Kinetic processes in Fermi-Luttinger liquids
ЖЖЭТФ
KINETIC PROCESSES IN FERMI-LUTTINGER LIQUIDS
Alex Levchenko a , Tobias Micklitz b a Department of Physics, University of Wisconsin–Madison, Madison, Wisconsin 53706, USA b Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil
In this work we discuss extensions of the pioneering analysis by Dzyaloshinskiˇi and Larkin [Sov. Phys. JETP ,202 (1974)] of correlation functions for one-dimensional Fermi systems, focusing on the effects of quasiparticlerelaxation enabled by a nonlinear dispersion. Throughout the work we employ both, the weakly interactingFermi gas picture and nonlinear Luttinger liquid theory to describe attenuation of excitations and explore thefermion-boson duality between both approaches. Special attention is devoted to the role of spin-exchangeprocesses, effects of interaction screening, and integrability. Thermalization rates for electron- and hole-likequasiparticles, as well as the decay rate of collective plasmon excitations and the momentum space mobilityof spin excitations are calculated for various temperature regimes. The phenomenon of spin-charge drag isconsidered and the corresponding momentum transfer rate is determined. We further discuss how momentumrelaxation due to several competing mechanisms, viz. triple electron collisions, electron-phonon scattering, andlong-range inhomogeneities affect transport properties, and highlight energy transfer facilitated by plasmonsfrom the perspective of the inhomogeneous Luttinger liquid model. Finally, we derive the full matrix ofthermoelectric coefficients at the quantum critical point of the first conductance plateau transition, and addressmagnetoconductance in ballistic semiconductor nanowires with strong Rashba spin-orbit coupling. For the special issue of JETP devoted to the 90th birthday jubilee of Igor E. Dzyaloshinskiˇi.
1. INTRODUCTION
The concept of quasiparticles plays a central rolein the condensed matter physics of strongly interactingmany-body quantum systems [1,2]. For instance, in thecontext of electrons in conductors, one typically viewsthe quasiparticle states as those evolving from the freeelectron gas to a Fermi liquid when adiabatically turn-ing on the interaction. In accordance with Landau the-ory [3], quasiparticles inherit some of the basic quan-tum numbers of bare electrons such as spin, charge,and momentum. Their respective dispersion relationsas well as thermodynamical and kinetic properties may,however, differ significantly due to interaction-inducedrenormalizations. A crucial advantage of the quasipar-ticle picture is that residual interactions are assumed tobe weak, and can be systematically and controllably ad-dressed by means of perturbation theory. The centralquestion related to the validity of the quasiparticle de-scription concerns their lifetime τ qp . Indeed, in the pro-cess of scattering quasiparticles decay, and their merenotion remains meaningful only if attenuation is weakenough so that they can be considered as sufficientlylong-lived collective excitations. In Fermi systems, thePauli principle severely limits the phase space available for quasiparticle collisions. The low temperature decayrate can then be estimated from the Golden rule as τ − qp ( ε, T ) ∝ ( νV ) ( ε + π T ) /ε F . (1)In this expression, the excitation energy ε = v F ( p − p F ) of a quasiparticle with momentum p is counted fromthe Fermi energy ε F , ν is the density of states and V is the characteristic strength of the short-range re-pulsive interaction. The dominant microscopic scat-tering channel leading to Eq. (1) involves quasipar-ticle decaying into three: another quasiparticle and aparticle-hole pair. The amplitude for this process isproportional to V , hence, the dimensionless factor of ( νV ) in the scattering probability entering Eq. (1).The factor ε is the phase space volume for scatter-ing of a quasiparticle with energy ε compatible withthe conservation of total energy and momentum. At fi-nite temperatures the smearing of states in the energystrip of order ∼ T per particle leads to the correspond-ing T dependence of τ − qp . Higher-order processes in-volving n + 1 quasiparticles, namely n > electron-hole pairs, are usually neglected as their respective rate Throughout the paper we use units with Planck and Boltz-mann constants set to unity (cid:126) = k B = 1 . a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r lex Levchenko, Tobias Micklitz ЖЭТФscales with higher powers of energy. In particular, atzero-temperature the rate for relaxation processes ofa quasiparticle with energy ε involving n particle-holepairs vanishes as τ − qp ∝ ε n . One notable property ofEq. (1) is that it predicts the same relaxation time forparticle-like and hole-like excitations. Another prop-erty is that the crossover from zero-temperature tofinite-temperature relaxation is governed only by onescale, viz. when the excitation energy compares to thetemperature itself ε ∼ T .In addition to the quasiparticle relaxation, whichis often viewed as an out-scattering rate from a par-ticular quantum state, one may address a more gen-eral question of relaxation of a nonequilibrium quasi-particle distribution function. In kinetic theory suchproblem is typically analyzed in the framework of thelinearized Boltzmann equation. The eigenvalues of thecorresponding collision operator define relaxation timesof different distribution function modes. In three di-mensional Fermi liquids this problem is exactly solv-able [4, 5] and one finds that all these rates are para-metrically the same, scaling respectively as ∝ T . Incontrast, in two-dimensional Fermi liquids, kinematicsof head-on collisions leads to a parametrically distinctrelaxation of odd and even momentum harmonics ofthe distribution function, in particular τ − even ∝ T /ε F while τ − odd ∝ T /ε F [6, 7].The role of dimensionality in quasiparticle relax-ation becomes the most dramatic in one-dimension(1D). This special case of electron liquids can be exper-imentally realized in quantum wires of GaAs/AlGaAsheterostructure [8] or carbon nanotubes [9] when par-ticle density is such that only the lowest sub-band oftransversal modes is occupied. It further requires thattemperature is sufficiently low and sample purity is suf-ficiently high, so that thermally- and disorder-inducedtransitions to higher sub-bands are suppressed. In ad-dition, edge modes formed at the boundaries of a 2Delectron gas when placed in a strong magnetic field inthe integer or fractional quantum Hall regime [10, 11],or edge states of 2D quantum spin Hall topological-insulators [12], provide other distinct examples of, re-spectively, chiral and helical quantum 1D electron liq-uids.In principle, all these systems can be successfullydescribed within the framework of Luttinger liquidtheory [13–15], which builds out of the Tomonaga-Luttinger (TL) model [16, 17]. As is known form pi-oneering works [18–20], in the asymptotic low-energylimit ε/ε F (cid:28) , the key properties of the TL modelare essenatially non-Fermi liquid like. A power-lawanomaly manifests in the suppression of the single par- ticle density of states ν ( ε ) = ν (cid:18) | ε | v F p Λ (cid:19) g sin( πg ) πg Γ(1 − g ) , (2)and collapse of the quasiparticle residue in the distri-bution function. At T → that is n ( ε ) = Γ( + g )2 √ π Γ(1 + g ) (cid:34) − Γ( − g )Γ( + g ) (cid:18) | ε | v F p Λ (cid:19) g sgn ( ε ) (cid:35) (3)where ν = 1 / (2 πv F ) , Γ( z ) is the Euler’s gamma func-tion, and p Λ is the momentum cutoff of the model(parametrically p Λ ∼ p F ). In the simplest spinless ver-sion of the TL-model with short-ranged interaction, asingle dimensionless coupling constant, g = 12 (cid:20) ν V √ ν V − (cid:21) , (4)can be related to the zero-momentum Fourier compo-nent of the bare interaction potential V . The limitof weak interaction corresponds to g (cid:28) and Eqs.(2)-(3) are valid for g < / . However, a direct at-tempt to apply Luttinger liquid theory to the questionof quasiparticle lifetime meets formidable challenges.In a fermionic representation of the TL-model, elabo-rated explicitly by Dzyaloshinskiˇi and Larkin [19], theelectron self-energy vanishes on the mass shell in allorders of perturbation theory and, consequently, corre-lation functions assume power-law tails. These results,and the absence of relaxation, can be alternatively un-derstood from the Mattis and Lieb [18], and Luther andPeschel [20] bosonization construction, which maps in-teracting 1D fermions to a collection of decoupled har-monic modes of charge-density and spin-density oscil-lations. Notably, in both approaches the exact solutionrelies heavily on the linearization of the fermionic dis-persion relation, which is a cornerstone approximation.One is then left with the natural puzzle whether in-corporating curvature of the dispersion relation into theTL-model would cure the issue and yield a finite life-time of excitations, thus possibly restoring Fermi liquidlike properties of the system. This line of reasoning canbe also corroborated within the fermionic picture, not-ing that spectrum nonlinearity softens phase space re-strictions for quasiparticle scattering, thus making theirrelaxation possible. In Ref. [19] the limit of strong interactions, g > /2, wasalso considered, including the scenario when coupling betweenfermions of the same chirality is different from coupling be-tween fermions of different chirality. For additional details onthe derivation of Eq. (3) see also Ref. [21]. ЭТФ
Kinetic processes in Fermi-Luttinger liquids
Similarly, at the level of the bosonic description,nonlinear terms of the dispersion relation couple chargeand spin modes thus enabling their decay. However, itwas quickly recognized that curvature cannot be in-cluded perturbatively, and a naive expansion leads tospurious divergences. These and other related ques-tions to 1D kinetics, including the connection betweenthe two pictures of the fermion-boson duality, attractedsignificant recent interest. This has lead to the devel-opment of the nonlinear Luttinger liquid theory, alsoreferred to as Fermi-Luttinger liquid (FLL) theory (seeRefs. [22, 23] for comprehensive reviews and referencesherein). Specifically for the problem of quasiparticlerelaxation in quantum wires, various scattering rateswere calculated within different interaction models forboth, spinless [24–33] and spin- / fermions [34–40]. Inparts of the present work we review and extend theseresults.On the experimental forefront the hallmark sig-natures of Luttinger liquid behavior have been ob-served by means of various spectroscopic techniques.Namely, power-law anomalies in the density of states,tunneling conductance, and current-voltage character-istics [9, 10, 41, 42], spin-charge separation [43, 44], andcharge fractionalization [45, 46]. Besides GaAs quan-tum wires, carbon nanotubes, and edge modes, clearfeatures of Luttinger liquid physics have been identi-fied in many other systems such as bundles of NbSe [47] and MoSe [48] nanowires, polymer nanofibers [49]and conjugated polymers at high carrier densities [50],as well as atomically controlled chains of gold atoms onGe surfaces [51], just to name a few distinct examples.In the most recent report [52], relaxation processes inquantum wires were captured and bounds on the cor-responding timescales were determined, thus providingmeasurements of quasiparticle properties beyond theparadigm of linear Luttinger liquid theory. In a parallelline of developments [53–56], cooling of nonequilibriumquasiparticles in quantum Hall edge fluids was mea-sured and the corresponding lengths scales oforther-malization processes were quantified.The focus of this communication is on the descrip-tion of elementary kinetic processes inducing relax-ation in nonlinear Luttinger liquids and their emer-gent transport properties. Keeping forward scatter-ing electron-electron interactions and accounting fornonlinear contributions to the electron dispersion, thistheory is beyond the Dzyaloshinskiˇi-Larkin theorem.The latter relax kinematic constraints and open phasespace for multi-loop corrections to the electron self-energy, thereby providing a plethora of inelastic pro-cesses which affect equilibrium as well as nonequilib- rium properties of the 1D quantum electron liquids.The rest of this work is structured as follows. Sec. 2focuses on the hierarchy of relaxation times in Fermi-Luttinger liquids. We present results beyond para-metric estimates, including detailed computations ofa number of experimentally relevant interaction mod-els. The complimentary kinetic equation approach,applied to the quasiparticle picture of a weakly in-teracting Fermi gas, and spin- and charge-excitationsof a Luttinger liquid, are explored concurrently. Wepresent numerical estimates for experimentally mea-sured relaxation rates and provide detailed comparisonto previous results. In Sec. 3, the temperature depen-dence of kinetic coefficients is calculated, accounting forextrinsic mechanisms of momentum relaxation due tophonons or long-range inhomogeneities. The contribu-tion to heat transport mediated by plasmons in the in-homogeneous Luttinger liquid is elucidated. We devoteparts of the discussion to the thermoelectric propertiesat the first plateau transition of the quantum conduc-tance. Finally, we consider effects of strong spin-orbitcoupling and magnetoconductance in ballistic semicon-ductor nanowires. In Sec. 4, we provide concluding re-marks by sketching a broader picture, commenting onrelated topics as well as open questions relevant for chi-ral, helical, and spiral versions of 1D Fermi-Luttingerliquids. Several Appendices accompany our presenta-tion in the main text, providing additional technicaldetails of the presented analysis and formalism.
2. HIERARCHY OF RELAXATION PROCESSES
The physics of quasiparticle relaxation in 1D quan-tum electron liquids is perhaps a surprisingly rich andcomplicated problem. In part this has to do withthe fact that, in contrast to their higher dimensionalcounterparts, two-particle collisions, namely scatter-ing processes with the emission of a single particle-hole excitation, do not result in finite relaxation rates.This statement pertains to generic dispersion relations,i.e. including curvature, and not only applies to mod-els with linear dispersion. Indeed, kinematics of two-particle scattering in 1D is such that particles eitherkeep or swap their momenta, but neither of these op-tions causes relaxation. To allow for the redistributionof momenta and, at the same time, to comply with re-strictions of conservations laws one necessarily needsto consider triple electron collisions, or alternatively,assume some extrinsic mechanisms. In part this material was summarized in Sec. IV of theextensive review in Ref. [23]. lex Levchenko, Tobias Micklitz ЖЭТФThe analysis of 1D kinematics of multi-particle col-lisions resolving energy and momentum conservationsreveals a variery of possible scattering events. They ul-timately lead to a hierarchy of relaxation stages in thesystem and an emergent asymmetry between the relax-ation of particle-like and hole-like excitations. All pro-cesses can be broken down into several distinct classes.First are the forward scattering processes with softmomentum transfer that involve either (i) all parti-cles from the same branch, or (ii) particles from bothbranches such that all initial and final states are nearthe Fermi energy. Second are processes involving statesdeeper in the band. These latter are relevant for (iii)the drift of quasiholes and (iv) backscattering processesthat change the number of right and left moving exci-tations before and after the collision. We will referto thermalization when discussing relaxation processesthat proceed without backscattering. These processesdetermine the lifetime of quasiparticles associated tothe redistribution of excess energy, and affect thermaltransport properties of the system. In contrast, the no-tion of equilibration will be used to refer to relaxationprocesses involving the backscattering of quasiparticles,which ultimately govern electrical transport properties.
In the picture of a weakly nonideal Fermi gas, theprobabilities of particle collisions can be calculated per-turbatively in the interaction, employing the usual ˆ T -matrix formalism [57]. Within the Golden Rule, thescattering rate, W = 2 π | A | δ ( E − E (cid:48) ) δ P,P (cid:48) , (5)is expressed in terms of the scattering amplitude A ofthe corresponding quantum process. Here E ( E (cid:48) ) and P ( P (cid:48) ) label total energy and momentum of initial (fi-nal) states, and the delta-function δ ( E − E (cid:48) ) along withthe Kronecker delta δ P,P (cid:48) enforce energy and momen-tum conservations. In the semiclassical limit, the three-particle amplitude A was considered in Ref. [58]. Thegeneralization to the degenerate quantum limit waspresented in the work of Ref. [59], and exchange termswere carefully examined in Refs. [26, 35, 60]. The re-sulting amplitude takes the form A = 1 L (cid:88) PP (cid:48) sgn ( P ) sgn ( P (cid:48) ) V p (cid:48) a − p a V p (cid:48) c − p c ε p b + ε p c − ε p b + p c − p (cid:48) c Ξ σσ (cid:48) . (6)Here L is the system size and sums run overall possible permutations P of momenta p i with i = 1 , , starting from the direct scattering process ( p , p , p ) → ( p (cid:48) , p (cid:48) , p (cid:48) ) to all its exchange pro-cesses, with sgn ( P ) accounting for the sign of theparticular permutation (using the convention thatsgn (123) = +1 ). Each permutation comes with aspin-dependent factor Ξ σσ (cid:48) = δ σ a σ (cid:48) a δ σ b σ (cid:48) b δ σ c σ (cid:48) c re-flecting particle exchange. In the spinless case, theamplitude has an identical structure to Eq. (6) with Ξ σσ (cid:48) ≡ . The amplitude consists of 36 distinct termsthat can be split into groups of six, each representingone direct and five exchange scattering processes,respectively. Technically speaking Eq. (6) appearsfrom the iteration of the ˆ T -matrix, ˆ T = V + V ˆ G ˆ T ,to second order in the bare two-particle interactionpotential V . Here ˆ G is the resolvent operator (viz.the free particle Green’s function) and ε p denotes theenergy-momentum dispersion relation.For practical applications to quasiparticle scatter-ing in quantum wires, it is sufficient to assume the sim-ple dispersion of a parabolic band ε p = p / m ∗ witheffective mass m ∗ , and use a Coulomb interaction po-tential. Effects of screening due to nearby gates can bemodeled by a conducting plate placed at a distance d away from the wire. In this case the interaction poten-tial is of the form V ( x ) = e κ (cid:20) | x | − √ x + 4 d (cid:21) , (7)where κ is the dielectric constant of the host mate-rial. The diverging short-range behavior of this poten-tial needs to be regularized in order to evaluate thesmall-momentum Fourier components V p entering theamplitude in Eq. (6). To this end, we introduce thesmall width w of the quantum wire, w (cid:28) d , and replace / | x | → / √ x + 4 w . Upon 1D Fourier transform wethen find V p = (2 e / κ ) [ K (2 w | p | ) − K (2 d | p | )] , (8)where K ( z ) is the modified Bessel function ofthe second kind. Using the asymptotic ex-pression of the Bessel function at z (cid:28) , K ( z ) ≈ ln(2 /ze γ E ) + z ln(2 /ze γ E − ) , with γ E the Euler constant, one then finds the simplified formof the interaction potential V p ≈ (2 e / κ ) (cid:2) ln( d/w ) − ( pd ) ln( e − γ E / | p | d ) (cid:3) , (9)applicable to the screened limit of Coulomb interac-tion and valid for p (cid:28) /d . In the opposite regime, d − (cid:28) p (cid:28) w − , the second term in Eq. (8) can beneglected since K ( z ) ∝ e − z / √ z at z (cid:29) . One then ЭТФ
Kinetic processes in Fermi-Luttinger liquids arrives at the simplified form of the unscreened poten-tial V p ≈ (2 e / κ ) (cid:2) ln( e − γ E / | p | w ) + ( pw ) ln( e − γ E / | p | w ) (cid:3) . (10)A few comments are in order in relation to the inter-action model presented in this section. (i) It should benoted that retaining numerical pre-factors of the orderof unity under the logarithm in above expressions for V p would exceed the accuracy of further calculations,so they will be dropped and simply set to unity. (ii)However, retaining the sub-leading corrections contain-ing p in the main log-series expansion of both Eqs.(9) and (10) is actually crucial. Indeed, in the spin-less case, the model with contact interaction as wellas the Calogero-Sutherland model, are known to becompletely integrable [61]. This implies that all irre-ducible multi-particle scattering amplitudes must van-ish identically for a constant V p and V p ∝ | p | . Fur-thermore, the extended model of short-ranged interac-tion, V p ∝ p , corresponding to the real space poten-tial V ( x ) ∝ δ (cid:48)(cid:48) ( x ) , is also integrable. This is knownas Cheon-Shigehara model [62]. It is only due to theadditional logarithm ∝ p ln | p | in Eq. (9), that thereis partial non-cancellation between different terms inEq. (6) and the amplitude remains finite. (iii) In themodel of long-ranged Coulomb interaction the situationis more subtle. A priori this model is not known to beintegrable. Nevertheless, the amplitude in Eq. (6) van-ishes for pure logarithmic interaction V p ∝ ln | p | , sothat retaining an additional p ln | p | term in Eq. (10)is important to get a finite result.The triple electron scattering rate from Eq. (5) gen-erates the collision integral (Stosszahlansatz) of the cor-responding Boltzmann equationSt { n } = (cid:88) { p } , { σ } W [ n p (cid:48) (1 − n p ) n p (cid:48) (1 − n p ) n p (cid:48) (1 − n p ) − n p (1 − n p (cid:48) ) n p (1 − n p (cid:48) ) n p (1 − n p (cid:48) ] . (11)Here each pair of Fermi functions, n p (1 − n p (cid:48) ) , cap-tures statistical occupation probabilities, whereas thetwo terms of the collision integral correspond to incom-ing and outgoing processes. At thermal equilibriumthese terms nullify each other by virtue of the detailedbalance condition. At weak disequilibrium, one can lin-earize n p = f p + δn p in the external perturbation δn p around the equilibrium Fermi-Dirac distribution func-tion f p . The collision term can then be considered as alinear integral operator, acting on δn p = f p (1 − f p ) ψ ,and one can formulate the eigenvalue problem for thisoperator, St { ψ n } = ω n ψ n . The spectrum of eigenval-ues ω n may be discrete or continuous, and captures all the information about the decay of different distribu-tion function modes. As solving this problem exactlyfor triple collisions presents a daunting task [29, 40],we here follow a simpler more pragmatic approach.Setting, for instance, δn p = δ p ,p F + ε/v F describes aninjected quasiparticle with excess energy ε . Neglect-ing then secondary collisions, the Boltzmann equationreduces to the simple relaxation time approximation, ( ∂ t + τ − qp ) δn p = 0 , with solution δn p ∝ exp( − t/τ qp ) .It is natural to identify the corresponding timescale fordecay with the quasiparticle life-time τ − qp = − ∂ St { n } /∂n p , (12)which follows from Eq. (11) by only retaining the out-scattering contribution. Alternatively, one may projectthe collision operator (11) onto either momentum or en-ergy modes and thus infer the relaxation time of inter-est. This approach is parametrically correct, however,may miss numerical factors of order unity when com-pared to the exact solution of the eigenvalue problem.We will employ both approaches in the forthcomingsections. Owing to one-dimensionality of the problem, it isconvenient to think of particles of different chirality,namely right-movers (R) and left-movers (L). It can bereadily checked that strictly at zero-temperature quasi-particle relaxation is only possible if collisions involveboth, right- and left-moving particles since otherwiseconservation laws cannot be satisfied. For this reason,consider first a process of relaxation that involves tworight-moving particles, with initial momenta p , p , anda left-moving particle labeled by momentum p . Theoutgoing momenta after the collision, p (cid:48) i = p i + q i , willbe labeled by momenta transfer q i for each of the par-ticle i = 1 , , . In these notations, the momentumconservation becomes q + q + q = 0 , and the energyconservation, for a simple parabolic band, can be castin the form p q + p q + p q )+ q + q + q = 0 . Theseconditions set the phase-space constraints for collisions.For an initial state with p = p F + ε/v F , the quasi-particle life-time corresponding to an RRL-process isthen τ − qp = (cid:88) p p p (cid:48) p (cid:48) p (cid:48) W (1 − f p (cid:48) ) f p (1 − f p (cid:48) ) f p (1 − f p (cid:48) ) , (13)where we first focused on a simpler case of spinlesselectrons. At this point it is convenient to shift mo-menta of left- and right-movers from the respective lex Levchenko, Tobias Micklitz ЖЭТФFermi points, p , = p F + k , and p = − p F + k .In addition, it is sufficient to linearize the spec-trum in the distribution functions, approximating f ± p F + k → f ± k = ( e ± v F k/T + 1) − , but not in thescattering probability W . Indeed, an analysis of thekinematic constraints suggests that q ≈ − q and q ≈ ( q /p F )( k − k + q ) , implying that | q | (cid:28) | q , | .In other words, relaxation occurs in incremental stepsof momentum transfer q ∼ ε /v F p F from right-moversto left-movers. With these observations at hand, wenext need the corresponding three-particle scatteringamplitude. For the case of long-ranged Coulomb inter-action Eq. (10), one finds from Eq. (6) after a laboriousexpansion A ≈ p F w ) L ε F (cid:18) e κ (cid:19) (cid:20) −
34 ln (cid:18) p F w (cid:19)(cid:21) ln (cid:18) q p F | q | (cid:19) . (14)This result is obtained to the leading loga-rithmic accuracy using two small parameters | q | /p F ∼ | q | / | q | (cid:28) in the expansion. Withthe same level of accuracy the momentum and energyconservations in Eq. (5) can be simplified to δ P,P (cid:48) δ ( E − E (cid:48) ) ≈ v F δ ( q − [ q ( k − k )+ q ] /p F ) δq , − q . (15)These approximations enable one to complete all fivemomentum integrations. Two integrations are removedby delta functions fixing values of q and q in termsof k , and q . Furthermore, in the zero tempera-ture limit, T → , Fermi occupations become step-functions, f k → θ ( − k ) . The integral over k then be-comes elementary, contributing by a pure phase spacefactor (cid:80) k f − k (1 − f − k − q ) = ( L/ π ) | q | θ ( − q ) . Theproduct of Fermi factors, f k (1 − f k − q ) , simply limitsthe domain of k to the range k ∈ [ −| q | , , while theremaining − f k + q dictates that q < k , and we re-call that in this setting k = ε/v F . Putting everythingtogether the RRL-process gives the life-time τ − qp = c ε F g λ ( p F w )( ε/ε F ) , (16)where g = e / κ v F is the dimensionless parameter ofinteraction strength in the model, and we introducednotation for λ ( z ) = z ln(1 /z ) . The numerical coeffi-cient c = (15 − π ) / π is obtained with help of theintegral (cid:90) (cid:90) x q ( x, y ) ln [ x/q ( x, y )] dxdy = 15 − π , (17)where q ( x, y ) = 1 − x (1 − y ) . We here notice that thenumerical factor c in Eq. (16) differs from that cal- culated in Refs. [26, 28] as different properties of theinteraction potential were assumed. We see that finite decay rate emerges in forth orderof the interaction strength. We also notice that the at-tenuation is inversely proportional to the cube of mass, τ − qp ∝ ( m ∗ ) − , and vanishes as the limit m ∗ → ∞ istaken at fixed band velocity. This limit corresponds tothe situation considered by Dzyaloshinskiˇi and Larkin.The energy scaling of the decay rate, ∝ ε , is consistentwith expectations based on the Fermi liquid picturefor a process involving two particle-hole pairs. How-ever, this result is not universal. This becomes evidentfrom repeating the above calculation for the model ofscreened short-range interaction, i.e. using the poten-tial given by Eq. (9). Expanding the amplitude in Eq.(6) under the same conditions as above, one then findsinstead of Eq. (14) the amplitude A ≈ − p F d ) L ε F (cid:18) e κ (cid:19) ln (cid:18) p F d (cid:19) × (cid:20) q p F (cid:20) (cid:18) | q | p F (cid:19)(cid:21) − q q (cid:20) (cid:18) | q || q | (cid:19)(cid:21)(cid:21) . (18)The crucial difference here compared to Eq. (14)is the appearance of the additional small parameter | q | /p F ∼ | q | / | q | ∼ ε/ε F (cid:28) , which can be relatedto the fact that this particular model is nearly inte-grable. A close inspection of the amplitude in Eq. (6)reveals that each term individually diverges as /q atsmall characteristic momentum transfer. However, allexchange terms combined together remove the singu-larity and partially cancel out all the way to ∼ q ln q order. The rest of the calculation carries through inexactly the same way as in the previous example, andone finds the decay rate τ − qp = c ε F g λ ( p F d )( ε/ε F ) ln ( ε F /ε ) (19)with c = 2445 / π and λ ( z ) = z ln(1 /z ) . Thefour extra powers in the energy dependence, can betraced back to the different asymptotic form of the am-plitude in Eq. (18). This demonstrates high sensitivityof decay rates in 1D systems to details of the interpar-ticle interaction. The result captured by Eq. (19) is ofcourse perturbative. For generic nonintegrable modelswith short-ranged interaction, it can be generalized toarbitrary interaction strength. It can further be shownthat τ − qp ∝ ε remains valid, and the pre-factor can be In Appendix we sketch derivation of Eq. (16) from thebosonization framework of a mobile impurity scattering in Lut-tinger liquids. ЭТФ
Kinetic processes in Fermi-Luttinger liquids expressed in terms of the exact spectrum, see Ref. [31]for details.As should be anticipated from the discussion above,electron spin plays a crucial role in the transition ma-trix element for the three-particle process, and thussignificantly affect the quasiparticle decay rate. Indeed,in the spinless case antisymmetry of the electron wavefunction dictates that its orbital component should beodd and therefore relevant exchange amplitudes aresuppressed by Pauli exclusion. Mathematically, onesees this in a cancellation of various terms that leadto Eq. (18). In contrast, for spinful electrons, singularparts of the amplitude do not cancel. They are domi-nated by p F exchange-processes between branches, inwhich left-movers are scattered into right-movers [35].Even though the strength of p F exchange interactionis weaker than that at small momentum scattering, V p F (cid:28) V , for Coulomb interaction the relative re-duction is only logarithmic. The gain in the ampli-tude, on the other hand, is much more substantial andcontrolled by the large factor ∼ ε F /v F q (cid:29) . Thisstatement can be verified explicitly from Eq. (6) whereafter spin summation one finds for the square of theamplitude of the RRL process (cid:88) σ σ σ (cid:48) σ (cid:48) σ (cid:48) | A | = 3 V p F ( V − V p F ) L ε F (cid:20) q q + 4 p F q (cid:21) . (20)To obtain this result we approximated V p − p ± q i ≈ V and V p , − p ± q i ≈ V p F in all the relevant terms since p , − p ≈ p F and q i (cid:28) | p i | . Again, by repeatingmomentum integrations, the decay rate is found to beof the form τ − qp = c ε F g λ ( p F w )( ε/ε F ) ln ( ε F /ε ) , (21)with c = 45 / π and λ ( z ) = ln(1 /z ) . To be con-sistent with the approximations that lead to Eq. (20),the difference V − V p F should be understood as a weaklogarithmic factor (cid:39) (2 e / κ ) ln( ε F /ε ) for the Coulombinteraction potential. This was incorporated into Eq.(21). The singularity of the amplitude was compen-sated by phase space factors, and perhaps surprisinglythis restores essentially the Fermi liquid form of the de-cay rate at T = 0 . We note that up to model dependentpre-factors, the quadratic dependence of the relaxationrate on energy of spin- / particles given by Eq. (21) isconsistent with predictions of previous studies [35, 39].We proceed with a discussion of the effects ofthermal broadening on relaxation processes. In theFermi liquid picture one expects a simple crossoverat excitation energies of the order of temperature ε ∼ T . For 1D liquids this is not the case, as evenat T < ε there are intermediate regimes and relax-ation shows nontrivial temperature dependence. In-deed, at finite temperatures each collision results ina typical momentum transfer q i ∼ T /v F allowed bythermal smearing of states near the Fermi energy. AsRRL relaxation is controlled by momentum transferbetween the branches, one needs to compare phasespaces available to left movers. Since at zero temper-ature q ∼ ε /v F p F , one deduces from comparison to q ∼ T /v F the crossover scale ε T ∼ √ ε F T . Tech-nically, this argument also becomes evident observingthat (cid:80) k f − k (1 − f − k − q ) = ( L/ π ) q ( e v F q /T − − ,and reducing to LT / πv F as q → . These consider-ations suggest that Eqs. (16), (19), and (21) are validfor T (cid:28) ε /ε F . Above this threshold one finds τ − qp = c ε F g λ ( p F w )( ε/ε F ) ( T /ε F ) , (22)instead of Eq. (16) for the spinless Coulomb case. Sim-ilarly, τ − qp = c ε F g λ ( p F d )( ε/ε F ) ( T /ε F ) ln ( ε F /ε ) , (23)instead of Eq. (19) for the spinless screened case, andfinally τ − qp = c ε F g λ ( p F w )( T /ε F ) ln ( ε F /ε ) (24)instead of Eq. (21) for the spin- / Coulomb case. Theset of coefficients c , , can be determined from numer-ical integrations, however, their specific values are ofno particular significance here.At elevated temperatures the above mechanism forrelaxation competes with another process involvingonly particles of the same chirality. As indicated ear-lier, this RRR- (or equivalently LLL-) process is kine-matically possible only at finite energies. It followsfrom the same amplitude Eq. (6), but admits differentconditions on the involved momenta. In this process, ahigh-energy particle with excess energy ε can relax ontwo other comoving particles, which during the collisionare scattered in opposite directions in energy. Namely,one is drifting slightly upwards in energy, whereas theother float downwards, closer to the Fermi point. A de-tailed calculation in the spinless Coulomb model showsthat the corresponding relaxation rate is given by τ − qp = c g ( p F w ) ( T /εε F ) ln ( ε w /ε ) , (25)where ε w = v F /w . This rate exceeds that given inEq. (22), provided that temperature is higher than ∼ ε (cid:112) ε/ε F . In the case of screened Coulomb interac-tion, the same mechanism is more strongly suppressed τ − qp = c g ( p F d ) ( T /εε F ) ln ( ε d /ε ) ln ( ε/T ) , (26) lex Levchenko, Tobias Micklitz ЖЭТФ τ − qp T < T T < T < T T < T < ε Coulomb ε /ε F T ε /ε F T /εε F Screened ε /ε F T ε /ε F T /εε F Spin- / ε /ε F T T ε T /ε ε d Table 1.
Energy and temperature dependencies of quasi-particle relaxation rates (only the leading parametric be-havior is indicated and logarithmic terms are omitted forbrevity). First two rows summarize results for spinlesselectrons interacting via Coulomb and screened short-range interaction models, respectively, and the last rowgives the result for the spin- / fermions. The firsttwo columns describe processes involving particles ofboth chiralities (e.g. the RRL process), and the lastcolumn describes the relaxation of comoving particleswith only same chirality (e.g. the RRR process). Inall cases T ∼ ε /ε F , while T ∼ ε (cid:112) ε/ε F in theCoulomb model, T ∼ ε (cid:112) ε/ε F in the screened model,and T ∼ ε ( ε d /ε F ) (cid:112) ε d /ε in the spinful model. where ε d = v F /d . In fact, ∝ T is a generic propertyfor any non-integrable finite-range interaction modelwith a sufficient degree of analyticity at small momenta[32, 33]. Lastly, in the case of spin- / chiral electronsone estimates the decay rate to be of the form τ − qp = c g ( T ε T /ε ε d ) ln ( d/w ) . (27)In addition to relaxation of particles with the samechirality, thermal broadening allows for the relaxationof hot quasiholes, a process kinematically forbidden atzero temperature. The derivation of the correspond-ing decay rate τ − qh proceeds in close analogy to thatfor the RRL process. Crucial modifications are (i) thesign of q , (ii) a smaller phase space volume, now sup-pressed by an additional factor ∼ T / ( ε /ε F ) , and (iii)that it takes ∼ ( ε/ε T ) steps to relax the excess energy.As a result, the quasihole relaxation rate e.g. for thespin- / model, τ − qh = c ε F g λ ( p F w )( T /ε ) ln ( ε F /ε ) , (28)is by a factor ( ε T /ε ) smaller than τ − qh defined in Eq.(21) when taken at the same energy. This pronouncedasymmetry in the relaxation rates of electron-like andhole-like excitations is a direct consequence of the 1Dkinematics of three-particle scattering with nonlinearspectrum. This feature marks a sharp distinction be-tween the quantum 1D Fermi-Luttinger liquids andhigher dimensional Fermi liquid counterparts.We summarize in Table 1 the discussed quasiparti-cle relaxation rates in the different regimes and models. Another common technique in kinetic theory ap-plied to the determination of relaxation rates is toproject the collision integral onto specific modes of in-terest, to infer their corresponding decay times. Forinstance, in the context of the present problem, onecan look at the thermal imbalance relaxation betweenleft- and right-movers. This amounts to projecting thecollision term onto the energy mode of the distributionfunction n p , which is even in momentum.To see the practical implementation of this method,consider a situation in which right-movers are hotterthan left-movers. The goal is then to derive an equa-tion which describes the relaxation of the difference intemperatures ∆ T = T R − T L of left- and right-movingelectrons. It should be noted that the physical settingwith imbalanced temperature is justified in 1D: whilethree-particle collisions generate both right- and left-moving particle-hole pairs the intrabranch relaxationinduced by these processes is faster, while interbranchis slow.We start from the Boltzmann equation, multiplyboth sides by ε p − ε F , and sum over p > (cid:88) p > ( ε p − ε F ) ∂ t n p = (cid:88) p > ( ε p − ε F ) St { n } , (29)where, as above, momentum p is that of a right-moving particle. We then assume n p to be ofFermi-Dirac form with nonequilibrium temperature T R = T + ∆ T of right-moving excitations, and lin-earize above equation in the left-hand-side with respectto ∆ T , ∂ t n p = ∂ T n p ∂ t ∆ T = ( ε p − ε F ) ∂ t ∆ T T cosh ( ε p − ε F T ) . (30)When computing the integral over p it is convenientto shift momentum to the respective Fermi point, p = p F + k . Linearizing further the dispersion re-lation in k , ε p − ε F ≈ v F k , one may use that theintegral is peaked at p F and rapidly converging. Not-ing that (cid:82) + ∞−∞ z dz/ cosh ( z ) = π / , one readily finds (cid:88) p > ( ε p − ε F ) ∂ t n p = πLT v F ∂ t ∆ T. (31)The next step is to also linearize the right-hand-side of Eq. (29) in ∆ T . To accomplish this task weparametrize n p = f p + f p (1 − f p ) ψ p , which allows to con-veniently take advantage of the detailed balance con-dition in the collision integral St { n } . For the thermal ЭТФ
Kinetic processes in Fermi-Luttinger liquids imbalance ψ p = ( ε p − ε F )∆ T /T , and one finds uponexpansion in ∆ T (cid:88) p > ( ε p − ε F ) St { n } = − ∆ TT (cid:88) { k,q,σ } ( v F k )( v F q ) W . (32)Here W = W f k (1 − f k + q ) f k (1 − f k + q ) f − k (1 − f − k − q ) , (33)and at intermediate steps we made use of the en-ergy conservation implicit in W , and approximated ε p − ε F ≈ v F k and ε p (cid:48) − ε p ≈ − v F q . It is nowevident that Eq. (29) can be cast in form of the usualrelaxation time approximation, ∂ t ∆ T = − ∆ T /τ th , (34)where we introduced the corresponding thermalizationtime. For the kinematics of the RRL process, the latterevaluates to τ − th = c ε F g λ ( p F w )( T /ε F ) ln ( ε F /T ) . (35)In a similar fashion one can find the relaxation rate forthe odd part of the imbalanced distribution. For thispurposes one may consider a boosted frame of refer-ence, ε p − pu , and derive the relaxation equation for u by projecting the collision integral onto the momentummode. Kinematics of the respective collision is differentthough, and will be considered in the next section.To get an idea of the order of magnitude of thedifferent timescales, it is instructive to consider the fol-lowing estimates for GaAs quantum wires using exper-imental parameters of Ref. [52]. For v F ∼ × m/sand κ ∼ , the interaction parameter is just within theapplicability criterion of the perturbative expressions g ∼ . For the typical electron density we use p F ∼ m − , w ∼ nm, and ε F ∼ meV. Then for ε ∼ ε F / ,which is a typical excess energy of injected particles intunneling experiments, and T ∼ . K one is securelyin the regime T (cid:28) ε /ε F . For this set of parameters τ − qp ∼ s − , τ − qh ∼ s − , and τ − th ∼ s − . Relaxation processes of low-energy excitations lead-ing to the decay of quasiparticles near the Fermi energydo not change the numbers of right- and left-movingparticles. Thus they are chirality conserving. It turnsout that it is also possible to have backscattering pro-cesses. The kinematics of these collisions involves statesdeep in the Fermi sea, and for this reason it is useful to consider the mobility of holes at the bottom of theband. These processes are commonly considered fromthe perspective of mobile impurities in a Luttinger liq-uid [63–69]. Here we will continue using the kineticequation approach for their description. The idea isthen to single out hole states at the bottom of the bandwith small momenta, and to derive an effective kineticequation capturing their dynamics and allowing the cal-culation of corresponding backscattering rates [70–72].For this purpose, let p and p (cid:48) be momenta near theband bottom, p and p (cid:48) lie near the right Fermi point (+ p F ) , and p and p (cid:48) be taken near the left Fermi point ( − p F ) . As before, the unprimed momenta correspondto incoming states whereas primed ones are associatedwith outgoing states. With these conventions, we intro-duce the hole distribution function, h p = 1 − n p , andthe collision integral for holes, St { h p } = − St { n p } .Starting from Eq. (11), the latter can be cast in theformSt { h p } = (cid:88) p (cid:48) (cid:2) P ( p , p (cid:48) ) h p (cid:48) − P ( p (cid:48) , p ) h p (cid:3) , (36)where P ( p , p (cid:48) ) = 12 (cid:88) { σ } (cid:88) p p p (cid:48) p (cid:48) W f p (1 − f p (cid:48) ) f p (1 − f p (cid:48) ) (37)is the rate for a transition in which a hole scatters fromsome state p (cid:48) into p , while P ( p (cid:48) , p ) denotes the ratefor the inverse process. In the above sums, all momentahave been restricted to the discussed ranges, which ex-plains the combinatorial overall factor of . Since both p and p (cid:48) lie near the bottom of the band, the distri-bution functions h p and h p (cid:48) are exponentially small ∝ e − ε F /T due to Pauli exclusion, and so is the collisionintegral of holes St { h p } . It is therefore unnecessaryto account for additional exponentially small contribu-tions in the transition rates P ( p , p (cid:48) ) and P ( p (cid:48) , p ) ,and this is why we replaced f p (cid:39) and f p (cid:48) (cid:39) inboth. As in the case of the forward scattering pro-cess, the typical scale for momentum change of all threeparticles in a hole backscattering is set by tempera-ture, q i = p (cid:48) i − p i ∼ T /v F . At the same time, thetypical momentum of a hole is p ∼ √ m ∗ T so that q /p ∼ (cid:112) T /ε F (cid:28) . This means that the net momen-tum change in each scattering event is small, and holeseffectively drift through the bottom of the band. Thusrelaxation occurs in multiple steps and the underlyingdynamics is diffusion in momentum space. Under theseconditions, the mobile impurity falls into the universalclass of problems described by Fokker-Planck equation[73]. The collision integral in Eq. (36) can then be lex Levchenko, Tobias Micklitz ЖЭТФsimplified by expanding in the small momentum step q (cid:28) p , and thus maps to the differential operatorSt { h p } ≈ − ∂ p [ A ( p ) h p ] + 12 ∂ p [ B ( p ) h p ] . (38)Here we introduced A ( p ) = − (cid:88) q q P q ( p ) , B ( p ) = (cid:88) q q P q ( p ) , (39)and used the short-hand notation P q ( p ) = P ( p (cid:48) , p ) .The diffusion coefficient in momentum space B ( p ) is afunction of the hole-momentum p varying on a scaleset by p F . For holes at the bottom of the band, onemay thus approximate B ( p ) by its value at p = 0 ,in the following simply denoted by B without argu-ment. Furthermore, the drift coefficient A ( p ) is read-ily obtained from noting that the collision integral (38)has to vanish for hole distributions of an equilibriumBoltzmann form. This condition leads to the relation A ( p ) = p B / m ∗ T .The rest of the calculation depends on the structureof the amplitude for a given kinematics of the three-particle process. In calculating A from Eq. (6) for themomentum configuration under consideration, and upto small corrections in T /ε F (cid:28) , it is sufficient toapproximate p ≈ , p ≈ + p F and p ≈ − p F . Mo-mentum and energy conservations provide additionalrestrictions on the transferred momenta, enforcing that q ≈ q ≈ − q / , again up to small corrections in T /ε F (cid:28) . As a result, the amplitude A can beparametrized only by a single momentum q . Expand-ing Eq. (6) and summing over spins one then finds (cid:88) { σ } | A | = 6 ε F L V p F ( V p F − V p F ) p F q . (40)The singularity of A at small momenta is cancelled inthe spinless case. The underlying reason is exactly thesame as discussed above in the context of quasiparti-cle thermalization processes. Specifically, for the long-range interaction model with Eq. (10) one finds A ≈ ε F L (cid:18) e κ (cid:19) λ ( p F w ) ln (cid:18) p F | q | (cid:19) , (41)whereas for the screened model A ≈ − ε F L (cid:18) e κ (cid:19) λ ( p F d ) . (42)In order to perform the remaining momentum integra-tions implicit in the definition of B , one can approx-imate delta functions in the scattering probability by δ P,P (cid:48) δ ( E − E (cid:48) ) ≈ v F δ ( q − q ) δ q , − q / . This removestwo integrations out of five, and gives B = 12 Lv F (cid:88) q k k q (cid:88) { σ } | A | f k − q (1 − f k ) f k + q (1 − f k ) , (43)where we shifted momenta p , to the respective Fermipoints, ± p F + k , , and linearized the dispersion rela-tion in all Fermi occupation functions. Finally, usingthe tabulated integral (cid:88) k f k + q (1 − f k ) = L π qb q , b q = 1 e v F q/T − , (44)where b q is the equilibrium Bose distribution, we arriveat the general expression B = 6 πv F (cid:18) L π (cid:19) (cid:88) q q (cid:88) { σ } | A | b q / (cid:0) b q / (cid:1) . (45)A notable feature of this expression is that it is entirelyexpressed in terms of bosonic modes. In essence, this isa manifestation of bosonization at the level of fermionickinetic theory, as the occupation of an electron-holepair near one of the Fermi points integrated over thecenter of mass momentum is equivalent to a collectiveboson emitted/absorbed in a course of hole diffusion.It will be shown in the subsequent section that struc-turally the same expression for B can be obtained froma purely bosonic formulation of the problem. Finally,inserting Eq. (40) into Eq. (45) one finds the momen-tum space diffusion coefficient of spin- / holes B = 768 ln (2) π g λ ( p F w ) (cid:18) Tε F (cid:19) p F ε F . (46)The corresponding backscattering relaxation rate canbe found from Einstein relation adopted to diffusion inmomentum space, ∆ p = B τ dh . The notation τ dh ismeant to emphasize kinetics of a deep hole as opposedto earlier notation τ qh describing quasiholes near Fermienergy. Thus for ∆ p (cid:39) m ∗ T the result is (omittingnumerical factosr for brevity) τ − dh (cid:39) g λ ( p F w )( T /ε F ) . (47)Finally we recall that the mobility of particles ς is re-lated to the diffusion constant by the simple kineticformula ς = T / B , and therefore ς ∝ /T .The result is different in the spinless case. FromEqs. (41), (42) and (45) one finds B ∝ T in both mod-els, modulo a logarithmic factor ln T in the Coulombcase, and thus τ − dh ∝ T and ς ∝ /T . The resultsdiscussed in this section are again perturbative in the ЭТФ
Kinetic processes in Fermi-Luttinger liquids interaction. The power laws in the temperature depen-dence of relaxation rates are, however, generic and alsoapply to the strongly interacting regime, as we furtherelaborate below, see also Refs. [67, 68].
Apart from the purely electronic mechanisms ofrelaxation electrons may scatter on phonons, disor-der, and sample imperfections thus relaxing their en-ergy and momentum. At extremely low temperaturesphonons are not expected to be efficient at cooling theelectronic sub-system. On the other hand, electron-phonon scattering has no such severe phase space re-strictions like the three-particle collisions consideredabove. It is thus instructive to estimate the tem-perature dependence for the corresponding relaxationrate. Unlike the previous studies of electron-phononrelaxation in multichannel quantum wires [74, 75], andphonon-induced backscattering relaxation [76, 77], wefocus on the complementary effect of soft collisions ina single-channel geometry of strictly 1D electrons and3D phonons.The coupling of electrons and phonons is describedby the collision integral [78]St { n p , N q } = (cid:88) p (cid:48) q W − [ n p (cid:48) (1 − n p ) N q − n p (1 − n p (cid:48) )(1 + N q )]+ (cid:88) p (cid:48) q W + [ n p (cid:48) (1 − n p )(1 + N q ) − n p (1 − n p (cid:48) ) N q ] , (48)where the scattering rate W ± ( p, p (cid:48) , q ) = (2 π ) | A ( q ) | δ ( ε p − ε p (cid:48) ± ω q ) δ p = p (cid:48) ± q x (49)describes phonon emission and absorption processeswith an amplitude A ( q ) = (cid:113) (cid:37) V ω q ( D | q | + i Λ) . Herewe took into account that at the level of the leadingBorn approximation, the probabilities of scattering fordirect and reverse processes are the same. In the am-plitude we include both deformation ( D ) and piezoelec-tric (Λ) couplings, (cid:37) is the mass density, q x the phononwave-vector along the wire, and V is the system vol-ume. For simplicity we assume only a single acousticbranch ω q = s | q | , with sound velocity s .For equilibrium Fermi and Bose distribution func-tions of electrons and phonons respectively, n p → f p and N q → b q , the collision integral in Eq. (48) van-ishes due to detailed balance condition. As in the aboveexample of the distribution imbalance relaxation, we then assume that electrons are hot. That is, at an ex-cess temperature T + ∆ T with respect to the temper-ature T of lattice phonons. Electron-phonon collisionstend to relax ∆ T , and the corresponding rate for relax-ation can be found by projecting the collision integralonto the energy mode, (cid:80) p (cid:15) p ˙ n p = − (cid:80) p (cid:15) p St { n p , N q } ,with (cid:15) p = ε p − ε F . To linear order in ∆ T one findsfrom the phonon emission processes of hot electrons, ∂ t ∆ T = − ∆ T /τ ep , where τ − = − v F πT L (cid:88) pp (cid:48) q W (cid:15) p ω q f ε p (1 − f ε p )( f ε p + ω q + b ω q ) . (50)Upon completion of the remaining momentum integra-tions, we then find to leading order in Tτ − = 9 ζ (3)8 π T (Λ /s v F (cid:37) ) . (51)The scattering rate due to the deformation potentialis parametrically weaker, scaling as τ − ∝ T . Thebackscattering mechanism results in an activated tem-perature dependence ∝ e − T A /T with T A = 2 sp F . Itis straightforward to generalize Eq. (51) to the casewhen electronic relaxation occurs via several acousticbranches. Notice also that the piezoelectric potentialmay have complicated angular dependence in case ofwires oriented arbitrarily with respect to the crystallo-graphical axis of the sample. A proper angular aver-aging would change then numerical factors in Eq. (51)where we took the simplest geometry. Luttinger liquideffects lead to renormalization of the linear- T behav-ior and transform it into a power-law with interactiondependent exponent ∝ T K , where K = v F /u is the ra-tio of Fermi and plasmon velocities. In the TL model u = v F (cid:112) V /πv F . The applicability of the Born approximation, usedto construct the quantum amplitude for triple particleprocesses captured by Eq. (6), requires that incom-ing spin- / quasiparticles have sufficiently high energycompared to the typical scale of interparticle interac-tion ε (cid:29) m ∗ v F V .In the generic interacting environment of a 1Dquantum fluid, quasiparticle excitations break downinto spin and charge modes. At the level of linear Lut-tinger liquid theory, spin-charge separation is an ex-act property of the model [14]. At weak coupling, thesplitting between velocities of collective spin ( v σ ) andcharge ( v ρ ) density waves is related to the forward scat-tering component of the interaction v ρ − v σ ∼ V (recall lex Levchenko, Tobias Micklitz ЖЭТФthat for repulsive interactions v ρ > v σ ). Assuming thenthermal excitations with ε ∼ T , the Born condition canbe equivalently formulated as T / ( m ∗ v F ) (cid:29) v ρ − v σ .In other words, for fermionic quasiparticles to preservetheir integrity the excitation energy (or temperature)should be bigger than the energy scale for spin-chargeseparation.The interplay of spectrum nonlinearities and in-teractions leads to spin-charge coupling [79, 80]. Al-though irrelevant in the renormalization group sense,the newly emerging higher order operators capture theattenuation of quasiparticles. The kinetic propertiesof 1D quantum liquids with spin-charge coupling arenot fully understood. There are basically two possibleapproaches one may pursue. The first is to refermion-ize the nonlinear bosonic theory to obtain an effectivedescription in terms of dressed quasiparticles: holonsand spinons. Holon relaxation was considered in Refs.[36, 37] based on non-Abelian bosonization [81]. Theadvantage of this complex theory is that, in principle,it allows to go beyond the weakly interacting limit forspinful fermions. Alternatively, one may choose to con-tinue working in the bosonic language. In the limitof weak backscattering one can then account for spin-charge interaction perturbatively in the basis of well-defined spin and charge modes. This second procedureis limited to weak interactions V p F (cid:28) V (cid:28) v F . Tocomplement previous studies, we follow in this sectionthe second path. In part this will enable us to ex-plore the fermion-boson duality. We delegate technicaldetails of bosonization to the Appendix and elucidatehere the impact of spin-charge scattering on variousdecay rates.The lowest order nonlinearity, compatible withSU(2) symmetry of the problem is cubic. It containsone charge and two spin operators. Treating this termin a perturbative Golden rule expansion generates acollision kernel that describes the decay of a plasmoninto two spin modes ρ → σσ . It readsSt { N ρ , N σ } = − (cid:88) q q W (cid:2) N ρq (1 + N σq )(1 + N σq ) − (1 + N ρq ) N σq N σq (cid:3) , (52)where N ρ/σ are the bosonic occupations of charge ( ρ )and spin ( σ ) excitations. The scattering probability W = 2 π | A | δ q = q + q δ ( ω ρq − ω σq − ω σq ) (53)contains an amplitude scaling cubically with mo-menta of the bosons | A | = ( π / L ) | q || q || q | Γ ρσσ .The perturbative result for the coupling constant is Γ ρσσ = V (cid:48) p F / √ π , where the prime denotes thederivative with respect to p F . Note that it thus van-ishes for the integrable case of constant interaction.At smallest momenta the dispersion relations are lin-ear ω ρ/σ = v ρ/σ | q | . The kinematics of this processuniquely fixes momenta in the final state. Indeed, forconcreteness let q > , then q = q ( v ρ + v σ ) / v σ and q = − q ( v ρ − v σ ) / v σ , which means that spin waves arecounterpropagating. From dimensional analysis it be-comes apparent that St { N ρ , N σ } defines the decay rateof a plasmon, and one can introduce the characteristicrate τ − ρ = (cid:80) q q W (cid:39) q ( V (cid:48) p F ) ( v ρ − v σ ) /v σ . For thesake of an estimate, one may now take V (cid:48) p F ∼ V p F /p F and replace v ρ/σ ∼ v F , except in their difference where v ρ − v σ ∼ V , and finds the life-time τ − ρ ∼ ε F ( V /v F )( V p F /v F ) ( q/p F ) . (54)Notice the nonanalytic dependence of interaction ∝ V .For thermal plasmons the relaxation rate can be calcu-lated from Eq. (52) by a projection onto an energymode. We observe that as | q | (cid:28) q the relaxation oc-curs by small energy transfer from right-movers to left-movers (or vise-versa) so that interbrach processes areslow. Assuming that right-moving excitations are hot-ter by ∆ T , and in complete analogy to the fermioniccase, we find (cid:88) q> ω ρq ∂ t N ρq = (cid:88) q> ω ρq St { N ρ , N σ } . (55)The left-hand-side is straightforwardly evaluated, not-ing that ∂ t N ρq = ∂ T N ρq ∂ t ∆ T = ω ρq ∂ t ∆ T T sinh ( ω ρq / T ) , (56)which after momentum integration gives a factor of ( πLT / v ρ ) ∂ t ∆ T . The right-hand-side can be linearizedwith the usual substitution N q = b q + b q (1 + b q ) φ q ,where φ q = ω q ∆ T /T for the case of a thermal imbal-ance. After some algebra one finds (cid:88) q> ω ρq St { N ρ , N σ } = − ∆ T (cid:88) qq q ( ω ρq /T ) W (1 + b ρq ) b σq b σq , (57)where we repeatedly used energy conservation and thedetailed balance condition. Performing the final inte-grations, we then arrive at τ − ρ = 3 π
16 Γ ρσσ ( T /v σ ) F ( v σ /v ρ ) , (58) ЭТФ
Kinetic processes in Fermi-Luttinger liquids where the dimensionless function reads F ( x ) = x (1 − x ) (cid:90) ∞ z (1 + b z ) b z + b z − dz (59)with b z = ( e z − − and z ± = z (1 ± x ) / . One canreadily check that F → π / in the limit x → .The same scattering process can be alternativelyviewed as a mutual spin-charge friction. Physically,this is analogous to the electron-phonon drag effect,typically studied in the context of thermoelectricity, orCoulomb drag in double-layers [82] and spin Coulombdrag [83]. In each of these examples momentum trans-fer between interactively coupled systems leads to drag-ging of one sub-system by the flow of the other. Forinstance, in the context of spin physics in Luttinger liq-uids, generation of spin current is possible by Coulombdrag [84]. To estimate the spin-charge drag rate, onecan consider a boosted frame of reference for spin andcharge excitations with mismatched boost velocities u ρ/σ . The scattering leads to momentum exchange be-tween spins and charge and, as a result, to relaxation ∂ t u ρ = − ( u ρ − u σ ) /τ ρσ . To capture this effect, welinearize the collision integral for N ρ/σ ( ω q − uq ) withrespect to u , for both spin and charge occupations, andthen project onto the momentum mode to calculate therate of momentum loss by (e.g.) charge modes ∂ t P ρ = (cid:88) q q St { N ρ , N σ } = − (∆ u/T ) (cid:88) qq q q W (1 + b ρq ) b σq b σq . (60)When we compare this to ∂ t P ρ = ( πLT / v ρ ) ∂ t u ρ , wefind that thus defined drag relaxation rate τ − ρσ coin-cides with Eq. (58) up to a constant factor. It is per-haps useful to note that τ − ρσ ∝ T is consistent withthe expectation that Coulomb drag transresistivity be-tween double quantum wires due to interwire momen-tum transfer from spin-charge coupling at zero mag-netic field scales as ρ drag ∝ T [37]. Indeed, this rate isaccompanied by two thermal phase space factors ∼ T per wire, thus leading to T . In the drag problem,the factor q results from the width of the dynamiccharge structure factor and the underlying scatteringthat gives rise to its ∝ q width is precisely the decayof a charge boson into two spin bosons.The next in complexity is a quartic nonlinearity inspin-charge coupling which leads to two-boson scatter-ing ρσ → ρσ . In particular, we consider backscatteringof spin excitations on plasmons. Such scattering pro-cesses correspond to the diffusion of spin excitationsnear the spectral edge, and the goal is to calculate the corresponding diffusion constant. As alluded to earlier,the discussion parallels the previous calculation of thebackscattering of a deep hole in the fermionic language.The corresponding collision integral readsSt { N ρ , N σ } = − (cid:88) q q (cid:48) q (cid:48) W (cid:104) N σq (1 + N σq (cid:48) ) N ρq (1 + N ρq (cid:48) ) − N σq (cid:48) (1 + N σq ) N ρq (cid:48) (1 + N ρq ) (cid:105) . (61)The scattering rate for this process is given by W = 2 π | A | δ Q,Q (cid:48) δ ( E − E (cid:48) ) (62)with the amplitude | A | = (Γ ρσ / L ) | q q (cid:48) q q (cid:48) | , wherethe coupling constant at the perturbative level reads Γ ρσ = V (cid:48)(cid:48) p F . The notations for momentum and energyconservation here are Q = q + q and E = ω σq + ω ρq .Let momenta q and q (cid:48) correspond to the initial and fi-nal states of the spin excitation near the spectral edge.Kinematically each momentum is of the order of theFermi momentum, q ∼ q (cid:48) ∼ p F , while their difference, q (cid:48) − q ∼ T /v F , is small. This corresponds to a smallmomentum change in each collision, which is accom-panied by the excitation of plasmons at low momenta q ∼ q (cid:48) ∼ T /v F . For this reason the low-energy de-scription based on Eq. (61) is sufficient to capture thisphysics. Under the specified conditions and in com-plete analogy with the fermionic case, we can convertthe collision integral into a Fokker-Planck differentialoperator, thus describing the diffusion of spins. In-deed, for momenta q ∼ p F the occupation is small, N σq ∝ e − ε σ /T (cid:28) , and correspondingly N σ ≈ ,where ε σ is the band width of spin excitations. The lat-ter is parametrically of the order of the spin exchangecoupling. Viewing Eq. (61) as the collision integral forspins, we thus write St { N σq } = (cid:88) q (cid:48) [ P ( q , q (cid:48) ) N σq (cid:48) − P ( q (cid:48) , q ) N σq ] , (63)where P ( q (cid:48) , q ) = (cid:88) q q (cid:48) W N ρq (cid:48) (1 + N ρq ) , (64)is the transition rate for spin scattering processes.More specifically, it describes a collision with mo-mentum transfer δq , in which a spin is scattered outof the initial state q . It can thus be rewritten as P ( q (cid:48) , q ) = P δq ( q ) , and following the same prescrip-tion, the transition rate for the inverse process reads lex Levchenko, Tobias Micklitz ЖЭТФ P ( q , q (cid:48) ) = P − δq ( q + δq ) . Performing then a small-momentum expansion, P ( q , q (cid:48) ) N σq (cid:48) ≈ P − δq ( q ) N σq + δq∂ q [ P − δq ( q ) N σq ] + δq ∂ q [ P − δq ( q ) N σq ] , (65)the collision integral of spin excitations takes the sim-plified form St { N σq } = − ∂ q [ A ρσ ( q ) N σq ] + 12 ∂ q [ B ρσ ( q ) N σq ] , (66)where A ρσ = − (cid:88) δq δq P δq ( q ) , B ρσ = (cid:88) δq δq P δq ( q ) . (67)At this stage we focus on the derivation of P δq ( q ) . Themomentum conservation implicit in W removes the q (cid:48) integration. We then notice that distribution functionslimit the typical momentum transfer and momenta ofplasmons to q ∼ δq ∼ T /v F . At the same time,the typical momentum of spins at the spectral edge is q ∼ p F and it is sufficient to calculate P δq ( p F ) . Withthese observations at hand, we can now approximateenergy conservation by δ ( E − E (cid:48) ) ≈ v ρ δ ( q − δq/ .This removes the q integral, and we thus arrive at P δq = V ρσ Lv ρ ( δq/p F ) sinh ( v ρ δq/ T ) , (68)with the notation V ρσ = p F Γ ρσ . Finally, this definesthe diffusion coefficient of spins in momentum spaceassociated to ρσ → ρσ scattering channel B ρσ = π
30 ( V ρσ /v ρ ) ( T /p F v ρ ) p F v ρ . (69)Exactly the same result for the temperature depen-dence of the momentum space diffusion constant per-tains to scattering in the charge channel, proceedingvia the ρρ → ρρ scattering process. With the replace-ment of the respective coupling constant one concludesthat B ρρ ∝ T .In addition to spin-charge scattering, nonlinearitiesalso allow for spin-spin scattering. Importantly for themomentum space diffusion, scattering processes withspin-flips are enhanced. They are thus described by adifferent scaling of the probability with momentum ascompared to Eq. (68). That is, P δq = V σσ π Lv σ ( v σ δq/ T ) , (70)and this crucial detail is technically speaking tracedback to the non-commutativity of spin operators when calculating the corresponding amplitude. The impor-tance of spin flips is also apparent at the level offermions. Indeed, the ratio of scattering rates betweenspinless and spinful cases has exactly the same param-eter ( q/p F ) (cid:28) as the ratio between probabilities inEqs. (68) and (70). The resulting diffusion constant inthe spin-spin channel is then B σσ = 4 π
15 ( V σσ /v σ ) ( T /p F v σ ) p F v σ . (71)A microscopic calculation of the respective couplingconstants for the different scattering channels is a chal-lenging task. Known approaches include weak couplingresults obtained via mobile impurity model [69], resultsfor Kondo polarons [66], and calculations in the stronginteraction limit within the non-Abelian bosonizationframework [37], as well as a model departing from theWigner crystal limit [85].Two-boson processes also contribute to the thermal-ization rates [33, 86, 87]. For charge excitations this re-sults in a subleading correction to Eq. (58). In thespin sector the situation is, however, different sinceat the cubic level of nonlinearities spins are kinemati-cally forbidden to scatter. In both cases nonlinearity ofthe bosonic spectrum plays an important role to openphase space for such collisions. In order to general-ize the present model, consider first the charge sectorand assume a weakly anharmonic dispersion of plas-mons, ω ρq ≈ v ρ | q | (1 − ( (cid:96)q ) ) . Assume now that a right-moving boson with momentum q (cid:38) T /v ρ is injectedinto the Luttinger liquid. For this setting the collisionterm from Eq. (61), with replacement N σ → N ρ , dic-tates that the dominant process limiting the lifetimeof the injected boson is due to scattering with inter-branch momentum transfer. Indeed, for q , q , q (cid:48) > momentum conservation implies that q (cid:48) is order q ,since energy conservation fixes q (cid:48) ≈ − (cid:96) q q q (cid:48) . Curi-ously, even though finite dispersion curvature parame-ter (cid:96) is crucial to resolve the kinematic constraints itdrops out from the corresponding rate provided that q (cid:28) (cid:112) T /v ρ (cid:96) . In this regime v ρ | q (cid:48) | (cid:28) T , implyingthat N ρq (cid:48) ≈ T /ω ρq (cid:48) and q (cid:48) cancels out from W . Thedecay rate then scales parametrically as τ − ρ ∝ T q ,which is applicable as long as T /v ρ (cid:46) q (cid:28) (cid:112) T /v ρ (cid:96) .For thermal plasmons, this rate can be estimatedmore accurately by projecting the collision integralonto the energy mode. Assuming that the boson withmomentum q is more energetic, namely hotter by atemperature difference ∆ T , one finds upon repeating ЭТФ
Kinetic processes in Fermi-Luttinger liquids the steps from the previous similar calculations τ − ρ = 6 v ρ πLT (cid:88) q q q (cid:48) q (cid:48) ω q ω q T W N ρq N ρq (1 + N ρq (cid:48) )(1 + N ρq (cid:48) ) . (72)For the kinematics of the process specified above, onesum is removed by momentum conservation setting q (cid:48) = ( q + q ) . Energy conservation removes anotherintegral, setting q (cid:48) = − (cid:96) q q ( q + q ) . The remainingintegrals can, after rescaling of momentum variables inunits of temperature, be brought to a dimensionlessdouble-integral. This results in τ − ρ = 3 c (4 π ) ( V ρρ /v ρ ) T ( T /p F v ρ ) , (73)where the coefficient c = (cid:82) ∞ x y ( x + y ) e x + y dxdy ( e x − e y − e x + y − and V ρρ = p F Γ ρρ .Two-spin scattering can be analyzed in the sameway, starting out from Eq. (61) by changing N ρ → N σ .The crucial difference is in the momentum dependenceof the scattering rate, which is enhanced by spin-flipprocesses. The resulting spin wave thermalization ratedue to two-boson scattering processes reads τ − σ ∼ ( V σσ /v σ ) T ( T /p F v σ ) . (74)This final estimate exhausts all possible scattering pro-cesses emerging from the quartic corrections to the lin-ear Luttinger liquid model.
3. TRANSPORT PROPERTIES
Thermalization and equilibration processes bearconsequences on the temperature dependence of kineticcoefficients in Fermi-Luttinger liquids. Before turningto their discussion we open by briefly recapitulating theelectrical and thermal transport properties predictedby linear Luttinger liquid theory. For a more expandedreview see e.g. Refs. [15, 88].For a clean single-mode quantum wire, adiabat-ically connected to Fermi liquid leads, linear Lut-tinger liquid theory predicts that the conductance G remains unrenormalized by interactions. At zero tem-perature the conductance thus takes the same valueas for noninteracting electrons G = e /π [89–92]. Insuch a two-terminal setup, finite resistance results onlyfrom scattering of electrons from the boundary re-gions, where the wide Fermi liquid reservoirs are con-nected to the narrow quantum wire. This result holdsalso for smoothly inhomogeneous wires as long as thecorrelation radius ξ of the disorder potential is suffi-ciently large and electron backscattering processes are exponentially suppressed in e − ξp F (cid:28) . As we dis-cuss below, equilibration processes in nonlinear Fermi-Luttinger liquids lead to finite temperature correctionsto the quantized conductance. Since the interaction-induced equilibration occurs via backscattering of holesnear the bottom of the band, these corrections areof the activated form δ G ∝ e − ε F /T in short wires[59, 60, 72]. For sufficiently long wires, they becomehowever more pronounced scaling then as δ G ∝ T inboth, weakly and strongly interacting regimes [70, 93].The physical picture of heat transport in linearLuttinger-liquids is different. Electrons entering thewire from one of the leads break up into plasmons,which then propagate along the wire as in a waveguide,carrying the energy current. In contrast to electrons,plasmons experience backscattering at the ends of thewire due to mismatch of the plasmon velocity in thewire and the Fermi velocity in the leads. This leads inessence to an analog of the Kapitza boundary thermalresistance.Multiple plasmon back reflections from both endsof the wire give further rise to interference like in aFabry-Pérot interferometer. Transmission coefficientsfor plasmons through the wire are thus described bya Fresnel type formula, and the thermal conductancereads K = 2 g K / ( g + 1) [94, 95], where K = πT / isthe thermal conductance quantum. As a consequence,the Lorentz ratio L = K /T G in a linear Luttinger liquidtakes a non-universal and interaction dependent value L / L = 2 g/ ( g + 1) < , with L = π / e . This alsoimplies violation of the Wiedemann-Franz law.Going beyond the linear Luttinger liquid descrip-tion, thermalization processes enable energy exchangebetween right- and left-moving excitations. This re-sults in temperature dependent corrections to the ther-mal conductance, δ K ∝ T [96], for wires of inter-mediate length for which backscattering of deep holesis still negligible. For long wires, equilibration pro-cesses prevail and suppress thermal conductance as K ∝ /L , now vanishing inversely proportional withthe wire length. The thermal conductivity, however,remains well defined and exhibits a complex tempera-ture dependence [97–99].We next present the above predictions for Fermi-Luttinger liquids in a more detail fashion, and discussother related transport properties. Consider a quantum wire of length L adiabaticallyconnected to leads of non-interacting electrons, and bi-ased by a finite voltage V . As particles enter the wire, lex Levchenko, Tobias Micklitz ЖЭТФthey initially keep memory of the lead they departedfrom and specifically of the respective chemical poten-tial. Deeper inside the wire, electrons interact and ex-perience backscattering, equilibrating thus the differ-ence in chemical potentials of left- and right-movingelectrons. Particle number conservation then implies asimple relation between the applied voltage V , current I carried by the wire, and rate for backscattering G V = I − e ˙ N R , (75)where ˙ N R ≡ ∂ t N R here is the rate of change in thenumber of right-moving particles. In the absence ofbackscattering, ˙ N R = 0 , and Eq. (75) reduces to theLandauer conductance of a perfectly transmitting chan-nel. To make further progress, we notice that the cur-rent inside the wire comprises of two terms, I = 2 e ∆ µ π + enu, (76)resulting from the imbalance in the chemical potentialsof left- and right-moving electrons, ∆ µ = µ R − µ L , re-spectively, the drift of the electron flow with velocity u . Each contribution may individually vary along thewire, and current conservation implies that this cannothappen in an independent fashion. In fact, both ∆ µ and u can be expressed in terms of the backscatteringrate ˙ N R , and Eq. (75) then defines the conductance ofinteracting electrons.To calculate ˙ N R microscopically one has to inte-grate the right-hand-side of the Boltzmann equationover all positive momenta ˙ N R = (cid:80) p> St { n p } . Forthe kinematics of the leading backscattering process inshort wires (to be specified momentarily), the collisionintegral can be approximated by the Fokker-Planck ker-nel for holes. From Eq. (38) it then follows that ˙ N R = 2 (cid:88) p (cid:20) ∂ p ( A h p ) − ∂ p ( B h p ) (cid:21) = L B π ( ∂ p h p ) p =0 . (77)To determine the discontinuity in the derivative of thehole’s distribution function, one thus has to solve theFokker-Planck equation, ∂ p (cid:104) − pm ∗ T h p + ∂ p h p (cid:105) = 0 , (78)supplemented by the boundary conditions h p = e p / m ∗ T e − µ R/L /T with µ R for right-movers ( p > and µ L for left-movers ( p < ), and weused that A ( p ) = B p/ m ∗ T . To linear order in ∆ µ = µ R − µ L one finds ˙ N R = − ∆ µπ Ll bs , l − bs = B√ πm ∗ T e − ε F /T . (79) The exponential factor in the backscattering length l bs is due to the small statistical occupation probability ofholes at the bottom of the band, while the prefactor re-flects the fact that backscattering occurs in a diffusionprocess in momentum space.While the Fokker-Planck analysis holds in shortwires, backscattering processes in long wires transformthe unperturbed distribution of particles to a partiallyequilibrated form with finite boost velocity u . To iden-tify the relevant length scale and to establish a relationbetween ˙ N R and u one can employ energy conservation.In a fully equilibrated wire with ∆ µ = 0 and u = I/en ,the backscattering in momentum space from the rightto the left Fermi point does not cost energy but requiresa net momentum p F to be redistributed between right-and left-movers. This momentum is equally absorbedby particle-hole excitations at the Fermi level. That is,creating and destroying particle-hole pairs with energy v F p F at the right and left Fermi point, respectively.As a result of the backscattering process, the energybalance for the right-movers thus consists of a loss of ε F due to removal of one particle from the Fermi leveland a gain of ε F due to the redistribution of momen-tum. Thus per each act of backscattering ∆ N R = − the energy changes by ∆ E R = ε F , which gives a fixedrelationship between the respective rates ˙ E R = − ε F ˙ N R . (80)If we now introduce the heat current j Q = j E − µj ,as the energy current counted from the Fermi energywith j the particle current, the energy flux gives theequation j Q = ˙ Q R where ˙ Q R = ˙ E R − µ ˙ N R . The heatcurrent is then readily calculated from the fully equi-librated electronic Fermi distribution function in theboosted frame of reference with u = I/en . Carryingout a Sommerfeld expansion to leading order in
T /ε F ,one finds j Q = π T ε F nu, (81)where at our level of accuracy we we do not distinguishbetween µ and ε F . When combined with the precedingconsiderations on the energy balance this finally gives ˙ N R = − π T ε F nu. (82)This equation together with Eq. (79) fixes the relationbetween ∆ µ and u . In combination with Eqs. (75) and(76) this then determines the conductance of interact-ing electrons beyond the pure Luttinger liquid model G = G (cid:20) − π T ε F LL + l eq (cid:21) , (83) ЭТФ
Kinetic processes in Fermi-Luttinger liquids where we introduced the relevant length scale for equi-libration l eq = ( π T / ε F ) l bs . As alluded to in theintroduction of this section, the interaction correctionto the conductance of a Fermi-Luttinger liquid sat-urates to the value δ G / G ∼ − ( T /ε F ) in the longwire limit L (cid:29) l eq . For practical applications, weshould notice however that it is hard to exceed theequilibration length with the experimentally availablequantum wire micro-constrictions. In the currentlyaccessible domain of parameters therefore L < l eq ,and the correction remains activated in temperature, δ G / G ∼ − ( T /ε F ) ( L/l eq ) ∝ e − ε F /T . Let us next consider that the quantum wire is ther-mally biased by a temperature difference ∆ T appliedto the leads. A similar consideration of energy fluxescarried by left- and right-movers gives the Landauertype formula for the thermal conductance K ∆ T = j Q − ˙ Q R . (84)Here ˙ Q R is the rate at which right-movers release theirenergy to left-movers via multi-particle thermalizingcollisions.The rate of thermalization ˙ Q R receives contribu-tions from both, chirality conserving collisions andbackscattering processes. In a parametrically widerange of lengths, l th < L < l bs , backscattering isweak and can be neglected. In this regime the elec-trical conductance remains unchanged by interactions.The thermal conductance, in contrast, is modified bythermalization due to the RRL quasiparticle decayprocesses discussed in Sec. 2.2. More specifically,the correction due to these processes is found to be δ K / K (cid:39) − ( T /ε F ) [96].A more dramatic effect occurs in long wires due tobackscattering collisions. To explore this limit, recallthat the thermal conductance is defined as the propor-tionality coefficient between the heat flux current andtemperature difference across the wire, computed atzero electrical current j Q = K ∆ T | I =0 . It then followsfrom Eq. (76) that to nullify the current the local dif-ference in chemical potentials of left- and right-moversis locked to the drift velocity, ∆ µ/π = − nu . Using Eqs.(79) and (80) the rate of thermalization in short wires L (cid:28) l eq , can be brought to the form ˙ Q R = − ε F Ll bs nu, (85)and we notice again that the difference between thechemical potential µ and Fermi energy ε F is irrelevant for this discussion. Finally, expressing nu via the heatcurrent j Q from Eq. (81) and inserting this into Eq.(84), one finds the thermal conductance for arbitrarywire lengths [70] K = K l eq L + l eq . (86)A notable feature of this result is that the thermal con-ductance vanishes for long wires, L (cid:29) l eq . Neverthelessthe thermal conductivity κ = K L remains finite, κ = π T v F τ eq . (87)Clearly the large thermal conductivity is a direct mani-festation of the exponentially slow relaxation processesin the one-dimensional system. The result Eq. (87)also applies to the Luttinger liquid regime of stronglyinteracting fluids, upon replacing v F → v the velocityof bosonic excitations and a respective change of theequilibration time.One should keep in mind that the above expressionapplies to the thermal conductivity defined at the low-est frequencies ω (cid:28) τ − eq of measurement. However, theexistence of hierarchical stages of relaxation processesdiscussed in Sec. 2 opens a broad range of frequencies, τ − eq (cid:28) ω (cid:28) τ − qp , where the thermal conductivity isgoverned by the gas of elementary quasiparticle exci-tations of the quantum liquid. Its value can be calcu-lated from the collision integral of triple collisions Eq.(11), focusing on RRL and LLR processes that enableenergy exchange between excitations of different chiral-ity. When solving the kinetic equation, one has to takecare of projecting out all zero modes of the collision op-erator. These include in the present situation not onlythose related to conservation of particle number ( N ) ,total momentum ( P ) and energy ( E ) , but also con-servation of the difference in the number of right- andleft-movers N R − N L . A solution method of the cor-responding kinetic problem was presented in Ref. [96].To evaluate κ in this regime it is further sufficient toset τ − eq → first, and only then take the limit ω → .The calculation then gives κ (cid:39) T ( T /ε F ) v F τ qp (88)modulo a numerical factor of order unity, in agreementwith recent studies [98, 99] of the spinless model. Re-calling that for weakly interacting spin- / fermions τ − qp ∝ T , we thus arrive at the temperature dependence κ ∝ T for the thermal conductivity of elementary exci-tations in the Fermi-Luttinger liquid. This drasticallycontrasts e.g. the thermal conductivity κ ∝ / ( T ln T ) of a 2D Fermi gas [100]. lex Levchenko, Tobias Micklitz ЖЭТФ
The upshot of our discussion from the previous sec-tions is that thermal effects strongly influence electricaltransport in long wires. This is related to backscatter-ing processes and the need to redistribute momentumamong excitations. This physics becomes especially in-tricate in the Luttinger liquid regime as it relates to un-even energy partitioning between counterpropagatingspin and charge modes [101]. In this section we followRef. [93] and repeat their arguments generalizing theabove calculation of conductance to spin-charge cou-pled quantum fluids in the limit of arbitrary strengthof interaction.We start out by noting that the total momentumof a Luttinger liquid is given by a sum of two contri-butions P = p F J + P b [13]. Here the first term rep-resents the momentum of a filled Fermi surface with J = N R − N L extra electrons at the right Fermi point,and the second term P b = P ρ + P σ captures the netmomentum of (spin/charge) bosons. The momentumconservation combined with the particle number con-servation, N = N R + N L , dictates that the rate ofbackscattering is related to relaxation of bosonic mo-mentum ˙ N R = ˙ J/ − ˙ P b / p F . (89)The momentum of bosons can be calculated from re-spective distribution in a boosted frame P b = − (cid:88) q q ( qu )( ∂ ω N ρq + ∂ ω N σq ) = πLT (cid:18) v ρ + 1 v σ (cid:19) u. (90)The relaxation of velocity u of the boson gas towardsdrift velocity of the fully equilibrated wire v d = I/en ˙ u = − u − v d τ eq (91)is dominated by the backscattering in the spin-spinchannel τ − eq ∝ B σσ e − ε σ /T . It is also important to notethat the boost is the same for both spins and changesas spin-charge drag scattering occurs on a time scale τ ρσ , that is much faster than the equilibration time.Next, one notices that the heat flux density determinesthe rate of energy exchange in backscattering processesas j Q = ˙ E R = π T (cid:18) v ρ + 1 v σ (cid:19) u. (92)Finally, in order to close the loop of formulas we need arelation between ˙ E R and ˙ N R . Consideration of the en-ergy partitioning between spin and charge modes prop-agating at different velocities suggests [93, 101] ˙ E R = − v ρ v σ ( v ρ + v σ )( v ρ + v σ ) p F ˙ N R . (93) Here the fraction results from the momentum pro-portional to v − ρ/σ , transferred into the charge or spinbranch, whereas the corresponding energies scale as v − ρ/σ . Following then the calculation for the weak cou-pling limit, one can now combine the above relationsto determine ˙ N R to linear order in v d . From this wethen find the conductance with help of Eq. (75). Fo-cusing on long wires, L (cid:29) l eq , the interaction-inducedcorrection can be cast in the form δ GG = − π T v σ p F F ( v σ /v ρ ) , F = 1 + x (1 + x )(1 + x ) . (94)This result is structurally identical to Eq. (83) whentaken in the same limit. Eq. (94) indicates thatspin excitations give a dominant contribution to theinteraction-induced correction to the conductance, as F → for v ρ (cid:29) v σ . Notice also that in this limit δ G becomes independent of characteristics of the chargemode. For shorter wires, L (cid:28) l eq , the correction stillshows activated temperature dependence δ G ∝ e − ε σ /T .The activation energy ε σ (cid:28) ε F is set by spin exci-tations, and thus reduced by strong interactions com-pared to the weak coupling limit. For completeness, we next consider extrinsic mecha-nisms of momentum relaxation that compete with equi-libration processes and thus affect temperature depen-dence of resistance in quantum wires. Specifically, wehere focus on electron-phonon scattering and the effectof smooth long-range inhomogeneities. The latter isalso relevant to the hydrodynamic regime of transportin strongly correlated electron liquids [102]. We do nottouch upon relaxation in linear Luttinger liquids withpoint-like impurities, which has e.g. been studied inRef. [27].To calculate the phonon-induced resistivity weadopt the approach of Sec. 2.5, used there to deter-mine the relaxation time of hot electrons. Indeed, inthe presence of equilibration processes the distributionfunction of electrons in the wire develops a finite driftvelocity u , and right-movers become effectively hot-ter than left-movers. Technically, this becomes evidentnoting that the finite drift velocity enters the Fermi dis-tribution as pu , which for states near the Fermi points ± p F can be absorbed into a redefinition of distinct tem-peratures T ± ∆ T R/L = T ± T ( u/v F ) for right- and left-movers, respectively. We can then calculate from Eq. ЭТФ
Kinetic processes in Fermi-Luttinger liquids (48) the rate of momentum loss by e.g. right-movers.Focusing on the phonon emission, ˙ P R ep = 2 π (cid:88) pp (cid:48) q ( p (cid:48) − p ) W [ n p (1 − n p (cid:48) )(1+ N q ) − n p (cid:48) (1 − n p ) N q ] . (95)We find upon linearization in the drift velocity ˙ P R ep = − πuv F (cid:88) pp (cid:48) q q x T W (cid:15) p f (cid:15) p (1 − f (cid:15) p )(1 − f (cid:15) p − (cid:36) q − f (cid:15) p + (cid:36) q ) . (96)Here we employed momentum and energy conserva-tions implicit in W , and introduced the following no-tations for energy variables (cid:15) p = v F p and (cid:36) q = v F q x .The momentum integral over p (cid:48) is removed by the mo-mentum conservation. The other momentum integra-tion, (cid:80) p → L πv F (cid:82) d(cid:15) p , can be completed in the closedform with the help of the tabulated integral + ∞ (cid:90) −∞ xdx cosh ( x ) [tanh( x + y ) + tanh( x − y )] = 2 y sinh ( y ) . (97)Imposing a force balance condition on segments ofthe wire shorter than ξ , we define the resistivity ρ ep = − ˙ P R ep / ( e n uL ) . The latter is then found to beof the form ρ ep = 1 e n v F (cid:88) q q x T W ( q ) ( v F q x ) sinh ( v F q x / T ) . (98)Finally, we separate in the phonon dispersion momen-tum components along and perpendicular to the wire, ω q = s (cid:112) q x + q ⊥ , and notice that energy conservationfixes q x ≈ ( s/v F ) q ⊥ . That is, because of the large dif-ference of phonon and electron velocities, phonons areemitted predominantly in transverse direction with re-spect to the electron momentum. The final integralover phonon phase space is separately done for piezo-electric and deformation potentials contributing to thescattering rate W . The result is thus a sum of thefollowing two contributions, ρ ep = ρ Λ + ρ D , where ρ Λ = 3 ζ (3)8 π e (cid:18) Tv F (cid:19) (cid:18) Ts (cid:19) (cid:104) Λ (cid:105) n (cid:37)sv F , (99a) ρ D = 15 ζ (5) π e (cid:18) Tv F (cid:19) (cid:18) Ts (cid:19) D n (cid:37)s . (99b)Here we approximated the piezoelectric cou-pling function by its average over angles of theemitted phonons in transverse direction, namely (cid:104) Λ (cid:105) = (cid:82) π Λ (0 , q ⊥ cos φ, q ⊥ sin φ ) dφ .In addition to electron-phonon scattering, two-particle electron collisions may also provide a chan-nel for momentum relaxation when translational invari-ance of the system is broken. This happens e.g. when screening of the interaction is inhomogeneous. The cor-responding relaxation process is captured by the usualcollision integral ˙ P R ee = 2 π (cid:88) pp (cid:48) > kk (cid:48) < ( p (cid:48) + k (cid:48) − p − k ) W [ n p n k (1 − n p (cid:48) )(1 − n k (cid:48) ) − n p (cid:48) n k (cid:48) (1 − n p )(1 − n k )] , (100)where crucially the scattering rate W does not pre-serve momentum. Notice also that momenta inEq. (100) are restricted to include one particle-holeexcitation in each branch. To accommodate inhomo-geneous screening in the simplest model, one can as-sume an interaction potential with separable kernel V ( x, x (cid:48) ) → V ( x − x (cid:48) )Υ(( x + x (cid:48) ) /ξ ) . Here the first termrepresents the usual screened Coulomb interaction, andthe nonuniformity of the system is encoded in the di-mensionless function Υ( x ) . The latter has a spatialscale with correlation radius ξ , which we assume tobe large compared to both, the Fermi wavelength andrange of interaction V ( x − x (cid:48) ) . The matrix elementof interaction V pk,p (cid:48) k (cid:48) = (cid:104) k (cid:48) p (cid:48) | V ( x, x (cid:48) ) | kp (cid:105) entering thescattering rate W can be calculate in the basis of planewaves | pk (cid:105) = √ ( e ipx e ikx (cid:48) − e ipx (cid:48) e ikx ) . A more accurateanalysis based on quasiclassical wave functions leads toqualitatively the same result. In complete analogy toelectron-phonon scattering, one finds upon expansionof Eq. (100) to linear order in the boost velocity theresistivity ρ ee = ( T /ε F v F )2 π e n (cid:18) V − V p F v F (cid:19) ∞ (cid:90) −∞ ( ω/ T ) dω sinh ( ω/ T ) × x x +∆ x (cid:90) x dxdx (cid:48) Υ( x/ξ )Υ( x (cid:48) /ξ ) e − iω ( x − x (cid:48) ) /v F . (101)To arrive at the above expression wesplit the energy conserving delta func-tion in the scattering rate into two, δ ( ε p + ε k − ε p (cid:48) − ε k (cid:48) ) → (cid:82) dωδ ( ε p − ε p (cid:48) − ω ) δ ( ε k − ε k (cid:48) + ω ) ,and made use of the following identities applicable forequilibrium Fermi functions, f ε (1 − f ε ± ω ) = f ε − f ε ± ω − e ∓ ω/T , + ∞ (cid:90) −∞ dε ( f ε − f ε ± ω ) = ± ω. (102)These steps enable us to complete three out of thefour momentum integrations in Eq. (100). The fi-nal result for ρ ee depends on the relation between lex Levchenko, Tobias Micklitz ЖЭТФtemperature T and the characteristic energy scale ofthe inhomogeneity, E ξ = v F /ξ . In the limit ofhigh temperatures, T (cid:29) E ξ , the exponential underthe integral is a rapidly oscillating function. Onecan explore this fact to integrate by parts, usingthat ( ω/v F ) e − iω ( x − x (cid:48) ) /v F = ∂ xx (cid:48) e − iω ( x − x (cid:48) ) /v F , andthen notice that the remaining frequency integral givesa delta function δ ( x − x (cid:48) ) . As a result one arrives at ρ ee = 132 πe (cid:18) V − V p F v F (cid:19) Tnε F ( ∂ x Υ( x )) , (103)where the remaining position integral was expanded tofirst order in ∆ x . In the opposite limit of low tem-peratures, T (cid:28) E ξ , one finds the weaker dependence ρ ee ∝ T .The above calculation for the weakly interactinglimit can be generalized to the Luttinger liquid regime.The inhomogeneity induced resistivity is then given bya sum of two contributions, ρ ee = ρ ρ + ρ σ , as bothcharge and spin modes dissipate energy throughout thewire. At T (cid:29) E ξ both contributions remain linear intemperature ρ ρ/σ ∝ T [103].A crucial point to re-emphasize is that both extrin-sic mechanisms leading to resistivities ρ ep and ρ ee aremade possible only by electron backscattering whichestablishes a finite boost u in the distribution functionof partially equilibrated electrons. In order to find thecomplete wire resistance one needs to modify the en-ergy balance condition Eq. (80) to account for bothmomentum relaxing collisions ˙ E R = − µ ˙ N R + v F ˙ P R mr , ˙ P R mr = ˙ P R ep + ˙ P R ee . (104)Since both ˙ P R ep and ˙ P R ee are proportional to u this con-dition mixes ∆ µ and u in the energy balance. It is thenuseful to introduce the length scale l mr for momentumrelaxation ˙ P R mr = − unp F Ll mr , l − mr = e π ( ρ ep + ρ ee ) . (105)Recalling Eq. (103) and its weak coordinate depen-dence, ρ ee should here be understood as averaged overthe wire. Repeating then the steps from Sec. 3.1, thatis, excluding ∆ µ with help of energy balance equationsand backscattering rates, and using Eq. (76) to express u in terms of the current through the wire I , one finallyfinds the wire resistance R = V /I from Eq. (75), R = πe (cid:20) r bs ( r + r mr )( r bs + r + r mr ) (cid:21) . (106)Here we introduced the short-hand notations r = π T / ε F , r bs = L/l bs , and r mr = L/l mr . For long wires r (cid:28) { r mr , r bs } , and one finds a resis-tance scaling linearly with the wire length, R = ρL ,with resistivity given by ρ = πe l bs + l mr . (107)This formula is in stark contrast to Matthiessen’s rule,stating that resistivity ρ = ρ + ρ + . . . comprises of asum of partial resistivities ρ i ∝ l − i originating from dif-ferent scattering channels, each being inversely propor-tional to the respective scattering length l i . The reasonfor this significant difference is that Matthiessen’s rulemisses both quantum and classical correlational effects.In the present context, the channels of chirality chang-ing electron collisions and momentum relaxing electroncollisions are strongly intertwined and Matthiessen’srule does not apply. That is, resistivities for compet-ing relaxation channels in Eq. (107) add up in parallelrather than in series. To further elucidate an important difference be-tween charge and thermal transport in quantum 1Dliquids we consider thermal currents carried by bosonicexcitations. In this section we use the framework de-veloped in Refs. [95, 104] and our condense discussiononly to the spinless case of inhomogeneous Luttingerliquids.We start out from the Landauer formula for theheat flux of plasmons j E = (cid:90) ∞ dω π ω | t ω | (cid:2) N Rω − N Lω (cid:3) (108)where all the effects of interactions in the wire are cap-tured by the transmission coefficient of plasmons | t ω | .The notation N R/Lω refers to bosonic distributions forright/left moving excitations. For a small temperaturedifference ∆ T maintained between the reservoirs we ob-tain thermal conductance in the form K = j E ∆ T = (cid:90) ∞ dω π ( ω/ T ) sinh ( ω/ T ) | t ω | (109)In the noninteracting limit | t ω | → this expressiongives the thermal conductance quantum K of a per-fect channel. Another simple limit to check is for astep-like model, with interaction parameter g < inthe wire and g = 1 in the leads. The transmission coef-ficient of plasmons in this case is that of the Fabri-Pérotinterferometer | t ω | = 11 + (cid:16) g − g (cid:17) sin (2 πω/ω L ) . (110) ЭТФ
Kinetic processes in Fermi-Luttinger liquids
At high temperatures T (cid:29) ω L = v/L thetransmission coefficient in Eq. (109) strongly oscil-lates. It can then be replaced by its average value (cid:104)| t ω | (cid:105) = ω L (cid:82) ω L | t ω | dω = gg +1 , which leads to thethermal conductance K = 2 g K / ( g + 1) quoted at thebeginning of this section.With these elementary examples, we next discussinteraction correction to the thermal conductance inan inhomogeneous wire. We here focus on the modelformulated in Ref. [95] and restrict analysis to the low-est temperatures. To this end, we consider the waveequation for plasmons in the basis of scattering states − ˆ hφ R/Lk ( x ) = ω k φ R/Lk ( x ) , (111)where L/R labels left- and right-movers, ω k is the en-ergy of the given mode, and ˆ h = (cid:112) n ( x ) ∂ x (cid:0) V ( x ) /m + π n ( x ) /m (cid:1) ∂ x (cid:112) n ( x ) . (112)is the Hamiltonian operator of the inhomogeneous wirewith coordinate dependent interaction potential. Thescattering states of left-movers take the asymptoticform φ Lk ( x ) = (cid:40) e ikx + r ω k e − ikx x → −∞ t ω k e ikx x → + ∞ (113)and similar for the right-movers. To establish a linkto the model employed in the previous sub-section wefurther treat the electron density n ( x ) as constant andonly allow space dependence of the interaction in themodel of a separable kernel. In this case the scatteringproblem simplifies to ( v ∂ x + ω k ) φ L/Rk ( x ) = − vπ ∂ x ( V ( x ) ∂ x φ L/Rk ( x )) . (114)We next treat this problem perturbatively in interac-tion and calculate leading order correction to the trans-mission coefficient. For this purpose we first constructthe Green’s function of the wave equation ( v ∂ x + ω k ) G ( x, y ) = δ ( x − y ) , (115)which reads G ( x, y ) = 12 ikv e ik | x − y | . (116)We then recall that for weak interactions in the modelunder consideration, V → ( V − V p F )Υ( x ) , and thatfor our purposes we can replace v → v F . As a result,the solution for the scattering problem can be cast inthe form φ Lk ( x ) = e ikx + δφ Lk ( x ) ,δφ Lk ( x ) = − V − V p F πikv F (cid:90) dy e ik | x − y | ∂ y (Υ( y ) ∂ y e iky ) . (117) Since for the reflected wave δφ Lk ( x ) = r ω k e − ikx we takethe asymptotic of δφ k,L ( x ) at x → −∞ and determinethe reflection coefficient to be r ω k = iω k πv F ( V − V p F ) (cid:90) dy Υ( y ) e iky . (118)Noting that | t ω | = 1 − | r ω | , it is evident from Eq.(109) that corrections to the thermal conductance δ K are due to plasmon backscattering. Inserting now | r ω | under the integral we get δ KK = − c (cid:18) V − V p F v F (cid:19) (cid:18) TE ξ (cid:19) (119)where c Υ is dimensional model specific number. Thisresult applies to lowest temperatures, T (cid:28) E ξ , where e iω ( y − y ) /v F ∼ since ω ∼ T and y − y (cid:48) ∼ ξ . Thesame result can be derived in fermionic language fromthe kinetic equation, by calculating ˙ Q R based on two-particle momentum relaxing scattering enabled by theinhomogeneity. As compared to a similar calculation ofresistivity ρ ee , here the correction to the thermal trans-port coefficient emerges without electron backscatter-ing. The connection to a step-function model is clear aswell. Indeed, from Eq. (110) one can expand the trans-mission in the limit ω/ω L (cid:28) and retain the leadingorder interaction renormalization, g ∼ − V /v F , whichconsistently yields a correction δ K ∝ T .The complete picture of plasmon thermal trans-port in inhomogeneous 1D systems is rather compli-cated. Disorder tends to localize plasmons and theirtransmission then may get exponentially suppressed | t ω | ∼ e − L/ζ ω on the length scale ζ ω for plasmon local-ization. This process competes, however, with inelasticplasmon-plasmon scattering. For instance, two-boson ρρ → ρρ scattering, which as we discussed above inSec. 2.6, kinematically produce low energy excitations.Since the decay of plasmons diverges inversely propor-tional to the plasmon frequency, ζ ω ∝ /ω , such low-energy bosons avoid localization and propagate ballisti-cally throughout the wire with almost perfect transmis-sion. As a result, it should be expected that the ther-mal conductance depends in a non-trivial fashion onthe system size L (see Ref. [105] for recent progress onthis issue). A generalization of these effects to systemswith spin degrees of freedom is still an open problem. We next extend the analysis of the preceding sec-tions to determine the full matrix of linear transport lex Levchenko, Tobias Micklitz ЖЭТФ − µ/T L h / e fully equil. (analytical)fully equil. (numerical)non-interacting − µ/T L h / e − µ/T L h / T − µ/T L h / T Fig. 1.
Linear response transport coefficients at the QCPof the first plateau transition of fully equilibrated (solidline) and noninteracting electrons (dashed line). Numer-ical results (symbols) were obtained from the solution ofthe kinetic equation (see Ref. [106] for details). In allcalculations we assumed the limit of long wires, L (cid:29) l eq ,where l eq is the equilibration length at the QCP. coefficients, (cid:32) j c j Q (cid:33) = ˆ L (cid:32) eV ∆ T (cid:33) (120)relating electric current ( j c ) and heat current ( j Q )to voltage and temperature drops. We remind thatcomponents of this matrix define the thermopower S = L eL , thermal conductance K = L − L L L ,Peltier coefficient Π = L L , and electric conductance G = eL , with the Onsager relation Π = S T /e .The ideas that went into the calculation of the con-ductance of fully equilibrated quantum wires, viz. thelimit L (cid:29) l eq of Eq. (83), relay exclusively on con-siderations of conservation laws. Thus one expects theconclusions of such analysis to be fairly generic. Indeed,in the spinless case, microscopic details of the interac-tion were not important in establishing energy parti-tioning between left- and right-movers in a backscat-tering process as they only went into the definition ofthe equilibration length. As a result, the interaction-induced correction to the conductance took a univer-sal form. An interesting application of these ideas isto investigate the impact of equilibration processes ontransport coefficients near the transition from one con-ductance plateau to the next [106]. The latter is anexample of a quantum phase transition. Importantly,in the vicinity of each plateau transition the equili-bration length crosses from exponential to power-law behavior, thus making an experimental verification ofequilibration effects on the conductance more promis-ing. From a dimensional analysis we estimate that forgeneric nonintegrable short-range interaction near thequantum critical point (QCP) l − eq ∝ T / .Following the strategy outlined in Sec. (3.1) thetransport matrix is found to be of the form (cid:32) j c j Q (cid:33) = g2 π (cid:126) (cid:32) e eβT β T β (cid:33) (cid:32) eV ∆ T (cid:33) . (121)Here we restored (cid:126) , and introduced the dimensionlessconductance g( z ) = α ( z ) − α ( z ) α ( z )2 α ( z ) β ( z ) − α ( z ) − α ( z ) β ( z ) (122)with z = µ/T , where the set of functions α n ( z ) = (cid:104) χ n (cid:105) z , β ( z ) = (cid:104) χ (cid:112) χ/z (cid:105) z (cid:104) (cid:112) χ/z (cid:105) z , (123)is defined by the thermal average (cid:104) ... (cid:105) z = − (cid:90) ∞− z dχ ( ... ) df χ dχ , f χ = 1 e χ + 1 . (124)The components of the thermoelectric matrix L ij fora fully equilibrated wire at the QCP are shown in Fig.(1) in comparison to the noninteracting limit. It can bereadily checked that away from the QCP, z = µ/T (cid:29) ,one finds to leading order in /z α = 1 , α = 0 , α = π / , and β = π / z . This reproduces the satu-rated form of the conductance, g = 1 / (1 + π / z ) . Incontrast, one finds at the QCP z = µ/T → , α = 1 / , α = ln 2 , α = π / , and β = − ζ (3 / √ ζ (1 / . This im-plies that G QCP = g e π (cid:126) with g ≈ . , which is about below the result of the noninteracting limit.These results can be readily generalized to the QCPof the N th plateau transition, where chemical poten-tials µ i (cid:29) T ( i = 1 , . . . , N − ) while µ N ∼ T . We canthen determine that at the QCP z N = µ N /T → ,asymptotically G QCP ≈ e π (cid:126) (cid:104) N − − π ( N − (cid:105) , for N > . We notice that the conductance of quantumwires was carefully measured in Ref. [107] for differenttemperatures. For quantum critical points of the firstand second plateau transition, that correspond to thosepoints where curves for different temperatures inter-sect, a reduced value of the conductance as comparedto the value of noninteracting electrons was observed.This is in qualitative agreement with the presented herepicture of interaction effects. ЭТФ
Kinetic processes in Fermi-Luttinger liquids
Recent significant progress in fabrication of cleangated semiconducting InAs and InSb quantum wireswith strong Rashba-type spin-orbit coupling [108, 109]opens new avenues for exploring the consequences ofthe interplay in spin physics and electron interactionson the nanoscale 1D transport. The simplest modelthat describes these systems includes Rashba and Zee-man terms in the Hamiltonian. The former lifts spindegeneracy, shifting electron bands in momentum spaceaccording to the spin polarization, and the latter fur-ther shifts bands in energy, partially gapping the spec-trum. The resulting single particle energy bands are ε ± p = p m ± (cid:112) B z + ( α so p ) (125)where α so describes the strength of Rashba coupling,and B z defines the energy scale of Zeeman splittinglinear in magnetic field. From the stand point of trans-port properties, the most interesting configuration iswhen chemical potential is within the Zeeman gap − B z < µ < B z . Then only the lowest helicity sub-band is occupied and the model can be bosonized. Lin-ear Luttinger liquid theory predicts quantization of theconductance for such Rashba wires, adiabatically con-nected to noninteracting leads at the value e /h , irre-spective of the interaction strength in the wire [110].The reasoning is exactly the same as for conventionalquantum wires discussed earlier. Taking into accountcurvature of the spectrum around the Fermi pointswithin an extended model, the peculiar nonlinear en-ergy dispersion relation Eq. (125) allows for both two-particle and three-particle inelastic e-e scattering pro-cesses. The impact of the latter on interaction-inducedcorrections to the quantized conductance was investi-gated in the recent papers [111, 112] for short quantumwires.To quantify these effects one needs to single outchirality changing processes. That is, scattering pro-cesses that do not conserve the number of right/left-movers before and after the collision. The leading kine-matically allowed two-particle process involves a set ofstates within the same helicity band accompanied byan interband transition. Momentum and energy con-servations dictate that the bottleneck for this processis the existence of an unoccupied state near p ∼ .At finite chemical potential lying within the Zeemangap this costs a Boltzmann factor ∝ e − B z /T e −| µ | /T .Being a two-particle process, Golden rule dictates thecorrection to the conductance scaling in the interac-tion parameter ∝ V . As usual, thermal broaden-ing allows momentum transfer in the collision of or- der ∼ T /v F . However, for the dispersion relation withthe square root Eq. (125), we expect contribution withan enhancement ∝ / √ T of the scattering processes.Combining these observations with dimensional reason-ing, suggests that the quantum correction to conduc-tance scales as δ G / G ∼ − L/l bs with the backscatteringlength l − bs ∼ B z α so (cid:18) Vα so (cid:19) (cid:114) TB z e − B z /T e −| µ | /T . (126)This estimate is supported by a detailed calculation[111]. In a similar spirit, one may estimate the con-tribution from three-particle processes. In contrastto the two particle collisions, the latter only involvestates from the same helicity band. At the perturba-tive Golden rule level the correction to the conductancecomes with a factor ∝ V , a phase factor ∝ T , anda thermal factor ∝ e − B z /T e | µ | /T . As a result we ex-pect again δ G / G ∼ − L/l bs , now with a backscatteringlength l − bs ∼ B z α so (cid:18) Vα so (cid:19) (cid:18) TB z (cid:19) e − B z /T e | µ | /T . (127)In the above estimate we took advantage of the factthat spin-orbit coupling lifts integrability constraints,and point-like interaction already lead to a finite relax-ation rate. That is, there are no more subtle cancel-lations which could bring additional T -dependent pref-actors.For wires e.g. with L ∼ µ m, α so ∼ eVÅ, B z ∼ . H meV/T and V /α so ∼ , these correctionsbecome experimentally noticeable for magnetic fields H ∼ mT and temperatures T ∼ mK such that B z /T ∼ where δ G / G ∼ − . Furthermore, one canexpect that triple electron processes may dominate pro-vided that < | µ | (cid:46) B z . We argue that in this regime,and for long enough equilibrated wires, L (cid:29) l bs , theinteraction correction to conductance δ G saturates.The calculation of the interaction induced correc-tion to the conductance of fully equilibrated Rashbawires is exactly analogous to the previous example oftransport near a QCP. We only need to accommodatethe different dispersion relation Eq. (125). The con-ductance is thus still given by Eq. (122) but now withthe modified thermal factors α n ( z ) = (cid:104)(cid:104) χ n (cid:105)(cid:105) , β ( z ) = (cid:104)(cid:104) χp ( χ ) (cid:105)(cid:105) / (cid:104)(cid:104) p ( χ ) (cid:105)(cid:105) , (128)and a redefined average (cid:104)(cid:104) ... (cid:105)(cid:105) = − (cid:18)(cid:90) ∞− z − (cid:90) − z − z (cid:19) dχ ( ... ) df χ dχ . (129) lex Levchenko, Tobias Micklitz ЖЭТФHere z = ( µ + mα s0 / B z / mα so ) /T , z = ( µ + B z ) /T and p ( χ ) is the dimensionlessmomentum corresponding to the dispersion in Eq.(125). In the limit | µ | (cid:46) { T, B z } (cid:28) mα so , analyt-ical progress is possible by expanding α n and β incorrections algebraically small in { T, B z } /mα so , andneglecting all other exponential small terms. As a re-sult, we find the conductance for the fully equilibratedRashba wire G = G (cid:34) − π (cid:18) Tmα so (cid:19) + 5 π T B z ( mα so ) (cid:35) , (130)where we also retained the leading order field-dependent correction. This result is analogous toEq. (83) taken in the limit L (cid:29) l eq , but perhapssurprisingly it leads to a negative magnetoresistance.
4. FINAL REMARKS AND PERSPECTIVE
In this work we in part reviewed recent progresson theory of kinetic processes in Fermi-Luttinger liq-uids and presented new results. We provided a com-prehensive discussion of quasiparticle relaxation mech-anisms in single channel quantum wires, and unveiledtheir impact on the thermoelectric and magnetotrans-port properties. While the emphasis of this study wason electron liquids in quantum wires, there are otherinteresting variants of Luttinger liquids, left aside inthis paper, where similar physics can be explored. Toprovide a broader perspective we highlight a few inter-esting examples.We saw that the temperature and energy depen-dence of quasiparticle relaxation times is extremely sen-sitive to details of the spectrum nonlinearities. Thisaspect of the problem becomes extremely intricate inchiral Luttinger liquids of the integer quantum Halleffect. The interplay of interactions and confinementleads to either spin- or charge-dominated mechanismsof edge reconstruction. As a result, relaxation ratesof hot electrons injected into edge channels are sig-nificantly altered in different reconstruction scenarios[113]. Similar complications exist in the regime of thefractional quantum Hall effect, where the interplay ofcounterpropagating modes of reconstructed edges hasdramatic consequences on relaxation mechanism, andultimately the temperature dependence of the electricand thermal conductances [114, 115].The quantum spin Hall effect gives rise to a he-lical version of Luttinger liquids [116]. In these sys-tems potential scatterers alone cannot prevent electronsfrom ballistic propagation along the edges. That is, backscattering is not permissible since counterpropa-gating states of the same energy form a Kramers dou-blet if time-reversal symmetry is preserved. Break-ing, however, the latter by e.g. magnetic impurities,or breaking axial spin symmetry in the presence ofstrong spin-orbit effects, opens the possibility for vari-ous inelastic and spin-flip scattering processes, thus en-abling quasiparticle relaxation and ultimately affectingthe conductance [117–122].One could add to this list another member, namelythe so-called spiral Luttinger liquids [124]. This pecu-liar state may form in quantum wires where a spon-taneous ordering of nuclear spins at low temperaturesproduces an effective Rashba spin-orbit coupling, lead-ing to a strongly nonlinear single-particle spectrum. Asa consequence, inelastic scattering processes becomepossible, which should result in interaction-inducedcorrections to transport properties in these systems.We thus expect this field to continue evolving in var-ious fruitful directions. In particular, a comprehensiveunderstanding of kinetic processes and time-scales forrelaxation presented in this work provides a necessaryingredient in bridging to a hydrodynamic descriptionof strongly correlated electron liquids.
Acknowledgements
We acknowledge collaborations with A. Andreev,L. Glazman, T. Karzig, K. A. Matveev, F. von Oppen,J. Rech, M. T. Rieder, A. Rosch, and Z. Ristivojavic.We would like to thank Maxim Khodas for useful dis-cussions, for reading the manuscript prior to submis-sion and for providing comments. This work was sup-ported by the U. S. Department of Energy (DOE), Of-fice of Science, Basic Energy Sciences (BES) Programfor Materials and Chemistry Research in Quantum In-formation Science under Award No. DE-SC0020313.T. M. acknowledges financial support by Brazilianagencies CNPq and FAPERJ.
A. NOTES ON BOSONIZATION
This section is prepared as a supplementary mate-rial to the main text of the paper. Here we concentrateon the derivation of an effective Hamiltonian for non-linear Luttinger liquids and construction of the corre-sponding kinetic theory of spin-charge scattering pro-cesses detailed in Sec. 2.6. To the large extend we fol-low here Giamarchi textbook Ref. [14] for the bosoniza-tion procedure and notations, plus the Haldane reviewarticle Ref. [13] to include band curvature effects. Anadditional element presented here is a more detailed ЭТФ
Kinetic processes in Fermi-Luttinger liquids bosonization of the interaction part of the fermionicHamiltonian. It will be shown that similar to the band-curvature terms, interaction also generates anharmoniccouplings between the spin and charge modes. As dis-cussed in the main text of the paper these nuances haveimportant consequences for the relaxation in 1D includ-ing spin-charge drag and energy transport.The starting point is the usual form of fermionicHamiltonian: H = − iv F (cid:88) s (cid:90) dx (cid:104) ψ † Rs ( x ) ∂ x ψ Rs ( x ) − ψ † Ls ( x ) ∂ x ψ Ls ( x ) (cid:105) − m (cid:88) s (cid:90) dx (cid:104) ψ † Rs ( x ) ∂ x ψ Rs ( x ) + ψ † Ls ( x ) ∂ x ψ Ls ( x ) (cid:105) + 12 (cid:88) ss (cid:48) (cid:90) dxdx (cid:48) V ( x − x (cid:48) ) ψ † s ( x ) ψ † s (cid:48) ( x (cid:48) ) ψ s (cid:48) ( x (cid:48) ) ψ s ( x ) . (131)Here index s = ↑↓ stands for the spin projection, ψ Rs and ψ Ls are the annihilation operators for right- andleft-moving spin- s electrons, while ψ s = ψ Rs + ψ Ls isfull operator in the interaction part of the Hamilto-nian. The standard approximation is that low energyexcitations take place near the Fermi points, such thatelectron operator is decomposed as follows ψ s ( x ) = ψ Rs ( x ) + ψ Ls ( x ) = e ik F x R s ( x ) + e − k F x L s ( x ) (132)where new fields R ( L ) s ( x ) are assumed to vary slowlyon the scale of the Fermi wavelength. In the bosoniza-tion description these fields can be expressed in termsof dosonic displacement ϕ s ( x ) and conjugated phase ϑ s ( x ) R s ( x ) = κ s √ πa exp[ iϑ s ( x ) − iϕ s ( x )] L s ( x ) = κ s √ πa exp[ iϑ s ( x ) + iϕ s ( x )] , (133)where a is the short distance cut-off ∼ k − F and κ s arethe Klein factors that ensure proper anticommutationrelation between original fermionic operators. Theyobey { κ s , κ s (cid:48) } = 2 δ ss (cid:48) and satisfy κ † s = κ s . The bosonicfields obey commutation [ ϕ s ( x ) , ϑ s (cid:48) ( x )] = iπ sgn ( x − x (cid:48) ) δ ss (cid:48) (134)With these notations at hand fermionic densities forright- and left-moving electrons become ρ Rs ( x ) = R † s ( x ) R s ( x ) = − π ∂ x [ ϕ s ( x ) − ϑ s ( x )] ,ρ Ls ( x ) = L † s ( x ) L s ( x ) = − π ∂ x [ ϕ s ( x ) + ϑ s ( x )] . (135) The total density operator per spin ρ s ( x ) = ψ † s ( x ) ψ s ( x ) contains the sum of the long-wavelenght part, ρ (0) s ( x ) ,and oscillatory part ρ (2 k F ) s ( x ) : ρ s ( x ) = ρ (0) s ( x ) + ρ (2 k F ) s ( x ) = − π ∂ x ϕ s ( x ) + 1 πa cos[2 ϕ s ( x ) − k F x ] . (136)For the first two terms of Eq. (131) in the bosonizationdictionary we have ψ † Rs ( x ) ∂ x ψ Rs ( x ) − ψ † Ls ( x ) ∂ x ψ Ls ( x ) = iπ [ ρ Rs ( x ) + ρ Ls ( x )] ,ψ † Rs ( x ) ∂ x ψ Rs ( x ) + ψ † Ls ( x ) ∂ x ψ Ls ( x ) = − π ρ Rs ( x ) + ρ Ls ( x )] , (137)such that kinetic part of the Hamiltonian transformsinto H kin = v F π (cid:88) s (cid:90) dx (cid:2) ( ∂ x ϕ s ) + ( ∂ x ϑ s ) (cid:3) − πm (cid:88) s (cid:90) dx (cid:2) ( ∂ x ϕ s ) + 3( ∂ x ϕ s )( ∂ x ϑ s ) (cid:3) . (138)For the interaction part of the Hamiltonian in Eq.(131) we proceed as follows. Up to an additive constantit can be rewritten as H int = 12 (cid:88) ss (cid:48) (cid:90) dxdx (cid:48) V ( x − x (cid:48) )[ ψ † Rs ( x ) ψ Rs ( x ) + ψ † Ls ( x ) ψ Ls ( x ) (cid:124) (cid:123)(cid:122) (cid:125) q ∼ + ψ † Ls ( x ) ψ Rs ( x ) (cid:124) (cid:123)(cid:122) (cid:125) q ∼ k F ][ ψ † Rs (cid:48) ( x (cid:48) ) ψ Rs (cid:48) ( x (cid:48) ) + ψ † Ls (cid:48) ( x (cid:48) ) ψ Ls (cid:48) ( x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) q ∼ + ψ † Ls (cid:48) ( x (cid:48) ) ψ Rs (cid:48) ( x (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) q ∼ k F ] (139)thus separating explicitly different scattering channels,where q labels characteristic momenta transferred inthe collision. From here one can read out forwardand backward scattering parts of the interaction term,namely H int = H fsint + H bsint . The formed one may beeasily rewritten in the bosonization dictionary H fsint = 12 (cid:88) ss (cid:48) (cid:90) dxdx (cid:48) V ( x − x (cid:48) ) ρ (0) s ( x ) ρ (0) s (cid:48) ( x (cid:48) )= V π (cid:88) ss (cid:48) (cid:90) dx ( ∂ x ϕ s )( ∂ x ϕ s (cid:48) ) . (140)Here we expanded the density ρ (0) s (cid:48) ( x (cid:48) ) around x usingthe fact that interaction potential V ( x − x (cid:48) ) is short-ranged and that fields ϕ s ( x ) are slowly varying on the lex Levchenko, Tobias Micklitz ЖЭТФscale where interaction appreciably decays. By V wedenote zero momentum Fourier transform of the in-teraction potential and higher order gradients were ig-nored. We concentrate now on the backward scatteringpart of the Hamiltonian. In the bosonization languageit can be written as H bsint = 18 π a (cid:88) ss (cid:48) (cid:90) dxdy V ( y ) × (cid:104) e − ik F y e iϕ s ( x + y/ − iϕ s (cid:48) ( x − y/ + h.c. (cid:105) . (141)For the case of s (cid:54) = s (cid:48) we have sine-Gordon part of theHamiltonian H bs | ss (cid:48) int = V k F π a (cid:90) dx cos[2 ϕ ↑ ( x ) − ϕ ↓ ( x )] (142)A little more careful consideration is required for thecase when s = s (cid:48) . In this case one should expand thefields, which are in fact non-commuting operators. Todo the expansion procedure safely the operator has tobe normal-ordered: exp[ iϕ s ( x )] = (cid:104) exp[ iϕ s ( x )] (cid:105) : exp[ iϕ s ( x )] := exp[ −(cid:104) ϕ s ( x ) (cid:105) /
2] : exp[ iϕ s ( x )] : (143)such that one may apply usual Taylor series for theoperator under the normal-ordered sign : ( . . . ) : . Thebrackets (cid:104) . . . (cid:105) imply quantum averaging and to the low-est order in interaction (cid:104) ( ϕ s ( x ) − ϕ s ( x (cid:48) )) (cid:105) = ln (cid:20) | x − x (cid:48) | a (cid:21) . (144)With this formalism at hand we have H bs | ss int ≈ π a (cid:88) s (cid:90) dxdyV ( y ) (cid:104) e − ik F y e − | y | a × (cid:18) i Φ s ( x, y ) + (2 i ) s ( x, y )+ (2 i ) s ( x, y ) + (2 i )
24 Φ s ( x, y ) (cid:19) + h.c. (cid:21) (145)where Φ s ( x, y ) = ϕ s ( x + y/ − ϕ s ( x − y/ ≈ y∂ x ϕ s ( x ) and we carried gradient expansion to the lowest order.By neglecting now constant and full derivative termsand noticing that exp[ − | y | /a )] = ( a/y ) , that can-cels cut-off dependent prefactor, one finds H bs | ss int ≈ π (cid:88) s (cid:90) dxdyV ( y ) (cid:2) − k F y )( ∂ x ϕ s ) − y sin(2 k F y )( ∂ x ϕ s ) + 43 y cos(2 k F y )( ∂ x ϕ s ) (cid:21) . (146) After the integration by parts above expression reducesto the form H bs| ss int = − V k F π (cid:88) s (cid:90) dx ( ∂ x ϕ s ) + V (cid:48) k F π (cid:88) s (cid:90) dx ( ∂ x ϕ s ) − V (cid:48)(cid:48) k F π (cid:88) s (cid:90) dx ( ∂ x ϕ s ) . (147)The first term is conventional for the bosonization tech-nique while the last two are new additions responsiblefor the interaction of bosons. At this point we per-form transformation to the spin-charge representationfor the boson fields: ϕ ρ = 1 √ ϕ ↑ + ϕ ↓ ) , ϕ σ = 1 √ ϕ ↑ − ϕ ↓ ) , (148)and similar for the ϑ -field. The final result we split intofive parts H = H + H bc + H bs + H + H sg . (149)The quadratic part H = v ρ π (cid:90) dx (cid:20) K ρ ( ∂ x ϕ ρ ) + K ρ ( ∂ x ϑ ρ ) (cid:21) + v σ π (cid:90) dx (cid:20) K σ ( ∂ x ϕ σ ) + K σ ( ∂ x ϑ σ ) (cid:21) (150)corresponds to the usual linear Luttinger liquid modelof spin-charge separation explicit. The boson velocitiesand Luttinger liquid interaction constants are lockedby relations v ρ K ρ = v σ K σ = v F , and at the level ofperturbation theory K ρ = 1 − (2 V − V k F ) / πv F and K σ = 1 + V k F / πv F . The sine-Gordon term is alsobelongs to the linear Luttinger liquid theory H sg = V k F π a (cid:90) dx cos[2 √ ϕ σ ] , (151)The cubic order coupling terms can be split into twogroups. First group is due to band curvature H bc = − √ πm (cid:90) dx (cid:2) ( ∂ x ϕ ρ ) + 3( ∂ x ϕ ρ )( ∂ x ϕ σ ) + 3( ∂ x ϕ ρ )( ∂ x ϑ ρ ) +3( ∂ x ϕ ρ )( ∂ x ϑ σ ) + 6( ∂ x ϑ ρ )( ∂ x ϕ σ )( ∂ x ϑ σ ) (cid:3) (152)The second group is due to backscattering H bs = V (cid:48) k F √ π (cid:90) dx (cid:2) ( ∂ x ϕ ρ ) + 3( ∂ x ϕ ρ )( ∂ x ϕ σ ) (cid:3) . (153) ЭТФ
Kinetic processes in Fermi-Luttinger liquids
These terms couple spin and charge excitations andlead to ρ → σσ decay processes that we discussed inthe context of spin-charge drag equilibration rates. Thequartic order terms H = − V (cid:48)(cid:48) k F π (cid:90) dx (cid:2) ( ∂ x ϕ ρ ) + 6( ∂ x ϕ ρ ) ( ∂ x ϕ σ ) + ( ∂ x ϕ σ ) (cid:3) (154)lead to ρρ , ρσ , and σσ type boson scattering. Finally,to obtain kinetic equations for bosons we use canonicaloscillator representation in normal modes ∂ x ϕ ρ/σ = − (cid:88) q (cid:114) π | q | L e − iqx (cid:104) b † ρ/σ ( q ) + b ρ/σ ( − q ) (cid:105) ,∂ x ϑ ρ/σ = (cid:88) q (cid:114) π | q | L sgn ( q ) e − iqx (cid:104) b † ρ/σ ( q ) − b ρ/σ ( − q ) (cid:105) , (155)written in terms of creation and annihilation operators.It should be also borne in mind that fields ϕ ρ/σ ( x ) and ϑ ρ/σ ( x ) contain topological terms, N R ± N L , which areimportant in defining the momentum operator [13]. B. MOBILE IMPURITY MODEL
The purpose of this section is to illustrate a connec-tion between calculation of quasiparticle decay rates infermions via three-particle collisions and in bosons vianonlinear Luttinger liquid approach. For simplicity wecondense this discussion to the spinless case. Here weessentially follow the framework developed in Ref. [65]with an extension to include an additional interactionterm known from the context of impurity dynamics inLuttinger liquid [64] that enables a decay processes.The technical essence of the method can be sum-marized as follows. Starting from the initial fermionicmodel one introduces not only conventional low-energy sub-bands ψ R ( L ) at ± k F for right-moversand left-movers, but also the sub-band modes d around the momentum k of the high-energy par-ticle (or hole) whose energy defines the threshold.In this approach, the fermion operator is split as ψ ( x ) ∼ e ik F x ψ R ( x ) + e − k F x ψ L ( x ) + e ikx d ( x ) in whichthe high-energy particle acts as a mobile impurity cou-pled to the Luttiger liquid modes. The Hamiltonian for this model reads [65] H = H + H d + H int , (156a) H = v π (cid:90) dx (cid:2) K ( ∂ x ϑ ) + K − ( ∂ x ϕ ) (cid:3) , (156b) H d = (cid:90) dx d † ( x ) [ ε ( k ) − iv d ∂ x ] d ( x ) , (156c) H int = 12 π (cid:90) dx [( V R − V L ) ∂ x ϑ − ( V R + V L ) ∂ x ϕ ] ρ d . (156d)Here operator d ( x ) creates a mobile particle of momen-tum k and velocity v d = ∂ε/∂k , and ρ d ( x ) = d † ( x ) d ( x ) is the fermion density operator. In this treatment thecurvature of the fermion dispersion was kept explicit.When applied to the calculation of the spectral func-tion, this model captures power-law threshold singu-larities beyond the limit of linear Luttinger liquid the-ory and yields the universal description. However, thismodel does not yet capture relaxation processes as theinteraction term couples mobile particle either to theleft-movers or to the right-movers separately. At thelevel of fermionic description of the problem, we sawthat finite decay rate is generated by RRL process thatinvolves particle and two particle-hole pairs. This sug-gests that we need another coupling term of d -particlewith both left- and right-movers. We thus add H (cid:48) int = γ (cid:90) dx ρ d ( ∂ x ϕ R )( ∂ x ϕ L ) (157)which is inspired by Ref. [64] where friction of a heavyparticle moving through the Luttinger liquid was con-sidered. An estimate for the coupling constant wasgiven γ ∼ V /ε F which is qualitatively consistent withthree-particle scattering process.Let us return now to the effective hamilto-nian Eq. (156) and look for a single high-energy particle. From H d the time-ordered(retarded) free propagator of d -electron is G ret ( x, t ) = (cid:104) T d † ( x, t ) d (0 , (cid:105) = θ ( t ) e iεt δ ( x − v d t ) which simply describes ballistically propagatingparticle. The Fourier transform of the latter is G ret ( k, ω ) = ( ω − ε − kv d + iα ) − . The idea now isto apply perturbation theory in H (cid:48) int to determine theself-energy for the d -particle induced by collisions withright-movers and left-movers.The Dyson equation for the d -electron gives dressedpropagator G ret ( k, ω ) = 1 ω − ε − kv d − Σ ret ( k, ω ) (158) lex Levchenko, Tobias Micklitz ЖЭТФwhere self-energy appears to the second order in γ Σ ret = − iγ (cid:90) dxdte − ikx + iωt Π R ( x, t )Π L ( x, t ) G ret ( x, t ) (159)where free propagator for bosonic fields is Π R ( L ) ( x, t ) = (cid:104) ∂ x ϕ R ( L ) ( x, t ) ∂ x ϕ R ( L ) (0 , (cid:105) = − π ( x ∓ vt ± iα ) . (160)A slight comment here that following Ref. [65] it is agood idea to rescale bosonic fields first ϕ → √ Kϕ and ϑ → ϑ/ √ K , which removes LL interaction parameter K from H but renormalizes coefficients in H (cid:48) int . Wereabsorbed all factors in the redefinition of γ and so Π R ( L ) ( x, t ) is written above for the rescaled fields. Theparticle life-time is determined by the imaginary partof the self-energy, so that we look at the latter ImΣ ret = − γ π ∞ (cid:90) −∞ dt e i ( ω − ε − kv d ) t [( v d − v ) t + iα ] [( v d + v ) t − iα ] (161)Notice here that by assumption v d > v so that poles ofthe integrand are in the different parts of the complexplane. At the mass-shell we find d -particle relaxationrate τ − d = ImΣ ret ( k, ω = ε + v d k ) = γ ( v d − v )8 πv d α (162)We arrive at the finite result which, however, de-pends on the cut-off parameter α . To resolve this is-sue we appeal to the fact that for the model to bewell-defined there has to be a clear energy separa-tion between the sub-bands of d -particle and bosonizedright- and left-movers. 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