Interaction effects on thermal transport in quantum wires
Alex Levchenko, Tobias Micklitz, Zoran Ristivojevic, K. A. Matveev
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Interaction effects on thermal transport in quantum wires
Alex Levchenko,
1, 2
Tobias Micklitz, Zoran Ristivojevic, and K. A. Matveev Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA Dahlem Center for Complex Quantum Systems and Institut f¨ur Theoretische Physik,Freie Universit¨at Berlin, 14195 Berlin, Germany Laboratoire de Physique Th´eorique-CNRS, Ecole Normale Sup´erieure, 24 rue Lhomond, 75005 Paris, France (Dated: September 16, 2011)We develop a theory of thermal transport of weakly interacting electrons in quantum wires.Unlike higher-dimensional systems, a one-dimensional electron gas requires three-particle collisionsfor energy relaxation. The fastest relaxation is provided by the intrabranch scattering of comovingelectrons which establishes a partially equilibrated form of the distribution function. The thermalconductance is governed by the slower interbranch processes which enable energy exchange betweencounterpropagating particles. We derive an analytic expression for the thermal conductance ofinteracting electrons valid for arbitrary relation between the wire length and electron thermalizationlength. We find that in sufficiently long wires the interaction-induced correction to the thermalconductance saturates to an interaction-independent value.
PACS numbers: 72.10.-d, 71.10.Pm, 72.15.Lh
I. INTRODUCTION
The classical Drude theory of electronic transport pro-vides universal relation between electric and thermaltransport coefficients known as the Wiedemann-Franzlaw, K = π T e G . (1)Here K and G are, respectively, the thermal and elec-tric conductances, T is the temperature in energy units( k B = 1), and e is the electron charge. The relationrepresented by Eq. (1) is a natural one since for non-interacting particles both charge and energy are carriedby the electronic excitations. Furthermore, as long aselastic collisions govern the transport, the validity ofthe Wiedemann-Franz law has been confirmed in thecase of arbitrary impurity scattering. However, an ac-count of the electron-electron interaction effects withinthe Fermi-liquid theory gives corrections to both G and K . These lead to a deviation from the Wiedemann-Franz law [Eq. (1)], which is associated with the inelasticforward scattering of electrons. In general, violation ofthe Wiedemann-Franz law is a hallmark of electron in-teraction effects and thus is of conceptual interest.In one-dimensional conductors, such as quantum wiresor quantum Hall edge states, the electron system canno longer be described as a Fermi liquid but instead isexpected to form a Luttinger liquid. It has been shownthat for the perfect Luttinger liquid conductor, such as animpurity-free single-channel quantum wire, interactionsinside the wire neither affect conductance quantization G = 2 e h , (2) nor change the thermal conductance of the system K = 2 π T h . (3)Since both G and K remain the same as in the case ofnoninteracting electrons, the Wiedemann-Franz law (1)holds for an ideal Luttinger liquid conductor.There are two important exceptions known in the lit-erature. The first one is a Luttinger liquid with an impu-rity studied by Kane and Fisher. In that case electronbackscattering takes place which strongly renormalizesboth G and K , such that the Wiedemann-Franz law isviolated. The second case is the Luttinger liquid withlong-range inhomogeneities studied by Fazio et al . If thespatial variations related to these inhomogeneities occuron a length scale much larger than the Fermi wavelength,electrons will not suffer any backscattering. The electricconductance will, therefore, be given by its noninteract-ing value [Eq. (2)]. At the same time the thermal con-ductance K will be altered by interactions. The reasonfor this is as follows. For the system with broken trans-lational invariance, momentum is not conserved. As aresult, there are allowed certain pair collisions which con-serve the number of right and left movers independentlybut provide energy exchange between them. These areprecisely the scattering processes that thermalize elec-trons and thus lead to the violation of the Wiedemann-Franz law.Recent advances in the fabrication of tunable con-strictions in high-mobility two-dimensional electron gaseshave allowed precise and sensitive thermal measurementsin clean one-dimensional systems. These include ex-periments on the thermal transport of single-channelquantum wires, where a lower value of the thermalconductance than that predicted by the Wiedemann-Franz law was observed at the plateau of the elec-trical conductance. Another set of experiments re-ported enhanced thermopower in low-density quantumwires and quantum Hall edges. Remarkable ex-periments based on momentum-resolved tunneling spec-troscopy provided direct evidence for the electronic ther-malization in one-dimensional systems.
Clearly in-teraction effects are responsible for the observed features;however, Luttinger liquid theory does not provide an ad-equate description for these observations.Ongoing theoretical efforts in the study of one-dimensional electron systems focus on nonequilibriumdynamics and consequences of the nonlinear disper-sion in transport that lie beyond the scope of con-ventional Luttinger liquid paradigm. The kinetics of one-dimensional electrons with nonlinear dispersion is pecu-liar. Indeed, constraints imposed by the momentum andenergy conservation allow either zero-momentum trans-fer or exchange of momenta for pair collisions. Neitherprocess changes the electronic distribution function, andas a result they have no effect on transport coefficientsand relaxation. Therefore, three-particle collisions playa central role. In this paper we study the fate of the Wiedemann-Franz law and the origin of the electron energy relaxationin clean single-channel quantum wires, accounting for thescattering processes that involve three-particle collisions.The paper is organized as follows. In the next section,Sec. II, we place our work into the context of recent stud-ies on equilibration in quantum wires and explain theconcept of partially equilibrated electron liquids, whichis central for our study. We then develop in Sec. IIIthe theory of the thermal transport in one-dimensionalelectron liquids based on the Boltzmann equation withthree-particle collisions included via the correspondingcollision integral and scattering rate. We elucidate thescattering processes involved, discuss the role of the spinand limitations of our theory, and summarize our findingsin Sec. IV. Additional comments and directions for futurework are presented in Sec. V. Various technical aspectsof our calculations are relegated to Appendixes (A)-(C).
II. PARTIALLY EQUILIBRATEDONE-DIMENSIONAL ELECTRONSA. Noninteracting electrons
Noninteracting electrons propagate ballisticallythrough the wire. They keep a memory of the leadthey originated from and remain in equilibrium withthe corresponding lead electrons. If voltage ( V ) and/ortemperature differences (∆ T ) are applied across thewire, right- and left-moving particles are at differentequilibria, and the distribution function takes the form f p = θ ( p ) e εp − µlTl + 1 + θ ( − p ) e εp − µrTr + 1 , (4)where ε p = p / m is the energy of an electron withmomentum p and θ ( p ) is the unit step function. T l = T l T r m l m r L N R . Q R . FIG. 1: (Color online) Schematic picture of a quantum wireof length L . Electrons in the left and right leads are describedby Fermi distribution functions characterized by temperatures T l ( r ) and chemical potentials µ l ( r ) . Due to three-particle col-lisions electrons may backscatter and also exchange energybetween the subsystems of warmer right movers and colderleft movers. T + ∆ T / T r = T − ∆ T /
2, and µ l = µ + eV / µ r = µ − eV / V and ∆ T , one readily finds the electricand heat currents I = G V | ∆ T =0 and I Q = K ∆ T | I =0 ,with conductances G and K , which coincide with theearlier stated noninteracting values, Eqs. (2) and (3). In the presence of weak interactions the distribution(4) describes an out-of-equilibrium situation. Collisionslead to electrons exchanging energy and momentum, withsome particles experiencing backscattering. As a result,net particle ( ˙ N R ) and heat ( ˙ Q R ) currents flow betweenthe subsystems of right- and left-moving electrons, relax-ing V and ∆ T (see Fig. 1 for a schematic illustration).The effect of electron-electron collisions on the distribu-tion function depends strongly on the length of the wire.Short wires are traversed by the electrons relatively fast,leaving interactions only little time to change the distri-bution in Eq. (4) considerably. In the limit of a very longwire, on the other hand, one should expect full equilibra-tion of left- and right-moving electrons into a single dis-tribution, even in the case of weak interactions. It turnsout that there exists a hierarchy of three-particle scatter-ing processes, classified by the corresponding relaxationrates or, equivalently, inelastic scattering lengths, whichhave different effects on the electron distribution functionEq. (4). B. Partially equilibrated electrons
We start our discussion of the different scattering pro-cesses involved in the electronic relaxation with the pro-cess shown in Fig. 2(a). This three-particle collisionprovides intra -branch relaxation within the subsystemof right-moving electrons. A similar relaxation processfor the left movers is not shown in the figure but is im-plicit. This process can be described by the correspond-ing inelastic scattering length, which we denote in thefollowing as ℓ a . The precise form of ℓ a and its temper-ature dependence are model specific. They are deter-mined by the scattering amplitude for the given inter-action potential and the phase space available for thisscattering [Fig. 2(a)] to occur. Quite generally one mayargue that ℓ a scales as a power of T . Indeed, at low tem-peratures, T ≪ µ , all participating scattering states arelocated within the energy strip ∼ T near the Fermi level.What is important for the present discussion is that forwires with length L ≫ ℓ a intrabranch electron collisions[Fig. 2(a)] become so efficient that the initial distribu-tion function Eq. (4) will be modified by interactions.One can find the resulting distribution by employing thefollowing observation. Intrabranch collisions conserve independently six quantities. These are the number ofright and left movers, N R/L = P p ≷ f p , their momenta, P R/L = P p ≷ pf p , and energies, E R/L = P p ≷ ε p f p .The form of the resulting electron distribution function f p can be obtained from a general statistical mechan-ics argument by maximizing the entropy of electrons, S = − P p [ f p ln f p + (1 − f p ) ln(1 − f p )], under the con-straint of the conserved quantities f p = θ ( p ) e εp − puR − µR T R + 1 + θ ( − p ) e εp − puL − µL T L + 1 . (5)This distribution is characterized by six unknown param-eters (Lagrange multipliers) which have a transparentphysical interpretation. Indeed, in Eq. (5) T R/L are ef-fective electron temperatures for right and left moversdifferent from those in the leads. The parameters u R/L have dimension of velocity and account for the conserva-tion of momentum in electron collisions. Finally, µ L/R are unequilibrated chemical potentials of left- and right-moving particles. In principle, all these parameters maydepend on the position along the wire.For longer wires interbranch three-particle collisions[see Fig. 2(b)] become progressively more important.Unlike intrabranch relaxation these processes allow en-ergy and momentum exchange between the subsystemsof right and left movers; thus P R/L and E R/L are nolonger independently conserved. However, the full mo-mentum P = P R + P L and energy E = E R + E L areobviously conserved. Corresponding to Fig. 2(b), thescattering length ℓ b is model specific and calculated inAppendix C. We note here that for all interaction po-tentials we studied ℓ b /ℓ a ∼ µ/T ≫
1. In view of thisdistinct length scale separation, ℓ b ≫ ℓ a , the electrondistribution in Eq. (5) is established at the first stage ofthe thermalization process. However, for longer wires, L ≫ ℓ b , relaxation of counterpropagating electrons be-comes so efficient that the temperatures T R/L and boostvelocities u R/L of right and left movers become equal, T R = T L = T and u R = u L = u , due to energy andmomentum exchange. At the same time, the chemicalpotentials µ R/L are still unequilibrated in this regime,∆ µ = µ R − µ L = 0, since the numbers of right- and left-moving electrons are still independently conserved. Asa result, for wires with length L ≫ ℓ b the distribution p pp m mme p e p e p (c) (d)(b) p e p (a) m FIG. 2: (Color online) (a) Intra-branch relaxation of co-moving electrons that establishes the partially equilibratedform of the distribution function in Eq. (5). (b) Dominant in-terbranch three-particle process for the energy exchange ˙ Q R between counterpropagating electrons that contributes to thethermal conductance correction. (c) Leading three-particlecollision which results in a finite rate ˙ N R ∝ e − µ/T and thusa temperature-dependent correction to the conductance of ashort wire (Ref. 19). (d) Equilibration mechanism: multistepdiffusion through the bottom of the band of an electron fromthe right to the left Fermi point accompanied by the excita-tion of many electron-hole pairs (Refs. 20–22). function (5) transforms into f p = θ ( p ) e εp − pu − µR T + 1 + θ ( − p ) e εp − pu − µL T + 1 . (6)Because ∆ µ = 0 we refer to the states of electron systemdescribed by the distributions in Eqs. (5) and (6) as thestates of partial equilibration. C. Fully equilibrated electrons
Energy and momentum conservation allow for the scat-tering process in which an electron at the bottom of theband is backscattered by two other particles near theFermi level [see Fig. 2(c)]. This is the basic three-particleprocess that changes the numbers of right and left moversbefore and after collision. In particular, the exponentiallysmall discontinuity of the distributions Eqs. (5) and (6)at p = 0 will be smeared by collisions of this type.Complete equilibration of electrons, namely, relaxationof ∆ µ , relies on the electron backscattering from the rightto the left Fermi point. One should notice here that itis impossible to realize such scattering directly since itrequires momentum transfer of 2 p F while Fermi blockingrestricts typical momentum exchange in the collision to δp ∼ T /v F ≪ p F . As a consequence, complete electronbackscattering, and thus relaxation of the chemical po-tential difference ∆ µ , occurs via a large number of smallsteps δp in momentum space such that δp ≪ p F . In itspassage between the subsystems of right and left moversthe backscattered electron has to pass the bottleneck ofoccupied states at the bottom of the band [see Fig. 2(d)].As a result, backscattering of electrons is exponentiallysuppressed by the probability ∼ e − µ/T to find an un-occupied state at the bottom of the band, and thus theequilibration length ℓ eq for the relaxation of the differencein chemical potentials of left and right movers is expo-nentially large, ℓ eq ∝ e µ/T . For sufficiently long wires, L ≫ ℓ eq , the state of full equilibration is achieved anddescribed by the distribution f p = 1 e εp − vdp − µeq T + 1 , (7)where the chemical potential µ eq inside the equilibratedwire is, in general, different from µ l ( r ) in the leads. InEq. (7) v d = I/ne is the electron drift velocity, where I is the electric current and n is the electron density.The partially equilibrated distribution function given byEq. (6) smoothly interpolates to the state of full equi-libration, Eq. (7), when the length of the wire exceedsthe equilibration length ℓ eq . The fully equilibrated dis-tribution Eq. (7) is obtained from Eq. (6) by setting µ R = µ L = µ eq ; thus ∆ µ = 0, and also u = v d . D. Brief summary
The regime of partial equilibration described by thedistribution function in Eq. (6) covers a wide range oflengths, ℓ a ≪ L . ℓ eq . It is more likely to be realized inexperiments than the fully equilibrated regime (7), as thelength scale ℓ eq is exponentially large. Depending on thewire length L a particular state of the electron systemis characterized by the extent to which the difference inchemical potentials ∆ µ and temperatures ∆ T of left andright movers has relaxed. The recent works Refs. 21,22addressed transport properties of wires with lengths inthe range ℓ a ≪ ℓ b ≪ L ∼ ℓ eq , (8)which covers the crossover from the partially equilibratedregime Eq. (6) to the fully equilibrated regime Eq. (7).The major emphasis of these works was on the effect ofequilibration due to electron backscattering [Fig. 2(d)].The main focus of the present paper is on the transportproperties of partially equilibrated wires with length ℓ a ≪ L ∼ ℓ b ≪ ℓ eq . (9)In this regime the numbers N L and N R of the left- andright-moving electrons are individually conserved up to corrections as small as e − µ/T , so that electrons with en-ergies near the Fermi level pass through the wire with-out backscattering [Fig. 2(c)]. This automatically impliesthat the conductance G unchaged intact by interactionsand is still given by Eq. (2). However, electrons will expe-rience other multiple three-particle collisions [Fig. 2(b)],which allow momentum and energy exchange within andbetween the two branches of the spectrum, thus alteringthermal transport properties. The role of these processeswas not explored in previous studies devoted to the trans-port in partially equilibrated quantum wires. III. BOLTZMANN EQUATION FORMALISMA. Three-particle collision integral
Consider a quantum wire of length L , connected byideal reflectionless contacts to noninteracting leads whichare biased by a temperature difference ∆ T (see Fig. 1).In the following, we are interested only in the thermaltransport properties of the wire and assume that thereis no external voltage bias, V = 0. We describe weaklyinteracting one-dimensional electrons in the frameworkof the Boltzmann kinetic equation v p ∂ x f ( p, x ) = I{ f ( p, x ) } , (10)where v p = p/m is the electron velocity and the evolutionof the distribution function is governed by the collision in-tegral I{ f ( p, x ) } . We consider the steady-state setup inwhich the distribution function does not depend explic-itly on time. The collision integral, in general, is a nonlin-ear functional of f ( p, x ), whose form is determined by thescattering processes affecting the distribution function.As discussed above, in our case the dominant processesare three-particle collisions. Assuming that the collisionintegral is local in space, we have I{ f } = − X p ,p p ′ ,p ′ ,p ′ W ′ ′ ′ × [ f f f (1 − f ′ )(1 − f ′ )(1 − f ′ ) − f ′ f ′ f ′ (1 − f )(1 − f )(1 − f )] , (11)where W ′ ′ ′ is the scattering rate from the incomingstates { p , p , p } into the outgoing states { p ′ , p ′ , p ′ } ,and we used the shorthand notation f i = f ( p i , x ). TheBoltzmann equation [Eq. (10)] is supplemented by theboundary conditions stating that the distribution f ( p, p >
0) at the left end of thewire and f ( p, L ) of left-moving electrons ( p <
0) at theright end coincide with the distribution function in theleads, Eq. (4). We note here that although Eq. (11)is written for the spinless case our subsequent analy-sis and solution of the Boltzmann equation presented inSec. III B–III D is applicable to the spinful electrons aswell.An exact analytical solution of the Boltzmann equa-tion [Eq. (10)] is, in general, very difficult to find dueto the nonlinearity of the collision integral Eq. (11). Asimplification is, however, possible in the case of a linearresponse analysis in the externally applied perturbation(in our case, the temperature difference ∆ T ). Then thecollision integral can be linearized near its unperturbedvalue. It is convenient to present f ( p, x ) as f ( p, x ) = f p + f p (1 − f p ) ψ ( p, x ) , (12)where f p = ( e ( ε p − µ ) /T + 1) − is the equilibrium Fermidistribution function and ψ ( p, x ) ∝ ∆ T . When lineariz-ing Eq. (11) with respect to ψ ( p, x ) the factor f p (1 − f p )in Eq. (12) makes it convenient to use the detailed bal-ance condition f p f p f p (1 − f p ′ )(1 − f p ′ )(1 − f p ′ ) = f p ′ f p ′ f p ′ (1 − f p )(1 − f p )(1 − f p ) , (13)valid at ε p + ε p + ε p = ε p ′ + ε p ′ + ε p ′ . SubstitutingEq. (12) into the collision integral and using Eq. (13) onearrives at the linearized version of Eq. (11) I{ ψ ( p , x ) } = − X p ,p p ′ ,p ′ ,p ′ K ′ ′ ′ × [ ψ ( p , x ) + ψ ( p , x ) + ψ ( p , x ) − ψ ( p ′ , x ) − ψ ( p ′ , x ) − ψ ( p ′ , x )] , (14)with the kernel K defined as K ′ ′ ′ = W ′ ′ ′ f p f p f p × (1 − f p ′ )(1 − f p ′ )(1 − f p ′ ) . (15)The explicit form of the scattering rate W ′ ′ ′ is notimportant for the following discussion. It is discussed indetail in Appendix B. B. Solution strategy
Even after linearization the solution of the integralBoltzmann equation that satisfies given boundary con-ditions is still a complicated problem. However, our taskis simplified greatly since we already know the structureof the distribution function f ( p, x ). Indeed, we have dis-cussed in Sec. II that for wires with length ℓ a ≪ L ∼ ℓ b the electron system is in the regime of partial equili-bration with the distribution function given by Eq. (5).Thus, the class of functions we need to consider to solveour boundary problem is, in fact, rather narrow. The so-lution we are seeking is conveniently parametrized by sixunknowns: µ R ( L ) ( x ), u R ( L ) ( x ), and T R ( L ) ( x ). Insteadof solving one integro-differential equation for f ( p, x ) wewill reduce our task to solving a system of six linear ordi-nary differential equations that govern the spatial evolu-tion of the parameters defining the distribution function in Eq. (5). This is possible since the momentum depen-dence of the distribution function is fully determined byour ansatz (5), which allows completion of all p integra-tions in the Boltzmann equation analytically. Among thesix equations we need, four represent conservation laws:conservation of numbers of right- and left-moving elec-trons N R/L , of total momentum P , and of energy E ofthe electron system. The other two are kinetic equationsthat account for the momentum and energy exchange be-tween subsystems of right and left movers, thus capturingthe processes of thermalization.For the following analysis it is convenient to measurethe chemical potentials and temperatures from their equi-librium values: µ R ( L ) ( x ) = µ + δµ R ( L ) ( x ) and T R ( L ) ( x ) = T + δ T R ( L ) ( x ). Expanding now Eq. (5) to the linear or-der in u R ( L ) , δµ R ( L ) and δ T R ( L ) and using Eq. (12) wecan identify ψ ( p, x ) that enters the collision integral inEq. (14) as ψ ( p, x ) = ψ R ( p, x ) + ψ L ( p, x ) (16)where ψ R ( L ) ( p, x ) = θ ( ± p ) × h ψ R ( L ) µ ( p, x ) + ψ R ( L ) u ( p, x ) + ψ R ( L ) T ( p, x ) i . (17)Here the three contributions are ψ R ( L ) µ ( p, x ) = δµ R ( L ) ( x ) T , (18) ψ R ( L ) u ( p, x ) = pu R ( L ) ( x ) T , (19) ψ R ( L ) T ( p, x ) = ( ε p − µ ) δ T R ( L ) ( x ) T . (20)These functions evolve in the real space as prescribed bythe collision integral Eq. (14) while their boundary valuescan be extracted from the respective distributions in theleads [Eq. (4)]. Indeed, expanding Eq. (4) with V = 0one obtains f p = f p + ( ε p − µ )∆ T T f p (1 − f p )[ θ ( p ) − θ ( − p )] . (21)By matching this result to Eq. (12) with ψ ( p, x ) takenfrom Eqs. (16) and (17) we deduce the boundary condi-tions δµ R (0) = δµ L ( L ) = 0 , (22) u R (0) = u L ( L ) = 0 , (23) δ T R (0) = − δ T L ( L ) = ∆ T / . (24)Our task now is to derive the set of coupled ordinarydifferential equations that govern the spatial evolutionof the unknown parameters u R ( L ) ( x ), δµ R ( L ) ( x ), and δ T R ( L ) ( x ). This will give us complete knowledge of theelectron distribution function. Knowing all parametersin Eq. (5) we will be able to find the heat current andfinally the thermal conductance of the system. Before werealize this plan, the conservation laws must be discussed. C. Transport currents and conservation laws
Conservation of the total number of particles impliesthat in a steady state the particle current I ( x ) is uni-form along the wire. Correspondingly, we infer from theconservation of total momentum P and total energy E that in the steady state a constant momentum current I P and a constant energy current I E flow through thesystem. In the following it will be convenient to expressthese currents as the sums of individual contributions ofthe left- and right-moving electrons, e.g., I = I R + I L ,thus introducing: I R ( L ) ( x ) = Z + ∞−∞ dph θ ( ± p ) v p f ( p, x ) , (25) I R ( L ) P ( x ) = Z + ∞−∞ dph θ ( ± p ) pv p f ( p, x ) , (26) I R ( L ) E ( x ) = Z + ∞−∞ dph θ ( ± p ) ε p v p f ( p, x ) . (27)The positive sign in the step function corresponds to rightmovers and the negative one to left movers. Since weneglect small backscattering effects, the numbers of right-and left-moving electrons are conserved independently:˙ N R/L = 0 . (28)It follows then immediately from the continuity equationsthat particle currents are uniform along the wire, ∂ x I R ( x ) = 0 , ∂ x I L ( x ) = 0 . (29)Similarly we present the conservation of total momentumand energy, ∂ x (cid:2) I RP ( x ) + I LP ( x ) (cid:3) = 0 , (30) ∂ x (cid:2) I RE ( x ) + I LE ( x ) (cid:3) = 0 . (31)As the next step we express the currents (25)–(27) interms of the parameters defining the electron distributionfunction. Specifically, we use Eq. (12) and Eqs. (17)–(20)together with the current definition in Eq. (25) and thusfind from Eq. (29) dδµ R ( L ) dx = ∓ p F du R ( L ) dx (cid:18) − π T µ − π T µ (cid:19) . (32)When deriving this equation from Eq. (25) we had tocarry out a Sommerfeld expansion up to the fourth orderin T /µ ≪
1. Note here that even though backscatteringis neglected δµ R ( L ) must change in space to accommo-date the conservation laws for currents [Eqs. (29)–(31)],and µ R ( L ) ( x ) = µ + δµ R ( L ) ( x ) coincide with µ r ( l ) = µ only at the ends of the wire [see the boundary conditionsEq. (22)].We then perform a similar calculation for the momen-tum and energy currents. From the momentum conser- vation Eq. (30), we find p F T µ (cid:18) π T µ (cid:19) (cid:18) du R dx − du L dx (cid:19) + (cid:18) π T µ (cid:19) (cid:18) dδ T R dx + dδ T L dx (cid:19) = 0 , (33)while from the energy conservation, Eq. (31), p F T µ (cid:18) π T µ (cid:19) (cid:18) du R dx + du L dx (cid:19) + (cid:18) dδ T R dx − dδ T L dx (cid:19) = 0 . (34)When deriving these two equations we also made use ofEq. (32) to exclude the chemical potentials of the rightand left movers. D. Scattering processes and kinetic equations
Although the total momentum and energy currents I P and I E are conserved, such currents taken for the leftand right movers separately, I R/LP and I R/LE , are not. In-deed, the three-particle collisions shown in Fig. 2(b) in-duce momentum and energy exchange between counter-propagating electrons. Let us focus on a small segmentof the wire between the positions x and x + ∆ x , where0 < x < L . The difference I R/LP ( x + ∆ x ) − I R/LP ( x ) isequal to the rate of change of the momentum of right-moving electrons ˙ P R/L = ˙ p R/L ∆ x . Here ˙ p R/L is the rateper unit of length. As a result, the continuity equationfor the momentum exchange reads ∂ x (cid:2) I RP ( x ) − I LP ( x ) (cid:3) = 2 ˙ p R . (35)Here we used ˙ p L = − ˙ p R , which is ensured by theconservation of total momentum. In complete anal-ogy we can now relate the difference of energy currents I RE ( x +∆ x ) − I RE ( x ) to the corresponding energy exchangerate ˙ E R = ˙ e R ∆ x , which gives us ∂ x (cid:2) I RE ( x ) − I RE ( x ) (cid:3) = 2 ˙ e R , (36)where we also used ˙ e L = − ˙ e R guaranteed by the energyconservation. The right-hand side of Eqs. (35) and (36)can be calculated from the collision integral of the Boltz-mann equation [Eq. (14)].There are two basic processes which contribute to ˙ p R and ˙ e R . The first one includes two right movers that scat-ter off one left mover. This process is shown in Fig. 2(d).The other process, when two left movers scatter off oneright mover, is equally important. Keeping both terms,using Eqs. (14) and (16) we find for the relaxation rates(details of the derivation are given in Appendix A)˙ p R = − k F p F τ u R − u L v F , (37)˙ e R = − k F µτ δ T R − δ T L T , (38)1 τ = 3 k F ∆ x X p > ,p > ,p < p ′ > ,p ′ > ,p ′ < v F ( p ′ − p ) µT K ′ ′ ′ . (39)Having determined relaxation the rates we now returnto Eqs. (35) and (36). Computing the momentum andenergy currents of right and left movers in the same wayas we did in the previous section from Eqs. (26)–(27) andusing the relaxation rates from Eqs. (37)–(38) we findtwo additional equations which describe the relaxationof momentum: p F π T µ (cid:18) du R dx + du L dx (cid:19)(cid:18) π T µ (cid:19) + π T µ (cid:18) dδ T R dx − dδ T R dx (cid:19)(cid:18) π T µ (cid:19) = − hk F τ u R − u L v F (40)and energy p F π T µ (cid:18) du R dx − du L dx (cid:19)(cid:18) π T µ (cid:19) + π T µ (cid:18) dδ T R dx + dδ T L dx (cid:19) = − hk F τ δ T R − δ T L T . (41)Equations (32)–(34) and (40)–(41), together with theboundary conditions Eqs. (22)–(24), represent theclosed system of six coupled differential equationswhose solution fully determines the six parameters µ R/L , u
R/L , T R/L that define the electron distributionfunction (5). We now find these parameters explicitly.For that purpose let us introduce dimensionless variables η ± = u R ± u L v F , θ ± = δ T R ± δ T L T , (42)and the microscopic scattering length ℓ b = π T µ ( v F τ ) . (43)Calculation of ℓ b requires a detailed knowledge of thescattering rate, implicit in the kernel K ′ ′ ′ , as a functionof momenta transferred in a collision. In Appendix B weprovide this information for the case of the three-particlecollisions under consideration and in Appendix C find ℓ b explicitly for the spinless and spinful cases.After some algebra the coupled equations (33), (34) and (40), (41) can be reduced to the following form: ∂θ + ∂x = − α ∂η − ∂x , (44) ∂η + ∂x = − β ∂θ − ∂x , (45) ∂θ − ∂x = η − ℓ b , (46) ∂η − ∂x = θ − ℓ b , (47)which contain two dimensionless parameters α = 1 + 29 π T µ , β = 1 − π T µ , (48)where corrections of higher order in T /µ ≪ θ R ( x ) = ∆ T T α − e x/ℓ b + α + e ( L − x ) /ℓ b α − + α + e L/ℓ b , (49) θ L ( x ) = − ∆ T T α + e x/ℓ b + α − e ( L − x ) /ℓ b α − + α + e L/ℓ b , (50) η R ( x ) = ∆ T T β − ( e x/ℓ b − − β + ( e ( L − x ) /ℓ b − e L/ℓ b ) α − + α + e L/ℓ b , (51) η L ( x ) = − ∆ T T β + ( e x/ℓ b − e L/ℓ b ) − β − ( e ( L − x ) /ℓ b − α − + α + e L/ℓ b , (52)where α ± = 1 ± α , β ± = 1 ± β and θ R/L = ( θ + ± θ − ) / η R/L = ( η + ± η − ) /
2. This concludes our solution of theBoltzmann equation for three-particle collisions.
IV. HEAT CURRENT AND THERMALCONDUCTANCE
Complete knowledge of the distribution function (5)allows us to compute physical observables. Specifically,we are interested in the thermal conductance K . For thelatter we need to evaluate the heat current I Q ( x ) = I E ( x ) − µI ( x ) . (53)By using Eqs. (5) we carry out a Sommerfeld expan-sion for the particle and energy currents I and I E fromEqs. (25) and (27), up to the fourth order in T /µ ≪ I Q ( x ) = π T h (cid:20) η + ( x ) β + θ − ( x ) (cid:21) , (54)which is presented here in our notation defined inEq. (42). With the help of Eqs. (49)–(52) it can be read-ily checked that I Q is uniform along the wire. This fact is a priori expected and follows from the conservation laws,which we already explored above. By knowing I Q we canfinally find the thermal conductance K ( L ) = I Q / ∆ T asa function of the wire length, K ( L ) K = tanh( L/ ℓ b ) + βαβ tanh( L/ ℓ b ) + β . (55)This is the main result of our paper. Note here that K = π T / h for the case of spinless electrons, whereas K isgiven by Eq. (3) for electrons with spin. The functionalform of K ( L ) remains the same in both cases except forthe expressions for ℓ b , which we discuss below. Let usnow analyze limiting cases of Eq. (55) and discuss themicroscopic form of the scattering length ℓ b .Equation (55) interpolates smoothly between two dis-tinct limits. In short wires, L ≪ ℓ b , from the expansionof Eq. (55) one obtains for the interaction-induced correc-tion to thermal conductance, δK = K − K , the followingresult: δK ( L ) K = − π T µ Lℓ b , L ≪ ℓ b . (56)In such short wires electrons propagate from one leadto the other, rarely experiencing three-particle collisionsof the type shown in Fig. 2(b). Thus their distributionfunction is approximately determined by that in the leads[Eq. (4)]. Under this assumption one can adopt the strat-egy of Ref. 19, applied previously for the calculation ofconductance and thermopower in short wires, and treatthe collision integral of the Boltzmann equation pertur-batively, thus neglecting effects of thermalization on thedistribution function. Technically speaking, this corre-sponds to a lowest order iteration for the Boltzmannequation, which amounts to substituting distribution (4)into the collision integral (14) to calculate the correctionto I Q . This perturbative procedure immediately repro-duces Eq. (56).It is physically expected that in longer wires particlecollisions Fig. 2(b) should have a much more dramatic ef-fect on the distribution function and thus thermal trans-port. Indeed, once full thermalization has been achievedfor L ≫ ℓ b we find from Eq. (55) that the correction tothermal conductance saturates: δK ( L ) K = − π T µ , ℓ b ≪ L ≪ ℓ eq . (57)One interesting aspect of Eq. (57) is that δK is indepen-dent of the interaction strength. It means that no matterhow weak the interactions are, for sufficiently long wiresthermalization between right- and left-moving electronsis eventually established, which leads to saturation of δK .The behavior of δK ( L ) as a function of the wire lengthis summarized Fig. 3. L { b - ∆ K K FIG. 3: (Color online) Interaction-induced correction to thethermal conductance of a clean quantum wire as a function ofits length plotted for different values of temperature (from thebottom to the top curve):
T /µ = 0 . , . , . , .
2. For L ≪ ℓ b the correction scales with L and saturates to a constantvalue ∝ ( T /µ ) once ℓ b ≪ L in accordance with Eqs. (56)and (57). The interaction strength, however, sets the length scale ℓ b at which thermalization occurs. Its actual dependenceon temperature is determined by the phase space avail-able for a three-particle collision to occur and by the de-pendence of the corresponding scattering amplitude onmomenta transferred in a collision. For spinless electronsand Coulomb interaction we find (see Appendix C for thederivation and additional discussions) ℓ − b ≃ k F λ ( k F w )( e / ~ v F κ ) ( T /µ ) , (58)where w is thw wire width and λ ( z ) = z ln (1 /z ). Inthe case of spinful electrons, the scattering length changesto ℓ − b ≃ k F λ ( k F w )( e / ~ v F κ ) ( T /µ ) ln ( µ/T ) , (59)where λ ( z ) = ln (1 /z ). Contrasting Eqs. (58) and (59),one sees that the spin of the electron plays an importantrole since the inverse scattering length of spinful elec-trons is significantly larger, by a factor of ( µ/T ) ≫ ∼ k − F . When electrons are spinless the Pauli exclusionsuppresses the probability of such scattering. In contrast,the suppression is not as strong when the total spin ofthe three colliding particles is 1 / V. DISCUSSION
In this paper we studied the thermal transport prop-erties of one-dimensional electrons in quantum wires. Inthis system equilibration is strongly restricted by thephase space available for electron scattering and con-servation laws such that leading effects stem from thethree-particle collisions. This is in sharp contrast withhigher-dimensional systems where already pair collisionsprovide electronic relaxation. Although our theory is ap-plicable only in the weakly interacting limit, the resultspresented are still beyond the picture of the Luttinger liq-uid since three-particle collisions are not captured by thelatter. We have elucidated the microscopic processes in-volved in electron thermalization and developed a schemefor solving the Boltzmann equation analytically within alinear response analysis. Our approach allows us to findthe thermal conductance for arbitrary relations betweenthe wire length and microscopic relaxation length [seeEq. (55)].In order to establish a connection to previous work we emphasize that our solution of the kinetic equationsand the result for thermal conductance presented inEq. (55) rely on the simplifying assumption that electronbackscattering can be neglected. This is a good approx-imation except for the case of very long wires, L & ℓ eq ,where the small probability of backscattering ∼ e − µ/T iscompensated by the large phase space available for scat-tering to happen. Accounting for the backscattering pro-cesses, it was found in Ref. 21 that for wires with length L ∼ ℓ eq the thermal conductance is K ( L ) K = ℓ eq L + ℓ eq . (60)This result gives only an exponentially small correctionto the thermal conductance, δK/K = − L/ℓ eq ∝ e − µ/T ,in the limit L ≪ ℓ eq , since in the analysis of Ref. 21thermalization effects on the distribution function wereneglected. It is our result Eq. (57) that gives the leading-order correction to δK in this case. On the other hand,our expression (55) is not applicable for the long wires, L ∼ ℓ eq , whereas Eq. (60) works in this regime. It dis-plays an essentially different feature, which is solely dueto backscattering processes, namely, the vanishing ther-mal conductance δK ∝ /L as L → ∞ .Our work may be relevant for a number of recentexperiments. In particular, the authors of Ref. 10 re-ported thermal conductance measurements and a lowervalue of K than that predicted by the Wiedemann-Franz law, at the plateau of electrical conductance. Aswe explained, corrections to G are exponentially small, G = 2 e /h − O ( e − µ/T ), for wires with L ≪ ℓ eq . Thusthe conductance remains essentially unaffected by inter-actions, and its quantization is robust. In contrast, theeffect of three-particle collisions on the thermal conduc-tance is much more pronounced. Our equation (57) showsthat the thermal conductance is reduced by interactions, which is qualitatively consistent with the experimentalobservation. Apparent violation of the Wiedemann-Franz law is due to the fact that interaction-inducedcorrections δK and δG originate from physically dis-tinct scattering processes [see Figs. 2(b) and 2(c), re-spectively].Another experiment reported measurements of theelectron distribution function in one-dimensional wires.This experiment demonstrated that electrons thermalizedespite the severe constraints imposed by the conserva-tion laws and dimensionality on the particle collisions.We take the point of view that three-particle collisionsare responsible for relaxation and provide an explicit so-lution of the Boltzmann equation, thus uncovering thestructure of the distribution function [see Eqs. (5) and(49)–(52)], which in principle can be compared to exper-imental results. A related study provided us information about thetime scales of thermalization of one-dimensional elec-trons. Although we do not study the latter our results forthe relaxation lengths Eqs. (58) and (59) can be directlylinked to the experiment. Note also that the dramaticdifference between the relaxation lengths, and thus thetimes, of spinful and spinless electrons provides a distinctsignature of three-particle collisions that could be testedexperimentally.There is a very important limitation on the applica-bility of Eqs. (55) and (59) that we need to discuss inthe case of spinful electrons. From the point of viewof Luttinger liquid theory, electrons are not well-definedexcitations in one dimension and instead one should usea bosonic description in terms of charge and spin modes.The weakly interacting limit considered here and usage ofthe Boltzmann equation assumes that electrons maintaintheir integrity during collisions and thus neglects effectsof spin-charge separation. In order to quantify to whatextent such a description is valid, consider an electronwith excitation energy ξ above the Fermi energy µ . Forquadratic dispersion, ε p = p / m , the velocity of suchelectrons differs from that of the electrons in the Fermisea by an amount ∆ v = ξ/mv F . Spin and charge do notseparate appreciably if ∆ v ≫ v c − v s , where v c ( s ) arethe velocities of charge (spin) excitations. At finite tem-peratures the characteristic excitation energy is ξ ∼ T ,so that the above condition can be equivalently reformu-lated as T /µ ≫ ( v c − v s ) /v F . For weakly interactingelectrons the difference between the velocities of chargeand spin modes is related to the zero-momentum Fouriercomponent of the electron-electron interaction potential,namely, v c − v s ≃ V /π ~ ≪
1. This implies that at lowtemperatures when
T /µ ≪ V / ~ v F a description in termsof electrons breaks down and Eqs. (55) and (59) are nolonger applicable.Finally, our work also points to open issues and direc-tions for future research. It is of great interest to un-derstand the fate of energy relaxation and the nature ofthermal transport in the case of strong interactions whichsimultaneously have to be combined with nonequilibrium0conditions. At very low temperatures a description of aone-dimensional system in terms of electronic excitationsbecomes inadequate even if the interactions are weak.The effect of spin-charge separation has to be included,and thermal transport from plasmons and their relax-ation are central issues to consider. Acknowledgements
We would like to acknowledge useful discussions withA. V. Andreev, N. Andrei, N. Birge, P. W. Brouwer,L. I. Glazman, A. Imambekov, A. Kamenev, T. Karzig,and F. von Oppen. This work at ANL was supportedby the U.S. DOE, Office of Science, under Contract No.DE-AC02-06CH11357, and at ENS by the ANR GrantNo. 09-BLAN-0097-01/2 (Z.R.).
Appendix A: Derivation of ˙ P R and ˙ E R In this appendix we derive Eqs. (37)–(39) presented inthe main text of the paper. As explained in Sec. III D,when computing ˙ P R and ˙ E R we have to account for twotypes of scattering process. One is shown in Fig. 2(b) andthe other is similar and consists of a scattering of one leftmover and two right movers. We start by considering thequantity P n = P p > p n f . Its rate of change is˙ P n = X p > p n ˙ f = − X p > ,p ,p p ′ ,p ′ ,p ′ p n K ′ ′ ′ × ( ψ + ψ + ψ − ψ ′ − ψ ′ − ψ ′ ) , (A1)where we used the Boltzmann equation [Eq. (10)] andthe short-hand notation ψ i = ψ ( p i , x ). It is convenientto split each sum from the last equation into parts thatcontain positive and negative values of the momenta, sothat one gets˙ P n = X + −−−−− ( . . . ) + 3 X + −− ++ − ( . . . ) + 6 X ++ − + −− ( . . . ) + X + −− +++ ( . . . )+ X +++ −−− ( . . . ) + 3 X ++++ −− ( . . . ) + 3 X +++++ − ( . . . ) + 2 X ++ − +++ ( . . . )+2 X ++ −−−− ( . . . ) + 3 X + −− + −− ( . . . ) + 6 X ++ − ++ − ( . . . ) + X ++++++ ( . . . ) . (A2)The notations here are as follows X + −−−−− ( . . . ) = X p > ,p < ,p < p ′ < ,p ′ < ,p ′ < ( . . . ) , (A3)and analogously for the other terms. When derivingEq. (A2) we have used the following symmetry proper-ties of the kernel: (a) exchange of incoming and out-going momenta K ′ ′ ′ = K ′ ′ ′ , (b) pairwise exchange K ′ ′ ′ = K ′ ′ ′ , and (c) inversion of momenta p i → − p i , K ′ ′ ′ = K − ′ − ′ − ′ − − − . In the final expression for ˙ P n wekeep only the terms of Eq. (A2) that contain equal num-bers of positive incoming and outgoing momenta, i.e.,the last three terms. The other terms contain at leastone state near the bottom of the band and therefore givea contribution that is exponentially suppressed due tothe small probability of finding an unoccupied state. Af-ter employing Eqs. (16)–(20) combined with momentumand energy conservations, we end up with˙ P n = 3 X ++ − ++ − K ′ ′ ′ (cid:20) δ T R − δ T L T [2 p n − ( − p ) n ] × ( ε p − ε p ′ ) + u L − u R T [2 p n + ( − p ) n ] ( p − p ′ ) (cid:21) . For n = 1 from the last expression we easily get˙ P R = ˙ P = − u R − u L T X ++ − ++ − K ′ ′ ′ ( p ′ − p ) , (A4)which reduces to Eq. (37) in the main text. For n = 2˙ E R = ˙ P m = − δ T R − δ T L T X ++ − ++ − K ′ ′ ′ ( ε p ′ − ε p ) . (A5)One additional step is required to obtain Eq. (38). Sinceall three particles participating in a collision are locatednear the Fermi points it means that the characteristicmomentum of right movers is ∼ p F while for the leftmover it is ∼ − p F . In contrast, the momenta transferredin a collision q i = p i ′ − p i are much smaller ∼ T /v F ≪ p F , which stems from the temperature smearing of theoccupation functions implicit in the kernel K ′ ′ ′ . Since | q i | ≪ p F we approximate p ≈ − p F and linearize thespectrum near the Fermi points, in particular, ε p + q − ε p ≈ m [( − p F + q ) − p F ] ≈ − v F q , (A6)which then brings the last expression for ˙ E R to the formof Eq. (38) in the main text. Appendix B: Three-particle scattering amplitude
For most of our analysis the detailed form of the scat-tering rate entering kinetic equation (10) was not im-portant. However, for the calculation of the microscopicquantities, such as the scattering lengths ℓ a and ℓ b , weneed to know the precise form of the scattering rate W ′ ′ ′ introduced in Eq. (11). Below we give the de-tails of the structure of the scattering rate. We startwith the golden rule expression W ′ ′ ′ = 2 π ~ |A ′ ′ ′ | δ ( E − E ′ ) δ P,P ′ (B1)1 p me p p me p A(11 ,22 ,33 ) ’ ’ ’
A(12 ,21 ,33 ) ’ ’ ’ p me p p me p A(11 ,23 ,32 ) ’ ’ ’
A(13 ,22 ,31 ) ’ ’ ’ p me p p me p A(12 ,23 ,31 ) ’ ’ ’
A(13 ,21 ,32 ) ’ ’ ’ a) b)d)c)e) f) P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P FIG. 4: (Color online) Direct (a) and five exchange (b)–(f)terms in the three-particle amplitude A ′ ′ ′ [Eq. (B2)] thatcontribute to the finite momentum ˙ P R and energy ˙ E R ex-change rates between right and left movers. where A ′ ′ ′ is the corresponding scattering amplitude,while the δ -functions impose conservations of the to-tal energy E ( E ′ ) = P i ε p i ( p i ′ ) and total momentum P ( P ′ ) = P i p i ( i ′ ) of the colliding electrons. One shouldnote that in Eq. (B1) we include δ P,P ′ in the definitionof the scattering rate W ′ ′ ′ rather than the amplitude A ′ ′ ′ , which is in contrast to the usual convention. Thisstep simplifies our notations.Since the electrons interact with two-particle interac-tion potential V ( x ), the three-particle scattering ampli-tude is found to the second order in V ( x ). The details ofthis calculation were presented in Ref. 19. In the case ofspinless electrons the final result reads A ′ ′ ′ = X π (1 ′ ′ ′ ) sgn(1 ′ ′ ′ ) A (11 ′ , ′ , ′ ) . (B2)One should notice that Eq. (B2) contains the term A (11 ′ , ′ , ′ ), which is the amplitude of the direct scat-tering process [Fig. 4(a)], and the terms obtained by theremaining five permutations of the outgoing momenta,which are the exchange terms [Figs. 4(b)–4(f)]. Theycan be written compactly for the segment ∆ x of the wire as follows: A (1 a, b, c ) = a ab + a ac + a bc , (B3) a ab ≡ a p a p b p p = 1(∆ x ) V p a − p V p b − p × (cid:20) E − ε p − ε p b − ε P − p − p b + 1 E − ε p a − ε p − ε P − p a − p (cid:21) , (B4)where ( a, b, c ) is a particular permutation of (1 ′ , ′ , ′ ). InEq. (B2) the notations π ( . . . ) and sgn( . . . ) denote permu-tations of the final momenta and parities of a particularpermutation. Finally, V p is the Fourier-transformed com-ponent of the bare two-body interaction potential. Forthe calculations we take the Coulomb interaction betweenelectrons, V ( x ) = e κ (cid:20) √ x + 4 w − √ x + 4 d (cid:21) , (B5)screened by a nearby gate, which we model by a conduct-ing plane at a distance d from the wire. We also intro-duced a small width w of the quantum wire, w ≪ d , toregularize the diverging short-range behavior of this po-tential. This enables us to evaluate the small-momentumFourier components V p of the interaction potential V ( x ).To this end, we find in the limit ~ /d ≪ p ≪ ~ /wV p = 2 e κ ln (cid:18) p w | p | (cid:19) (cid:20) p p w (cid:21) , (B6)while in the limit of very small momenta p ≪ ~ /dV p = 2 e κ ln (cid:18) dw (cid:19) (cid:20) − p p d ln( p d / | p | )ln( d/w ) (cid:21) . (B7)In the last two equations we introduced the notations p w = ~ /w and p d = ~ /d . We also employed logarith-mic accuracy approximation for V p , meaning that nu-merical coefficients in the arguments of the logarithms inEqs. (B6) and (B7) are neglected. In the following dis-cussions we refer to Eq. (B6) as the unscreened Coulombpotential and to Eq. (B7) as the screened one. The com-plete expression for the amplitude (B2) with the inter-action potential taken in the form (B6) or (B7) is fairlycomplicated. However, major simplification is possible bystudying the kinematics of the three-particle collisions,which in a way allows us to obtain the approximatedform of the amplitude for specific scattering processes,such as the one in Fig. 2(b), which determines the scale ℓ b .It is convenient to label the outgoing momenta as p i ′ = p i + q i for i = 1 , , q i transferred in a collision. Momentumconservation then reads q + q + q = 0 . (B8)2while energy conservation E = E ′ can be equivalentlyrewritten as2 p q + 2 p q + 2 p q + q + q + q = 0 . (B9)At low temperatures, T ≪ µ , the Fermi occupation func-tions constrain particles participating in the collision tolie in a momentum strip of the order of T /v F ≪ p F nearthe Fermi level. This means in practice that the typi-cal momentum transferred in a collision will not exceedmax {| q |} . T /v F . To leading order in T /µ ≪ q ≈ − q + O{ [( p − p ) , q ] /p F } , (B10a)and q ≈ q ( q + p − p )2 p F + O{ [( p − p ) , q ] /p F } , (B10b)where we used p − p ∼ T /v F and set p ≈ − p F . Fromthis analysis one concludes that energy transfer betweenthe right and left movers occurs via small portions ofmomentum q exchange such that {| q | , | q |} ∼ T /v F , | q | ∼ T /v F µ ≪ {| q | , | q |} . (B11)Having two small parameters at hand, | q | /p F ≪ | q | / | q | ≪
1, and accounting for all the exchange con-tributions, one can expand the amplitude (B2) to theleading nonvanishing order. In the course of this ex-pansion we observed that exchange contributions resultin severe cancellations between different scattering pro-cesses. The result of the calculations for the model ofunscreened interaction (B6) is |A ′ ′ ′ | = (cid:18) e κ (cid:19) λ ( k F w )64 µ (∆ x ) ln (cid:18) q p F | q | (cid:19) (B12)where the amplitude is written for a segment of the wireof length ∆ x and the function λ ( k F w ) was introducedearlier [see the definition after Eq. (58)]. For the screenedcase we find |A ′ ′ ′ | = (cid:18) e κ (cid:19) λ ( k F d )4 µ (∆ x ) × (cid:20) q p F ln (cid:18) p F | q | (cid:19) − q q ln (cid:18) | q || q | (cid:19)(cid:21) , (B13)where λ ( z ) = z ln (1 /z ). Both amplitudes (B12) and(B13) are written in logarithmic accuracy approximation.As argued above, the typical scattering processes studiedhere involve only small-momentum transfer, of the orderof q ∼ T /v F . As a result, from the conditions of appli-cability of the interaction potential Eq. (B6) it followsthat the corresponding amplitude Eq. (B12) applies for T ≫ ~ v F /d . Similarly, the screened interaction potentialEq. (B7) and corresponding amplitude Eq. (B13) applyat lower temperatures T ≪ ~ v F /d . There are several general remarks we need to make re-garding the scattering amplitude in Eq. (B2). It is knownfrom the context of integrable quantum many-body prob-lems that for some two-body potentials, N -body scat-tering processes factorize into a sequence of two-body col-lisions. In the context of this work, this means that three-particle scattering for the integrable potentials may resultonly in permutations within the group of three momentaof the colliding particles; all other three-particle scatter-ing amplitudes must be exactly zero for such potentials.We have checked explicitly that the three-particle scatter-ing amplitude in Eq. (B2) is nullified for several specialpotentials: for the contact interaction, V p = const, forthe Calogero-Suthreland model, V p ∝ | p | , and also for thepotential V p ∝ − p /p which is dual to the bosonic Lieb-Liniger model. Surprisingly, we have also noticed thatthe logarithmic interaction potential V p ∝ ln | p | gives ex-actly zero for the three-particle amplitude in Eq. (B2)although we are unaware of any exactly solvable modelfor that case. This is the reason to keep the next leading-order term ∼ ( p/p w ) ≪ A ′ ′ ′ = X π (1 ′ ′ ′ ) sign(1 ′ ′ ′ )Ξ σ σ σ σ ′ σ ′ σ ′ A (11 ′ , ′ , ′ ) , (B14)where Ξ σ σ σ σ ′ σ ′ σ ′ = δ σ σ ′ δ σ σ ′ δ σ σ ′ . One can repeatthe expansion of the amplitude for | q | /p F ≪ | q | / | q | ≪ X { σ } |A ′ ′ ′ | = (cid:18) e κ (cid:19) λ ( k F w )32 µ (∆ x ) (cid:20) p F q + q q (cid:21) ln (cid:18) p F | q | (cid:19) , (B15)which is by a factor of ( p F / | q | ) ≫ λ ( k F w ) was defined under Eq. (59). Appendix C: Intra-branch and inter-branchrelaxation lengths
In this appendix we estimate the scattering lengths ℓ a and ℓ b . Our starting point for evaluation of the inter-branch length ℓ b is the expression ℓ − b = 1080 µ π T v F k F ∆ x X ++ − ++ − ( v F q ) µT W ′ ′ ′ F { f } , (C1)3which follows from Eqs. (39) and (43), where in additionwe introduced the notation F { f } = f p (1 − f p + q ) f p (1 − f p + q ) f p (1 − f p + q ) . (C2)In view of the kinematic constraints (B10a) and (B10b),conservation of momentum and energy in the expression(B1) for the scattering rate W ′ ′ ′ can be presented as δ ( E − E ′ ) δ P,P ′ ≈ v F δ (cid:18) q − q ( q + p − p )2 p F (cid:19) δ q , − q , (C3)which eliminates two out of six momentum integrationsin Eq. (C1). The other four integrals can be completedanalytically with logarithmic accuracy. This amountsto replacing the weak logarithmic parts of the ampli-tude in Eqs. (B12), (B13), and (B15) by their typi-cal values taken at characteristic momenta q ∼ T /v F and q ∼ T /v F µ . We thus treat ln( q / p F | q | ) inEq. (B12) as a constant of order unity and approximateln( p F / | q | ) ≃ ln( | q | / | q | ) ≃ ln( µ/T ) in Eqs. (B13) and(B15). After this step we can integrate in Eq. (C1) explic-itly by linearizing the electron dispersion relation insidethe Fermi functions and get X p p p q F { f } = (∆ x ) T h v F q sinh (cid:0) v F q T (cid:1) , (C4) X p p p q ( p − p ) F { f } = − (∆ x ) T h v F q sinh (cid:0) v F q T (cid:1) , (C5) X p p p q ( p − p ) F { f } = (∆ x ) T h v F q (cid:16) q + π T v F (cid:17) sinh (cid:0) v F q T (cid:1) . (C6)For the spinless case and high-temperature regime T ≫ ~ v F /d , where the Coulomb interaction is unscreened, weobtain ℓ − b ≃ λ ( k F w ) p F T ∆ x (cid:18) e ~ κ (cid:19) X q q (cid:16) q + π T v F (cid:17) sinh (cid:0) v F q T (cid:1) . (C7)Note here that we do not keep track of the numerical co-efficient in the expression for ℓ b since within the adoptedcalculation with logarithmic accuracy this coefficient isnot determined. After the remaining q integration onerecovers Eq. (58), presented in the main text of the paper.At lower temperatures T ≪ ~ v F /d , screening effectsbecome important and one should use Eq. (B13) in theexpression for the scattering length Eq. (C1). Estimationof ℓ b in this case gives ℓ − b ≃ k F λ ( k F d )( e / ~ v F κ ) ( T /µ ) ln ( µ/T ) . (C8) In the spinful case this calculation is completely anal-ogous to the one above; we just need to use a differentexpression for the scattering amplitude. With the helpof Eq. (B15), which is applicable for the model of anunscreened Coulomb potential interaction, we get at theintermediate step with logarithmic accuracy, ℓ − b ≃ λ ( k F w ) T ∆ x (cid:18) e ~ κ (cid:19) ln (cid:16) µT (cid:17) X q q (cid:16) q + π T v F (cid:17) sinh (cid:0) v F q T (cid:1) . (C9)After the final integration this translates into Eq. (59).We turn now to discussion of the intra-branch relax-ation length ℓ a introduced in Sec. II. Unlike the case ofinterbranch relaxation, Fig. 1(b), here all three collidingparticles are near the same Fermi point; see Fig. 1(a).In this case, the typical momentum change for the threeelectrons is the same, | q | ∼ | q | ∼ | q | ∼ T /v F . (C10)At this point we should emphasize that for the processesthat determine the length scale ℓ b , a new energy scale T /µ appeared in the problem purely from the kinematicconstraints based on the conservation laws. This scaledetermined the typical momentum transfer of the parti-cle that was alone at one side of the Fermi surface; seeEqs. (B10b) and (B11).Another important quantity is the scattering ampli-tude. For Coulomb interaction and for the process whereall three particles are near the same Fermi point, itis a relatively complicated expression, but similarly toEqs. (B12) and (B13) it depends on momenta only weakly(logarithmically) for the intra-branch processes.These two observations help us to estimate ℓ a usingthe known result Eq. (58) for ℓ b . Namely, by replacingthe energy scale T /µ in ℓ b by T , we obtain the estimate ℓ − a ≃ k F λ ( k F w )( e / ~ v F κ ) ( T /µ ) , (C11)for the unscreened Coulomb case, T ≫ ~ v F /d . At lowertemperatures, T ≪ ~ v F /d it changes to ℓ − a ∝ T . It isimportant to emphasize that regardless of the interactionmodel we use there exists a distinct separation betweenthe scales of the relaxation lengths, namely, ℓ a /ℓ b ∼ T /µ ≪ . (C12)This fact justifies our ansatz for the distribution function[see the discussion after Eq. (5)]. The detailed calculationof ℓ a will be presented elsewhere. A. A. Abrikosov,
Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 1988). G. V. Chester and A. Thellung, Proc. Phys. Soc. London , 1005 (1961). Electron-Electron Interaction In Disordered Systems ,edited by A. J. Efros and M. Pollak (Elsevier, Amsterdam,1985). G. Catelani and I. L. Aleiner, JETP , 331 (2005). T. Giamarchi,
Quantum Physics in One Dimension ,(Claredon Press, Oxford, 2003). D. L. Maslov and M. Stone, Phys. Rev. B , R5539(1995); V. V. Ponomarenko, Phys. Rev. B , R8666(1995); I. Safi and H. J. Schulz, Phys. Rev. B , R17040(1995). R. Fazio, F. W. J. Hekking, and D. E. Khmelnitskii, Phys.Rev. Lett. , 5611 (1998). C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. , 3192 (1996). K. Schwab, E. A. Henriksen, J. M. Worlock, andM. L. Roukes, Nature Phys. , 974 (2000). O. Chiatti, J. T. Nicholls, Y. Y. Proskuryakov, N. Lump-kin, I. Farrer, and D. A. Ritchie, Phys. Rev. Lett. ,056601 (2006). J. T. Nicholls and O. Chiatti, J. Phys.: Condens. Matter , 164210 (2008). F. Sfigakis, A. C. Graham, K. J. Thomas, M. Pepper,C. J. B. Ford, and D. A Ritchie, J. Phys.: Condens. Matter , 164213 (2008). G. Granger, J. P. Eisenstein, and J. L. Reno, Phys. Rev.Lett. , 086803 (2009). C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna,D. Mailly, and F. Pierre, Nature Phys. , 34 (2009). Y. F. Chen, T. Dirks, G. Al-Zoubi, N. O. Birge, and N. Ma-son, Phys. Rev. Lett. 102, 036804 (2009). G. Barak, H. Steinberg, L. N. Pfeiffer, K. W. West,L. I. Glazman, F. von Oppen, and A. Yakoby, Nature Phys. , 489 (2010). D. B. Gutman, Y. Gefen, and A. D. Mirlin, Phys. Rev.Lett. , 126802 (2008); Phys. Rev. B , 085436 (2010). S. Takei, M. Milletari, and B. Rosenow, Phys. Rev. B ,041306 (2010). A. M. Lunde, K. Flensberg, and L. I. Glazman, Phys. Rev.B , 245418 (2007). J. Rech, T. Micklitz, and K. A. Matveev, Phys. Rev. Lett. , 116402 (2009). T. Micklitz, J. Rech, and K. A. Matveev, Phys. Rev. B ,115313 (2010). A. Levchenko, T. Micklitz, J. Rech, and K. A. Matveev,Phys. Rev. B , 115413 (2010). K. A. Matveev, A. V. Andreev, and M. Pustilnik, Phys.Rev. Lett. , 046401 (2010). T. Karzig, L. I. Glazman, and F. von Oppen, Phys. Rev.Lett. , 226407 (2010). A. Levchenko, Z. Ristivojevic, and T. Micklitz, Phys. Rev.B , 041303(R) (2011). Y. M. Sirenko, V. Mitin, and P. Vasilopoulos, Phys. Rev.B , 4631 (1994). To be accurate we point out that for the noninteractingelectrons G = (2 e /h )(1 + e − µ/T ) − , where the exponen-tially small term comes from the states at the bottom of theband. At low temperatures e − µ/T ≪ For the estimates we use typical parameters from the ex-periment of Ref. 15. In particular, v F ∼ × m/s, k F ∼ m − , κ ∼
10, and w ∼ e / ~ v F κ ∼ µ = ~ v F k F / ∼
25 meV so that forthe temperature T ∼ ℓ b ∼ µ m. Z. Ristivojevic, A. Levchenko and K. Matveev, (unpub-lished). B. Sutherland,