eqs. (D.3), (D.5), and (D.2), we obtain B++ (x, τ ) given by Eq. (7.9). Note that the factor vF /u in η+ (x, τ ) [Eq. (7.13)] comes from the ratio Λ/Λv in Eq. (D.5). The remaining cutoff in L(x, τ ) is given by Λ. The Green’s function [Eq. (7.1)] of a right-mover then reads for |x| ≫ d: (0) g+ (x, τ ) = g+ (x, τ ) exp [−B++ (x, τ ) ] 1 iT = − 2u sinh [ πT (x/u + iτ ) ] (πT /Λ)αb × . {sinh [ πT (x/u + iτ ) ] sinh [ πT (x/u − iτ ) ]}αb /2 Note that the Fermi velocity vF drops out completely in (0) the product g+ (x, τ )η+ (x, τ ). The Green’s function in the real-time domain, Eq. (5.16), is obtained by means of the Wick rotation τ → it + sgn t/Λ. The calculation of B+− (x, τ ) is performed in a similar way: substituting Eq. (A.8) in Eq. (7.8), we get B+− (x, τ ) = −T n In this Appendix, we demonstrate that the correlators (7.7) in the presence of disorder are modified according to Eq. (7.8) with the effective RPA interaction Vµν (7.20),(7.21) whose dynamical properties reflect the disorder-induced backscattering (“dirty RPA”). For this purpose, we directly calculate the disorder-induced correction to the averages of the type exp{i[θµ (x, τ ) − θν (0, 0)]} in the framework of the diagrammatic technique formulated in Sec. VII B. In the expansion in powers of γ, the rules of the technique prescribe that impurity vertices are connected with each other via the correlators Bµν (x, τ )|γ=0 in a spatially homogeneous system. To obey Wick’s theorem, multiple fermionic loops can only be present in this technique if each of them contains backscattering vertices, since all fermionic loops without impurities have been accounted for by the “clean RPA” propagators. We show below that the result for Bµν obtained this way coincides for α ≪ 1 with the result of the “dirty RPA”, Eq. (7.32). (x1,τ1) -- (x1,τ1) ′ -- + dq eiqx−iΩn τ − 1 g2 2 2 . (D.6) 2π q u + Ω2 n Ω2 n + + (0,0) -(x,τ) 2 q 2 vF Note that the denominator + in Eq. (7.8) is canceled by the numerator in Eq. (A.8), so that vF disappears from the problem right away and the integration over q in Eq. (D.6) is determined at Ωn = 0 solely by the plasmon poles q = ±iΩn /u. For |x| ≫ d, we arrive at Eq. (7.10). In Sec. VII and in Appendix E, we do contour integrals in the complex plane of the Matsubara time τ and should be careful about the behavior of Bµν (x, τ ) at x → 0 for τ → 0. When integrating in the complex plane of τ , it is convenient to have the ultraviolet cutoffs in Bµν (x, τ ) that are independent of τ . Our choice is to use Eq. (D.4), which yields a cutoff which depends on x: u x → x + sgn x = x + d sgn x , Λ (D.7) i.e., |x| → |x| + d in the propagators Bµν (x, τ ) in Eqs. (7.9)–(7.10), and similarly in the disorder-induced corrections to them in Eqs. (7.30),(7.30). FIG. 23: Diagram in the functional-bosonization technique, which describes the addition of disorder in the RPA interaction propagator. Two backscattering vertices (0, 0) and (x, τ ) are connected to each other via the fermionic bubble with a single impurity at point x1 . Let us consider the average of first order in γ, ei[θ+ (x,τ )−θ+(0,0)] − ei[θ+ (x,τ )−θ+(0,0)] ≃ γ=0 dx1 I(x, τ, x1 ) , (E.1) given by the diagram in Fig. 23. We are interested in the pairings between the fields θ+ (0, 0) and θ+ (x, τ ) that connect up the points (0, 0) and (x, τ ) via the fermionic bubble that contains two backscatterings at point x1 : I(x, τ, x1 ) = vF γ0 −B++ (x,τ ) e 2 1/T 1/T dτ1 0 0 ′ dτ1 35 (0) (0) ′ ′ × g+ (0, τ1 − τ1 )g− (0, τ1 − τ1 ) ×e sinh(πT z), J(x, τ, x1 ) is written as ′ ′ ′ −B++ (0,τ1 −τ1 )−B−− (0,τ1 −τ1 )+2B+− (0,τ1 −τ1 ) ′ ′ × e−B++ (x1 ,τ1 )+B+− (x1 ,τ1 )+B++ (x1 ,τ1 )−B+− (x1 ,τ1 ) − 1 ′ ′ × eB++ (x−x1 ,τ −τ1 )−B+− (x−x1 ,τ −τ1 ) × e−B++ (x−x1 ,τ −τ1 )+B+− (x−x1 ,τ −τ1 ) − 1 J1 τ’ 1 J2 J3 × . (E.2) 0 2 1/T 1/T dτ1 0 0 1/T τ1 τ 1 1/T ′ ′ s(|x1 |/u + iτ1 )s(|x1 |/u − iτ1 ) s(|x1 |/u + iτ1 )s(|x1 |/u − iτ1 ) s(|x − x1 |/u + iτ − iτ1 ) ′ s(|x − x1 |/u + iτ − iτ1 ) × ′ dτ1 1 ′ ′ s1−αe [i(τ1 − τ1 )]s1−αe [−i(τ1 − τ1 )] × τ 0 −i|x−x1|/u τ T 2 × i|x −x1|/u −i|x1|/u i|x1|/u J(x, τ, x1 ) = − s(|x − x1 |/u − iτ + iτ1 ) ′ s(|x − x1 |/u − iτ + iτ1 ) αe /2 χ1 − 1 αe /2 αe /2 χ2 − 1 , (E.3) where a) b) FIG. 24: Analytical structure of the integrand in Eq. (E.2) ′ in the complex plane of τ1 and τ1 . (a) The contour of integra′ tion over τ1 is closed upwards and runs around three branch cuts [J1,2,3 are the corresponding terms in Eq. (E.4)]. (b) The contour of integration over τ1 for the first two terms in Eq. (E.4) is closed downwards and runs around two branch cuts. All functions Bµν in Eq. (E.2) are understood as taken at zero disorder. The analytical structure of the integrand ′ in the complex plane of τ1 and τ1 is similar to that in the integrals analyzed in Secs. VII E,VII F. Specifically, in ′ the plane of τ1 we have six vertical branch cuts starting at ′ ′ ′ τ1 = ±i|x1 |/u, τ1 = ±i|x − x1 |/u, and τ1 = τ1 ± i/Λ, and ′ going to Im τ1 = ±i∞, Fig. 24a. On top of that, there are singularities associated with the functions ηµ (7.13). Note that the strong singularity (a double pole if α → 0) ′ at τ1 = τ1 (if Λ is sent to infinity) that comes from the (0) ′ product of two free Green’s functions g± [ 0, ±(τ1 − τ1 ) ] is largely canceled by the vanishing of the rest of the integrand at this same point. Evaluation of the integrals (E.2) at arbitrary α leads to rather cumbersome expressions which are not particularly interesting for our purposes. Let us extract the factor that renormalizes the impurity strength, vF γ0 Λ 2u2 πT × J(x, τ, x1 ) , I(x, τ, x1 ) = − 2αe e−B++ (x,τ )|γ=0 and analyze the behavior of the factor J(x, τ, x1 ) at small α. By introducing a short-hand notation s(z) = ′ s(x1 /vF + iτ1 )s(x1 /u + iτ1 ) ′ )s(x /v + iτ ) , s(x1 /u + iτ1 1 F 1 s[(x − x1 )/vF + i(τ − τ1 )] = s((x − x1 )/u + i(τ − τ1 )] χ1 = χ2 × ′ s[(x − x1 )/u + i(τ − τ1 )] . ′ s[(x − x1 )/vF + i(τ − τ1 )] In Sec. VII C, while calculating the disorder-induced correction to Bµν to order O(γα2 ), we neglected the difference between vF and u in the interaction propagators. To compare with the result of Sec. VII C, we neglect the difference between the two velocities here as well; in particular, by putting χ1 = χ2 = 1. We discuss the role of such factors in the calculation of the dephasing rate in Appendix F. ′ The integral over τ1 in Eq. (E.3) can be represented as a sum of three terms, J =− T 2 1/T dτ1 (J1 + J2 + J3 ) , (E.4) 0 each of which is the integral around a branch cut in the ′ complex plane of τ1 , Fig. 24a. Inspection of Eq. (E.3) shows that J3 , the term related to the branch cut start′ ing at τ1 = τ1 + i/Λ, can be omitted for αe ≪ 1, since J3 ∼ O(α3 ), which gives a subleading contribution to J e [one power of αe comes from the difference of the integrand on two sides of the cut and two more come from the expansion of the expressions in the curly brackets in ′ Eq. (E.3) in τ1 − τ1 ]. The integrals J1,2 are written for αe ≪ 1 as 36 ∞ πT J1 ≃ αe 2 dt′ 1 0    s ×   s t′ +  1 s2(1−αe ) t′ + 1 |x−x1 | vF |x−x1 | vF |x1 | vF + iτ − iτ1 s |x1 | vF + + iτ s ∞ πT J2 ≃ −αe 2   s t′ +  1 ×    1 |x−x1 | vF |x−x1 | vF t′ 1  − |x1 | vF s |x1 | vF + s2(1−αe ) |x1 | vF − |x1 | vF s t′ + 1 |x−x1 | vF |x−x1 | vF − iτ s t′ + 1 + iτ1 s |x1 | vF |x1 | vF +  − iτ1 αe /2 eiπαe /2 − 1  + iτ  + iτ1 − iτ |x1 | vF + iτ1 s − iτ + iτ1 |x−x1 | vF αe /2 1 s(t′ )s t′ + 2 |xF | 1 1 v 1 dt′ 1 0 + iτ1  |x−x1 | vF s     ,   |x−x1 | vF + iτ − iτ1 s s(t′ )s 1 − iτ + iτ1 2 |x−x1 | vF t′ 1 +    e−iπαe /2 − 1 ,   αe /2 − iτ  − iτ1 αe /2  In turn, each of the integrals of J1 and J2 over τ1 can be split into a sum of two integrals along branch cuts in the complex plane of τ1 (Fig. 24b), so that in total we have J ≃ (J1a + J1b ) + (J2a + J2b ), where for αe ≪ 1: πT 2 J1a ≃ −α2 e   s t +  1 ×   s t′ +  1 J1b ≃ α2 e  × × ∞ ∞ dt1 0 0 dt′ 1 1 s2(1−αe ) + |x1 | vF − iτ s t1 − |x−x1 | vF + |x1 | vF |x−x1 | vF + |x1 | vF + iτ s t′ − 1 |x−x1 | vF + |x1 | vF 2 ∞ ∞ dt1 0 0 dt′ 1 1 s t1 − |x1 | vF πT 2 s t′ + 1 + |x−x1 | vF 2 |x−x1 | vF + iτ s t′ + 1 ∞ ∞ dt1 0 + 0 |x1 | vF dt′ 1 |x−x1 | vF + |x1 | vF + |x−x1 | vF + iτ s t′ − 1   + iτ |x−x1 | vF 1 s2(1−αe ) t1 + t′ + 1 − iτ s t1 + |x1 | vF + |x−x1 | vF  1 s (t1 ) s t1 + 2 |xF | v αe /2    − iτ  −1 ,   + iτ |x1 | vF s2(1−αe ) t1 + t′ + 1 αe /2 1 s(t′ )s t′ + 2 |xF | 1 1 v  1 t1 + t′ + 2 |xF | 1 v |x−x1 | vF πT 2  1 s t′ + 2 |xF | s (t1 ) s t1 + 2 |x−x1| 1 v vF J2a ≃ α2 e  2 |x1 | vF + |x−x1 | vF − iτ s t1 + − iτ |x1 | vF − (E.5) s(t′ ) 1 |x1 | vF s t1 + +  |x1 | vF − + iτ |x−x1 | vF − iτ αe /2   , (E.6) |x1 | vF − iτ s (t1 ) 1 s t1 + 2 |xF | s (t′ ) s t′ + 2 |x−x1| 1 1 v vF and J2b is obtained from J1a by interchanging x1 ↔ x − x1 . The terms J1a and J2b on one hand and J1b and J2a on the other give essentially different contributions to J(x, τ, x1 ). The integrals J1a and J2b , as a function of the position of the impurity x1 , are not exponentially small only in a close vicinity of width ∼ u/T around x1 = 0 and x1 = x, respectively. After averaging over x1 , these “local” terms yield a contribution of order O(α2 ) e which does not depend, in the leading approximation, on x and τ . On the contrary, J1b and J2a for small αe do not  |x−x1 | vF  |x−x1 | vF αe /2 + iτ s t′ + 1 + αe /2 αe /2  , (E.7) depend on x1 for x1 between points x1 = 0 and x1 = x. The integration over x1 for |x|/vF ≫ 1/T yields then the leading contribution to Eq. (E.1). Only the latter terms are thus of importance to us. In the limit of small αe , their contribution to J is represented as J(x, τ, x1 ) ≃ α2 e × sinh−2 πT πT 2 2 ∞ ∞ dt1 0 t1 + t′ + 1 0 dt′ 1 |x1 | |x − x1 | + + iτ vF vF 37 + (τ → −τ ) } . (E.8) Analytically continuing J(x, τ, x1 ) onto the imaginary axis of τ , we observe that J(x, it, x1 ) is exponentially small for large positive values of (T /vF )(|x1 | + |x − x1 | ± vF t) and is a linear function of this sum when it is large and negative. Since we are interested in the limit of |x|/vF , |t| ≫ 1/T , we can approximate J(x, it, x1 ) as |x1 | |x − x1 | π − J(x, it, x1 ) ≃ − α2 T t − e 2 vF vF × Θ (−|x1 | − |x − x1 | + vF t) + ( t → −t ) . (E.9) dx1 I(x, it, x1 ) = −e c3 (x, τf − τi , x1 , x2 , x3 ) = 1/T × 1/T dτ1 0 vF γ0 2 1/T d¯1 τ 0 3 1/T dτ2 0 1/T d¯2 τ 0 1/T dτ3 0 d¯3 τ 0 (0) × g+ (x1 , τ1 − τi ) ei[ θ+ (x1 ,τ1 )−θ− (x1 ,τ1 ) ] × g− (x2 − x1 , τ2 − τ1 ) e−i[ θ+ (x2 ,τ2 )−θ− (x2 ,τ2 ) ] (0) (0) × Integrating over x1 we obtain −B++ (x,it)|γ=0 potentially dangerous factors η± (x, τ ) in the calculation of the WL dephasing. The function c3 in Eq. (7.61), which gives the threeimpurity Cooperon loop, is explicitely written as b(x, it) , (E.10) where b(x, it) is given by Eq. (7.54). We thus provided a direct verification of the RPA form of Eq. (7.8) by deriving the first term in the expansion of exp(−B++ ) in powers of γ without using the disorder-renormalized effective interaction (7.20). This procedure can be extended to higher powers of γ and to the correlator B+− . An important point concerns the factors sinh±αe /2 (πT z), where z denotes the arguments of the functions s(z) in Eqs. (E.6),(E.7). These were omitted in the limit of small αe in Eq. (E.8). However, even for αe ≪ 1, this step cannot be justified for arbitrary x, x1 , and τ , e.g., if αe |x|T /vF ≫ 1. What is important to us is that all these factors actually cancel each other after the analytical continuation onto the imaginary axis of τ → it for large positive values of −|x1 | − |x − x1 | ± ut. As Eq. (E.9) shows, only this range of t is relevant to J(x, it, x1 ), which is the quantity used in the calculation of the dephasing rate. The RPA for pairing of all four fields at points (0, 0) and (x, τ ), shown in Fig. 24, namely of the combinations θ+ (x, τ ) − θ− (x, τ ) and θ+ (0, 0) − θ− (0, 0) with each other, via the fermionic bubble with two backscatterings at x = x1 can be reproduced in a similar way. APPENDIX F: CLASSICAL TRAJECTORY FOR THE COOPERON In this Appendix, we substantiate an important step in our derivation of the interaction-induced action SC of the Cooperon within the functional-bosonization scheme. Specifically, we analyze the role of the factors η± (x, τ ), Eq. (7.13), which contain both the Fermi velocity vF and the plasmon velocity u. We demonstrate that in the spinless case, considered in this paper, the velocity of electrons on the “Cooperon path” which contributes to ∆σwl is given by u and not by vF . Put another way, the velocity of the interfering quasiclassical trajectories is renormalized by the interaction and coincides with that of the RPA propagators. As a result, one can omit the g+ (x3 − x2 , τ3 − τ2 ) ei[ θ+ (x3 ,τ3 )−θ− (x3 ,τ3 ) ] × g− (x − x3 , τf − τ3 ) × × (0) (0) τ τ g− (x1 − x, τ1 − τf ) e−i[ θ+ (x1 ,¯1 )−θ− (x1 ,¯1 ) ] ¯ (0) τ τ g+ (x2 − x1 , τ2 − τ1 )ei[ θ+ (x2 ,¯2 )−θ− (x2 ,¯2 ) ] ¯ ¯ (0) × τ τ g− (x3 − x2 , τ3 − τ2 ) e−i[ θ+ (x3 ,¯3 )−θ− (x3 ,¯3 ) ] ¯ ¯ × g+ (−x3 , τi − τ3 ) ¯ (0) , (F.1) where we write the external times of the bubble as τi and τf [in Eq. (7.61), τf − τi = τ ]. To examine the role of the factors η± , it is convenient to introduce the function 1/T C3 = 1/T dτf c3 eiΩn (τf −τi ) dτi 0 0 by integrating first over the external times. The averaging . . . over the fluctuating fields θ± (x, τ ) yields the factor exp [−2(Mf f − Mf b ) ] , defined in Eqs. (7.37)–(7.39), in the integrand of Eq. (F.1). In the absence of interaction (Mf f = Mf b = 0), the integrals over the internal times are determined by the poles of the free Green’s functions [specified by Eq. (F.4) below with u → vF ]. For given coordinates x1 , x2 , x3 , and x, we refer to this set of times as the “classical trajectory” of the Cooperon with velocity vF . Each interaction-induced term M (N, N ′ ) in Eqs. (7.37),(7.38) contains the combination B++ + B−− , and hence depends on both u and vF . If one neglects disorder in the RPA propagators (“clean RPA”), the factor exp[−M (N, N ′ )] is represented as e−M(N,N ′ ) = e−αe L(xN −xN ′ ,τN −τN ′ ) × ζ(xN − xN ′ , τN − τN ′ ) , (F.2) where ζ(x, τ ) = η+ (x, τ ) η− (x, τ ), αe = 1 − K, and the functions L(x, τ ) and η± (x, τ ) are given by Eqs. (7.12),(7.13). The dependence on vF comes from the factors η± . The “dirty RPA” does not affect the latter. We thus see that each pair of impurities N and N ′ in Eq. (F.1) generates two poles associated with the 38 functions η± : vF 2 u sinh [ πT (x/vF + iτ ) ] sinh [ πT (x/vF − iτ ) ] . × sinh [ πT (x/u + iτ ) ] sinh [ πT (x/u − iτ ) ] ζ(x, τ ) = The singularities in Eq. (F.1) coming from the functions ζ(x, τ ) are analogous to those in the bare Green’s func(0) tions g± (x, τ ). However, they can appear either as “upoles” at iτ = ±x/u or as “vF -poles” at iτ = ±x/vF , de′ ¯ pending on the sign of the factor (−1)N +N or (−1)N +N in Eqs. (7.37),(7.38). We can readily integrate over τi and τf the free Green’s functions that connect to the external vertices. Since there are no interaction-induced factors at the external vertices, the integration yields again the free Green’s functions. The function C3 then becomes C3 = vF γ0 2 1/T × × × × × × × × × × × × × × 3 1 2 vF 1/T d¯1 τ 1/T dτ2 1/T d¯2 τ 1/T dτ3 0 0 (0) g+ (x1 (0) g− (x2 d¯3 τ − x3 , τ1 − τ3 ) ζ(x1 − x3 , τ1 − τ3 ) ¯ ¯ (0) g+ (x3 (0) g− (x1 (0) g+ (x2 (0) g− (x3 ζ −1 (x1 ζ −1 (x1 ζ −1 (x2 iτ1 iτ2 iτ3 i¯3 τ i¯2 τ i¯1 τ = = = = = = −x1 /u , (x2 − 2x1 )/u , (2x2 − 2x1 − x3 )/u , −x3 /u , (x2 − 2x3 )/u , (2x2 − 2x3 − x1 )/u . 0 0 0 0 − x1 , τ2 − τ1 ) ζ(x2 − x1 , τ2 − τ1 ) − x2 , τ3 − τ2 ) ζ(x3 − x2 , τ3 − τ2 ) − x3 , τ1 − τ3 ) ζ(x1 − x3 , τ1 − τ3 ) ¯ ¯ − x1 , τ2 − τ1 ) ζ(x2 − x1 , τ2 − τ1 ) ¯ ¯ ¯ ¯ − x2 , τ3 − τ2 ) ζ(x3 − x2 , τ3 − τ2 ) ¯ ¯ ¯ ¯ − x3 , τ1 − τ3 ) ζ −1 (x1 − x3 , τ1 − τ3 ) ¯ ¯ −1 − x2 , τ1 − τ2 ) ζ (x3 − x2 , τ3 − τ2 ) ¯ ¯ − x1 , τ2 − τ1 ) ζ −1 (x2 − x3 , τ2 − τ3 ) ¯ ¯ ζ(0, τ1 − τ1 ) ζ(0, τ2 − τ2 ) ζ(0, τ3 − τ3 ) ¯ ¯ ¯ R(x1 , x2 , x3 , τ1 , τ2 , τ3 , τ1 , τ2 , τ3 ) ¯ ¯ ¯ W (x1 , τ1 , x3 , τ3 , Ωn ) ¯ W (x − x3 , τ3 , x − x1 , τ1 , −Ωn ) , ¯ (F.3) where ζ(0, τ ) = (vF /u)2 , the function R combines all the factors coming from the first factor in Eq. (F.2), and W (x1 , τ1 , x3 , τ3 , Ωn ) ¯ = e−|Ωn x1 |−iΩn τ1 [ Θ(Ωn )Θ(x1 ) − Θ(−Ωn )Θ(−x1 ) ] ¯ + e−|Ωn x3 |−iΩn τ1 [ Θ(Ωn )Θ(x3 ) − Θ(−Ωn )Θ(−x3 ) ] . In the single-particle Green’s function, the Fermi velocity vF drops out completely in the product (0) g± (x, τ )η± (x, τ ). One can see that each free Green’s function in Eq. (F.3) is multiplied by a corresponding factor η± , so that all the vF -poles associated with the free Green’s functions in Eq. (F.3) are replaced by upoles. One might think that vF disappears completely (F.4) However, the impurity vertices are all interconnected by the interaction propagators. Among the latter, there are propagators that connect vertices which are not connected by the single-particle Green’s functions [e.g., points (x1 , τ1 ) and (x2 , τ2 )]. As a result, the vF -poles in ¯ the corresponding functions η± remain uncompensated. In the noninteracting case, the sequence of points in real space 1→2→3→1→2→3→1 1/T dτ1 from the problem, and hence the integration over the internal times in Eq. (F.3) straightforwardly yields a new classical trajectory defined by (F.5) forms a closed loop with segments given by the Green’s (0) functions g± . Now that we have the interaction-induced functions η± that connect every pair of impurities, other trajectories can contribute to C3 , whose segments are given either by the Green’s function or by the function η± . In view of this, the notion of a single classical trajectory contributing to ∆σwl requires a more accurate justification. As we show below, the interaction itself “chooses” a unique classical trajectory which is defined in Eq. (F.4), i.e., the one that is determined by a combination of the Green’s functions with the renormalized velocity vF → u. The contributions of all other trajectories are exponentially suppressed. Note that within the fermionic path-integral approach used in Sec. VI, the “correct” velocity is chosen automatically on the saddlepoint trajectory of the total action. When analyzing the role of the factors η± , we can, in the first approximation, neglect other interactioninduced factors in Eq. (F.1) by setting the exponent αe = 0 in Eq. (F.2), i.e., by putting R = 1 in Eq. (F.3) while keeping the difference between vF and u in the functions ζ. This simplifies the analytical structure of the integrand in Eq. (F.3) by reducing the integration along branch cuts to the integration around simple poles only. The integration around all the poles then yields C3 represented as a sum over a whole set of trajectories, not only the one given by Eq. (F.5). The integrand of Eq. (F.3) contains products of the (0) type g± (x, it)ζ(x, it) which, as a function of t, have poles on the classical trajectory (F.4) at t = ±x/u with a residue equal to (0) [ ut ∓ (x + u sgn x/Λ) ] g± (x, it) ζ(x, it)|t→±x/u ≃− 1 vF exp 2πi u u − vF πT |x| uvF (F.6) 39 for |x|/u ≫ 1/T . Writing the remaining factors ζ which are not singular at this pole as vF 2 exp πT |tv | + |tv | − |tu | − |tu | − + − + u v v u u × sgn t+ sgn t− sgn t+ sgn t− (F.7) ζ(x, it) ≃ for |tv,u | ≫ 1/T , where tv = t ± x/v and tu = t ± x/u, ± ± ± we observe that all the exponential factors coming from Eqs. (F.6),(F.7) cancel out on the classical trajectory (F.4). On the contrary, any other sequence of poles has at least some of the segments of the trajectory passed with velocity vF and no such compensation occurs, which leads to an exponential decay of the contribution to C3 . 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Note that if we were to start with a quadratic dispersion and then linearize, an additional term in the current vertices would appear proportional to ±∂τ θ± , as pointed out in Ref. 93. This, however, only yields an ultraviolet renormalization of the bare Fermi velocity vF by e-e interactions with high-energy transfers. In this sense, in our model vF already includes all “Fermi-liquid-like” renormalizations of the electron spectrum.