ЖЭТФ, 2021, том 159, вып. 4, стр. 776-793

KINETIC PROCESSES IN FERMI-LUTTINGER LIQUIDS

A. Levchenkoa*, T. Micklitz^b

^a Department of Physics, University of Wisconsin-Madison

53706, Madison, Wisconsin, USA

^b Centro Brasileiro de Pesquisas Físicas

22290-180, Rio de Janeiro, Brazil

Received January 9, 2021,

revised version January 25, 2021.

Accepted for publication January 25, 2021

Contribution for the JETP special issue in honor of I. E. Dzyaloshinskii’s 90th birthday

DOI: 10.31857/S0044451021040210

of strongly interacting many-body quantum systems

[1, 2]. For instance, in the context of electrons in con-

ductors, one typically views the quasiparticle states as

Abstract. The present work discusses extensions

of the pioneering analysis by Dzyaloshinskii and Larkin

those evolving from the free electron gas to a Fermi

liquid when adiabatically turning on the interaction.

[Sov. Phys. JETP 38, 202 (1974)] of correlation func-

tions for one-dimensional Fermi systems, focusing on

In accordance with Landau theory [3], quasiparticles

inherit some of the basic quantum numbers of bare

the effects of quasiparticle relaxation enabled by non-

linear dispersion. Throughout the work we employ

electrons such as spin, charge, and momentum. Their

respective dispersion relations as well as thermodynam-

both, the weakly interacting Fermi gas picture and non-

ical and kinetic properties may, however, differ signifi-

linear Luttinger liquid model to describe attenuation

of excitations and explore the fermion-boson duality

cantly due to interaction-induced renormalizations. A

crucial advantage of the quasiparticle picture is that

between both approaches. A special attention is de-

voted to the role of spin-exchange processes, effects

residual interactions are assumed to be weak, and can

be systematically and controllably addressed by means

of interaction screening, and integrability. Thermal-

ization rates for electron- and hole-like quasiparticles,

of perturbation theory. The central question related

to the validity of the quasiparticle description concerns

as well as the decay rate of collective plasmon excita-

their lifetime τ_qp. Indeed, in the process of scattering

tions and the momentum space mobility of spin exci-

quasiparticles decay and their mere notion is meaning-

tations are calculated for various temperature regimes.

ful only if attenuation is weak enough and they can

The phenomenon of spin-charge drag is considered and

the corresponding momentum transfer rate is deter-

be considered as sufficiently long-lived collective exci-

tations. In Fermi systems, the Pauli principle severely

mined. In the context of transport properties, momen-

tum relaxation due to several competing mechanism,

limits the phase space available for quasiparticle col-

lisions. The low temperature decay rate can then be

viz. triple electron collisions, electron-phonon scat-

tering, and long-range inhomogeneities is addressed.

estimated from the Golden rule as

Energy transfer facilitated by plasmons is highlighted

from the perspective of inhomogeneous Luttinger liq-

ε² + π²T²

uid. The full matrix of thermoelectric coefficients is

τ-1qp(ε, T) ∝ (νV₀)²

(1)

ε_F

found at the quantum critical point of the first conduc-

tance plateau transition.

1. Introduction. The concept of quasiparticles

In this expression, the excitation energy ε = v_F (p-p_F )

plays a central role in the condensed matter physics

of a quasiparticle with momentum p is counted from

the Fermi energy ε_F , ν is the density of states and

^* E-mail: levchenko@physics.wisc.edu

V₀ is the characteristic strength of the short-range re-

776

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

pulsive interaction¹⁾. The dominant microscopic scat-

ficiently high, so that thermally- and disorder-induced

tering channel leading to Eq. (1) involves quasiparti-

transitions to higher sub-bands are suppressed. In ad-

cle decaying into three: another quasiparticle and a

dition, edge modes formed at the boundaries of a 2D

particle-hole excitation. The amplitude for this process

electron gas when placed in a strong magnetic field in

is proportional to V₀, hence, the dimensionless factor

the integer or fractional quantum Hall regime [10, 11],

of (νV₀)² in the scattering probability entering Eq. (1).

or edge states of 2D quantum spin Hall topological-in-

The factor ε² is the phase space volume for scattering

sulators [12], provide other distinct examples of, respec-

of a quasiparticle with energy ε compatible with the

tively, chiral and helical quantum 1D electron liquids.

conservation of total energy and momentum. At finite

In principle, all these systems can be successfully

temperatures the smearing of states in the energy strip

described within the framework of Luttinger liquid

of order ∼ T per particle leads to the corresponding

theory [13-15], which builds out of the Tomonaga-Lut-

T² dependence of τ-1qp. Higher-order processes involv-

tinger (TL) model [16, 17]. As is known form pioneer-

ing 2n + 1 quasiparticles, namely n > 1 electron-ho-

ing works [18-20], in the asymptotic low-energy limit

le pairs, are usually neglected as their respective rate

ε/ε_F

≪ 1, the key properties of the TL model are

scales with higher powers of energy. In particular, at

manifestly non-Fermi liquid like. A power-law anomaly

zero-temperature the rate for relaxation processes of

manifests in the suppression of the single particle den-

a quasiparticle with energy ε involving n particle-hole

sity of states

pairs vanishes as τ-1qp ∝ ε2n. One notable property

(

)2g

|ε|

sin(πg)

of Eq. (1) is that it predicts the same relaxation time

ν(ε) = ν₀

Γ(1 - 2g),

(2)

v_F p_Λ

πg

for particle-like and hole-like excitations. Another pro-

perty is that the crossover from zero-temperature to

and collapse of the quasiparticle residue in the distri-

finite-temperature relaxation is governed only by one

bution function. At T → 0 that is

scale, viz. when the excitation energy compares to the

temperature itself ε ∼ T .

Γ(1/2 + g)

n(ε) =

In addition to the quasiparticle relaxation, which

2√πΓ(1 + g)×

[

]

(

)2g

is often viewed as an out-scattering rate from a par-

Γ(1/2 - g)

|ε|

× 1-

sgn(ε)

(3)

ticular quantum state, one may address a more gene-

Γ(1/2 + g) v_F p_Λ

ral question of relaxation of a nonequilibrium quasi-

particle distribution function. In kinetic theory such

where ν₀ = 1/(2πv_F ), Γ(z) is the Euler’s gamma func-

problem is typically analyzed in the framework of the

tion, and p_Λ is the momentum cutoff of the model

linearized Boltzmann equation. The eigenvalues of the

(parametrically p_Λ ∼ p_F ). In the simplest spinless ver-

corresponding collision operator define relaxation times

sion of the TL-model with short-ranged interaction, a

of different distribution function modes. In three di-

single dimensionless coupling constant,

mensional Fermi liquids this problem is exactly solvab-

[

]

1+ν₀V₀

le [4, 5] and one finds that all these rates are para-

√

-1 ,

(4)

1 + 2ν₀V₀

metrically the same, scaling respectively as ∝ T². In

contrast, in two-dimensional Fermi liquids, kinematics

can be related to the zero-momentum Fourier compo-

of head-on collisions leads to a parametrically distinct

nent of the bare interaction potential V₀. The limit of

relaxation of odd and even momentum harmonics of

weak interaction corresponds to g ≪ 1 and Eqs. (2),

the distribution function, in particular τ-1even ∝ T²/ε_F

(3) are valid for g < 1/22). However, a direct at-

while τ-1odd ∝ T⁴/ε3F [6, 7].

tempt to apply Luttinger liquid theory to the question

The role of dimensionality in quasiparticle relaxa-

of quasiparticle lifetime meets formidable challenges.

tion becomes the most dramatic in one-dimension (1D).

In a fermionic representation of the TL-model, elabo-

This special case of electron liquids can be experimen-

rated explicitly by Dzyaloshinskii and Larkin [19], the

tally realized in quantum wires of GaAs/AlGaAs hete-

electron self-energy vanishes on the mass shell in all

rostructure [8] or carbon nanotubes [9] when parti-

orders of perturbation theory and, consequently, corre-

cle density is such that only the lowest sub-band of

lation functions assume power-law tails. These results,

transversal modes is occupied. It further requires that

²⁾ In Ref. [19] the limit of strong interactions, g > 1/2, was

temperature is sufficiently low and sample purity is suf-

also considered, including the scenario when coupling between

fermions of the same chirality is different from coupling bet-

¹⁾ Throughout the paper we use units with Planck and Boltz-

ween fermions of different chirality. For additional details on

mann constants set to unity ℏ = k_B = 1.

the derivation of Eq. (3) see also Ref. [21].

777

A. Levchenko, T. Micklitz

ЖЭТФ, том 159, вып. 4, 2021

and the absence of relaxation, can be alternatively un-

measurements of quasiparticle properties beyond the

derstood from the Mattis and Lieb [18], and Luther and

parading of linear Luttinger liquid theory. In a parallel

Peschel [20] bosonization construction, which maps in-

line of developments [53-56], cooling of nonequilibrium

teracting 1D fermions to a collection of decoupled har-

quasiparticles in quantum Hall edge fluids was mea-

monic modes of charge-density and spin-density oscil-

sured and corresponding lengths scales of thermaliza-

lations. Notably, in both approaches the exact solution

tion processes were quantified.

relies heavily on the linearization of the fermionic dis-

The focus of this communication is on the descrip-

persion relation.

tion of elementary kinetic processes inducing relax-

One is then left with the natural puzzle whether in-

ation in nonlinear Luttinger liquids and their emergent

corporating curvature of the dispersion relation into the

transport properties. Keeping forward scattering elect-

TL-model would cure the issue and yield a finite life-

ron-electron interactions and accounting for nonlinear

time of excitations, thus possibly restoring Fermi liquid

contributions to the electron dispersion, this theory is

like properties of the system. This line of reasoning can

beyond the Dzyaloshinskii-Larkin theorem. The lat-

be also corroborated within the fermionic picture, not-

ter relax kinematic constraints and open phase space

ing that spectrum nonlinearity softens phase space re-

for multi-loop corrections to the electron self-energy,

strictions for quasiparticle scattering, thus making their

thereby providing a variety of inelastic processes which

relaxation possible.

affect equilibrium as well as nonequilibrium proper-

Similarly, at the level of the bosonic description,

ties of the 1D quantum electron liquids. The rest of

nonlinear terms of the dispersion relation couple charge

this work is structured as follows (see the full text).

and spin modes thus enabling their decay. However, it

Section 2 focuses on the hierarchy of relaxation times

was quickly recognized that curvature cannot be in-

in Fermi-Luttinger liquids. We present results be-

cluded perturbatively, and a naive expansion leads to

yond parametric estimates, including detailed compu-

spurious divergences. These and other related ques-

tations of a number of experimentally relevant interac-

tions to 1D kinetics, including the connection between

tion models³⁾. The complimentary kinetic equation ap-

the two pictures of the fermion-boson duality, attracted

proach, applied to the quasiparticle picture of a weakly

significant recent interest. This has lead to the devel-

interacting Fermi gas, and spin- and charge-excitations

opment of the nonlinear Luttinger liquid theory, also

of a Luttinger liquid, are explored concurrently. We

referred to as Fermi-Luttinger liquid (FLL) theory (see

present numerical estimates for experimentally mea-

Refs. [22, 23] for comprehensive reviews and references

sured relaxation rates and provide detailed comparison

herein). Specifically for the problem of quasiparticle

to previous results. In Sec. 3, the temperature depen-

relaxation in quantum wires, various scattering rates

dence of kinetic coefficients is calculated, accounting

were calculated within different interaction models for

for extrinsic mechanisms of momentum relaxation due

both, spinless [24-33] and spin-1/2 fermions [34-40]. In

to phonons or long-range inhomogeneities. The contri-

parts of the present work we review and extend these

bution to heat transport mediated by plasmons in the

results.

inhomogeneous Luttinger liquid is elucidated. Finally,

On the experimental forefront the hallmark sig-

we devote parts of the discussion to the thermoelectric

natures of Luttinger liquid behavior have been ob-

properties at the first plateau transition of the quan-

served by means of various spectroscopic techniques.

tum conductance. In Sec. 4, we provide a summary of

Namely, power-law anomalies in the density of states,

main findings and open questions, sketching a broader

tunneling conductance, and current-voltage characte-

picture and commenting on related topics relevant for

ristics [9,10,41,42], spin-charge separation [43,44], and

chiral, helical, and spiral versions of 1D quantum fluids.

charge fractionalization [45, 46]. Besides GaAs quan-

Several Appendices accompany our presentation in the

tum wires, carbon nanotubes, and edge modes, clear

main text, providing additional technical details of the

features of Luttinger liquid physics have been identi-

presented analysis and formalism.

fied in many other systems such as bundles of NbSe₃

2. Hierarchy of relaxation processes. The

[47] and MoSe [48] nanowires, polymer nanofibers [49]

physics of quasiparticle relaxation in 1D quantum elec-

and conjugated polymers at high carrier densities [50],

tron liquids is perhaps a surprisingly rich and compli-

as well as atomically controlled chains of gold atoms on

cated problem. In part this has to do with the fact

Ge surfaces [51], just to name a few distinct examples.

that, in contrast to their higher dimensional counter-

In the most recent report [52], relaxation processes in

quantum wires were captured and bounds on the cor-

³⁾ In part this material was summarized in Sec. IV of the ex-

responding timescales were determined, thus providing

tensive review in Ref. [23].

778

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

parts, two-particle collisions, namely scattering pro-

ization to the degenerate quantum limit was presented

cesses with the emission of a single particle-hole ex-

in the work of Ref. [59], and exchange terms were care-

citation, do not result in finite relaxation rates. This

fully examined in Refs. [35, 60]. The resulting ampli-

statement pertains to generic dispersion relations, i. e.

tude takes the form

including curvature, and not only applies to models

with linear dispersion. Indeed, kinematics of two-par-

1^∑

sgn(P)sgn(P^′)×

ticle scattering in 1D is such that particles either keep

L²

PP^′

or swap their momenta, but neither of these options

Vp′_a-pa Vp′c -pc

causes relaxation. To allow for the redistribution of

Ξ_σσ′ .

(6)

ε_p

+εpc -ε_p

momenta and, at the same time, to comply with re-

b+pc -p^′

strictions of conservations laws one necessarily needs

to consider triple electron collisions, or alternatively,

Here L is the system size and sums are over all possible

assume some extrinsic mechanism.

permutations P of momenta p_i with i = 1, 2, 3 star-

ting from the direct scattering process (p₁, p₂, p₃) →

The analysis of 1D kinematics of multi-particle col-

lisions resolving energy and momentum conservations

→ (p′1, p′2, p′3) to all its exchange processes, with

sgn(P) accounting for the sign of the particular per-

reveals a plethora of possible scattering events. They

ultimately lead to a hierarchy of relaxation stages in the

mutation (using the convention that sgn(123) = +1).

system and an emergent asymmetry between the relax-

Each permutation comes with a spin-dependent factor

ation of particle-like and hole-like excitations. All pro-

Ξ_σσ′ = δσaσ′_a δσ_bσ^′

δσcσ′ reflecting particle exchange. In_c

cesses can be broken down into several distinct classes.

the spinless case, the amplitude has an identical struc-

First are the forward scattering processes with soft

ture to Eq. (6) with Ξ_σσ′ ≡ 1. The amplitude con-

momentum transfer that involve either (i) all parti-

sists of 36 distinct terms that can be split into groups

of 6, each representing one direct and five exchange

cles from the same branch, or (ii) particles from both

branches such that all initial and final states are near

scattering processes, respectively. Technically speak-

the Fermi energy. Second are processes involving states

ing Eq. (6) appears from the iteration of the

T-matrix,

Ĝ₀

deeper in the band. These latter are relevant for

T =V+V

T, to second order in the bare two-particle

(iii) the drift of quasiholes and (iv) backscattering pro-

interaction potential V . Here

Ĝ₀ is the resolvent op-

cesses that change the number of right and left moving

erator (viz. the free particle Green’s function) and ε_p

excitations before and after the collision. We will refer

denotes the energy-momentum dispersion relation.

to thermalization when discussing relaxation processes

For practical applications to quasiparticle scatter-

that proceed without backscattering. These processes

ing in quantum wires, it is sufficient to assume the sim-

determine the lifetime of quasiparticles associated to

ple dispersion of a parabolic band ε_p = p²/2m^∗ with

the redistribution of excess energy, and affect thermal

effective mass m^∗, and use a Coulomb interaction po-

transport properties of the system. In contrast, the no-

tential. Effects of screening due to nearby gates can be

tion of equilibration will be used to refer to relaxation

modeled by a conducting plate placed at a distance d

processes involving the backscattering of quasiparticles,

away from the wire. In this case the interaction poten-

which ultimately govern electrical transport properties.

tial is of the form

2.1. Quasiparticle interaction model. In the

[

]

picture of a weakly nonideal Fermi gas, the probabil-

e²

V (x) =

√

(7)

ities of particle collisions can be calculated perturba-

|x|

x² + 4d²

tively in the interaction, employing the usual

T-matrix

formalism [57]. Within the Golden Rule, the scattering

where κ is the dielectric constant of the host mate-

rate

rial. The diverging short-range behavior of this poten-

W = 2π|A|²δ(E - E^′)δ_P,P′

(5)

tial needs to be regularized in order to evaluate the

entering the

small-momentum Fourier components V_p

is expressed in terms of the scattering amplitude A of

amplitude in Eq. (6). To this end, we introduce the

the corresponding process. Here E(E^′) and P (P^′) la-

small width w of the quantum wire, w ≪ d, and replace

√

bel total energy and momentum of initial (final) states,

1/|x| → 1/

x² + 4w². Upon 1D Fourier transform we

and the delta function δ(E - E^′) along with the Kro-

then find

necker delta δ_P,P ′ enforce energy and momentum con-

servations. In the semiclassical limit, the three-particle

2e²

amplitude A was considered in Ref. [58]. The general-

V_p =

[K₀(2w|p|) - K₀(2d|p|)] ,

(8)

779

A. Levchenko, T. Micklitz

ЖЭТФ, том 159, вып. 4, 2021

where K₀(z) is the modified Bessel function of the

The triple electron scattering rate from Eq. (5) gen-

second kind. Using the asymptotic expression of the

erates the collision integral (Stosszahlansatz) of the cor-

Bessel function at z ≪ 1,

responding Boltzmann equation

(

)

(

)

∑

z²

St{n} =

W [n_p′

(1-np1)n_p′

(1-np2)n_p′ ×

K₀(z) ≈ ln

ze^γE

ze^γE-1

{p},{σ}

× (1-np3)-np1(1-n_p′

)np2 (1-n_p′

)np3 (1-n_p′ )].

(11)

with γ_E the Euler constant, one then finds the simpli-

fied form of the interaction potential

Here each pair of Fermi functions, n_p(1 - n_p′ ), cap-

tures statistical occupation probabilities, whereas the

[

⁽d⁾

⁽e1-γE^)]

two terms of the collision integral correspond to in-

Vp ≈

- (pd)² ln

(9)

|p|d

coming and outgoing processes. At thermal equilib-

rium these terms nullify each other by virtue of the de-

applicable to the screened limit of Coulomb interac-

tailed balance condition. At weak disequilibrium, one

tion and valid for p ≪ 1/d. In the opposite regime,

can linearize n_p = f_p + δn_p in the external perturba-

d-1 ≪ p ≪ w-1, the second term in Eq. (8) can be

tion δn_p around the equilibrium Fermi-Dirac distribu-

neglected since K₀(z) ∝ e^-z/√z at z ≫ 1. One then

tion function f_p. The collision term can then be con-

arrives at the simplified form of the unscreened poten-

sidered as a linear integral operator, acting on δn_p =

tial

= f_p(1 - f_p)ψ, and one can formulate the eigenvalue

. The spec-

problem for this operator, St{ψ_n} = ω_nψ_n

trum of eigenvalues ω_n may be discrete or continuous,

Vp ≈

and captures all the information about the decay of

[

)

⁽e^-γ

⁽e1-γE^)]

different distribution function modes. As solving this

× ln

+ (pw)² ln

(10)

|p|w

problem exactly for triple collisions presents a daunting

task [29, 40], we here follow a simpler more pragmatic

A few comments are in order in relation to the in-

approach. Setting, for instance, δnp1 = δ_p

1,pF +ε/vF de-

teraction model presented in this section. (i) It should

scribes a quasiparticle with excess energy ε. Neglect-

be noted that retaining numerical pre-factors of the

ing then secondary collisions, the Boltzmann equation

order of unity under the logarithm in above expres-

reduces to the simple relaxation time approximation,

sions for V_p would exceed the accuracy of further cal-

(∂_t + τ-1qp)δn_p = 0, with solution δn_p ∝ exp(-t/τ_qp).

culations, so they will be dropped and simply set to

It is natural to identify the corresponding timescale for

unity. (ii) However, retaining the sub-leading correc-

decay with the quasiparticle life-time

tions containing p² in the main log-series expansion of

∂St{n}

both Eqs. (9) and (10) is actually crucial. Indeed, in

τ-1qp = -

(12)

∂n_p

the spinless case, the model with contact interaction as

well as the Calogero-Sutherland model, are known to

which follows from Eq. (11) by only retaining the

be completely integrable [61]. This implies that all irre-

out-scattering contribution. Alternatively, one may

ducible multi-particle scattering amplitudes must van-

project the collision operator (11) onto either momen-

ish identically for a constant V_p and V_p ∝ |p|. Fur-

tum or energy modes and thus infer the relaxation time

thermore, the extended model of short-ranged interac-

of interest. This approach is parametrically correct,

tion, V_p ∝ p², corresponding to the real space poten-

however, may miss numerical factors of order unity

tial V (x) ∝ δ^′′(x), is also integrable. This is known

when compared to the exact solution of the eigenvalue

as Cheon-Shigehara model [62]. It is only due to the

problem. We will employ both approaches in the forth-

additional logarithm ∝ p² ln |p| in Eq. (9), that there

coming sections.

is partial non-cancellation between different terms in

Eq. (6) and the amplitude remains finite. (iii) In the

2.2. Quasiparticle decay rates. Owing to one-

model of long-ranged Coulomb interaction the situa-

dimensionality of the problem, it is convenient to think

tion is more subtle. A priori this model is not known

of particles of different chirality, namely right-movers

to be integrable. Nevertheless, the amplitude in Eq. (6)

(R) and left-movers (L). It can be readily checked

vanishes for pure logarithmic interaction V_p ∝ ln |p|, so

that strictly at zero-temperature quasiparticle relax-

that retaining an additional p² ln |p| term in Eq. (10) is

ation is only possible if collisions involve both, right-

important to get a finite result.

and left-moving particles since otherwise conservation

780

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

laws cannot be satisfied. For this reason, consider first

momentum and energy conservations in Eq. (5) can be

a process of relaxation that involves two right-moving

simplified to

particles, with initial momenta p₁, p₂, and a left-moving

particle labeled by momentum p₃. The outgoing mo-

δ_P,P′ δ(E - E^′) ≈

(

)

menta after the collision, p^′i = p_i +q_i, will be labeled by

q₁(k₁ - k₂) + q²¹

momenta transfer q_i for each of the particle i = 1, 2, 3.

≈

δ q₃ -

(15)

δq1,-q2 .

v_F

p_F

In these notations, the momentum conservation be-

comes q₁ + q₂ + q₃ = 0, and the energy conservation,

These approximations enable one to complete all five

for a simple parabolic band, can be cast in the form

momentum integrations. Two integrations are removed

by delta functions which fix values of q₂ and q₃ in terms

2(p₁q₁ + p₂q₂ + p₃q₃) + q²¹ + q²² + q²³ = 0.

of k1,2 and q₁. Furthermore, in the zero temperature

limit, T → 0, Fermi occupations become step-func-

These conditions set the phase-space constraints for

tions, f_k → θ(-k). The integral over k₃ then becomes

collisions.

elementary, contributing by a pure phase space factor

For an initial state with p₁ = p_F + ε/v_F , the quasi-

particle life-time corresponding to an RRL-process is

∑

then

f-k3 (1 - f-k2-q3 ) =

|q₃|θ(-q₃).

2π

τ-1qp =

The product of Fermi factors, fk2 (1 - fk2-q1 ), simply

∑

limits the domain of k₂ to the range k₂ ∈ [-|q₁|, 0],

W (1 - f_p′

)fp2 (1 - f_p′

)fp3 (1 - f_p′ ),

(13)

while the remaining 1-fk1+q1 dictates that q1 < k1. Fi-

p2p3

p′1p′2p^′

nally, we recall that in this setting k₁ = ε/v_F . Putting

everything together the RRL-process gives the life-time

where we begin analysis from the spinless case. At

(

)₄

this point it is convenient to shift momenta of left- and

τ-1qp = c₁ε_F g⁴λ²¹(p_F w)

(16)

right-movers from the respective Fermi points, p1,2 =

ε_F

= p_F + k1,2 and p₃ = -p_F + k₃. In addition, it is

sufficient to linearize the spectrum in the distribution

where g

= e²/κv_F is the dimensionless interac-

functions, approximating

tion strength of the model and we introduced

[ (

)

λ₁(z) = z2 ln(1/z). The numerical coefficient c1 =

]-1

v_F k

= (15 - π²)/32π³ is obtained with help of the following

f±pF +k → f±k = exp

integral

but not in the scattering probability W . Indeed,

∫∫

(

)

an analysis of the kinematic constraints suggests that

15 - π²

x²g(x, y)ln²

dx dy =

(17)

q1 ≈ -q2 and q3 ≈ (q1/pF )(k1 - k2 + q1), implying

g(x, y)

that |q₃| ≪ |q1,2|. In other words, relaxation occurs in

incremental steps of momentum transfer q₃ ∼ ε²/v2F p_F

where g(x, y) = 1 - x(1 - y). Notice that the numeri-

from right-movers to left-movers. With these observa-

cal factor in Eq. (16) differs from the one calculated in

tions at hand, we next need the corresponding three-

Refs. [26, 28] as different properties of the interaction

particle scattering amplitude. For the case of long-

potential were assumed⁴⁾.

ranged Coulomb interaction Eq. (10), one finds from

We see that finite decay rate emerges in forth order

Eq. (6) after a laborious expansion

of the interaction strength. We also notice that the at-

tenuation is inversely proportional to the cube of mass,

2(p_F w)

⁽2e² )²

τ-1

∝ (m^∗)-3, and vanishes as the limit m^∗ → ∞ is

A≈

L²ε_F

taken at fixed band velocity. This limit corresponds to

[

(

)]

(

)

the situation considered by Dzyaloshinskii and Larkin.

q²¹

The energy scaling of the decay rate, ∝ ε⁴, is consistent

× 1-

(14)

p_F w

p_F |q₃|

with expectations based on the Fermi liquid picture for

This result is obtained to leading logarithmic accuracy

⁴⁾ In Appendix (see full text) we sketch derivation of Eq. (16)

using two small parameters |q₁|/p_F ∼ |q₃|/|q₁| ≪ 1 in

from the bosonization framework of an impurity scattering in

the expansion. With the same level of accuracy the

Luttinger liquids.

781

A. Levchenko, T. Micklitz

ЖЭТФ, том 159, вып. 4, 2021

a process involving two particle-hole excitations. How-

They are dominated by 2p_F exchange-processes be-

ever, this result is not universal. This becomes evi-

tween branches, in which left-movers are scattered into

dent from repeating the above calculation for the model

right-movers [35]. Even though the strength of 2p_F

of screened short-range interaction, i.e. using the po-

exchange interaction is weaker than small momentum

tential given by Eq. (9). Expanding the amplitude in

scattering, V2pF ≪ V₀, for Coulomb interaction the rel-

Eq. (6) under the same conditions as above, one then

ative reduction is only logarithmic. The gain in the

finds instead of Eq. (14) the amplitude

amplitude, on the other hand, is more substantial and

q ≫ 1. This

controlled by the large factor ∼ ε_F /v_F

(

)

5(p_F d)

⁽2e² )²

statement can be verified explicitly from Eq. (6) where

A≈-

3L²ε_F

p_F d

after spin summation one finds for the square of the

[

amplitude for the RRL-process

q²¹

⁽ |q₁|)]-

1 + 6ln

]

4p2F

p_F

∑

3V22p

(V₀ - V2pF )²

^[q²¹

4p2F

|A|² =

(20)

[

32L⁴ε²

q²³

q²

σ2σ3

q²³

⁽ |q₃|^)]]

1 + 6ln

(18)

σ′1σ′2σ^′

q²¹

|q₁|

The crucial difference here compared to Eq. (14) is the

To obtain this result we approximated Vp1-p2±qi ≈ V0

appearance of the additional small parameter |q₁|/p_F ∼

and Vp1,2-p3±qi ≈ V2pF in all the relevant terms since

∼ |q₃|/|q₁| ∼ ε/ε_F ≪ 1, which can be related to the fact

p1,2 - p₃ ≈ 2p_F and q_i ≪ |p_i|. Again, by repeating

momentum integrations, the decay rate is found to be

that this particular model is nearly integrable. A close

inspection of the amplitude in Eq. (6) reveals that each

of the form

term individually diverges as 1/q at small characteris-

(

)₂

⁽ε_F )

tic momentum transfer. However, all exchange terms

τ-1qp = c₃ε_F g⁴λ²³(p_F w)

ln²

(21)

combined together remove the singularity and partially

ε_F

cancel out all the way to ∼ q² ln q order. The rest of the

calculation carries through in exactly the same way as

with c₃ = 45/32π³ and λ₃(z) = ln(1/z). To be consis-

in the previous example, and one finds the decay rate

tent with the approximations that lead to Eq. (20),

(

)₈

the difference V₀ - V2pF should be understood as a

⁽ε_F )

τ-1qp = c₂ε_F g⁴λ²²(p_F d)

ln²

(19)

weak logarithmic factor ≃ (2e2/κ) ln(εF /ε) for the

ε_F

Coulomb interaction potential. This was incorporated

with c₂ = 2445/3584π³ and λ₂(z) = z⁴ ln(1/z). The

into Eq. (21). The singularity of the amplitude was

four extra powers in the energy dependence, can be

compensated by phase space factors, and perhaps sur-

traced back to the different asymptotic form of the

prisingly this restores essentially the Fermi liquid form

amplitude in Eq. (18). This demonstrates the high

of the decay rate at T = 0. We note that up to model

sensitivity of decay rates in 1D to details of the in-

dependent pre-factors, the quadratic dependence of the

teraction. The result captured by Eq. (19) is of course

relaxation rate on energy of spin-1/2 particles given by

perturbative. For a generic nonintegrable models with

Eq. (21) is consistent with predictions of previous stu-

short-ranged interaction, it can be generalized to ar-

dies [35, 39].

bitrary interaction strength. It can further be shown

We proceed with discussion of the effects of thermal

that τ-1qp ∝ ε⁸ remains valid, and the pre-factor can be

broadening on relaxation processes. In the Fermi liquid

expressed in terms of the exact spectrum [31].

picture one expects a simple crossover at excitation en-

As should be anticipated from the discussion above,

ergies of the order of temperature ε ∼ T. For 1D liquids

electron spin plays a crucial role in the transition ma-

this is not the case, as even at T < ε there are inter-

trix element for the three-particle process, and should

mediate regimes and relaxation shows nontrivial tem-

thus significantly affect the quasiparticle decay rate.

perature dependence. Indeed, at finite temperatures

Indeed, in the spinless case antisymmetry of the elect-

each collision results in a typical momentum trans-

ron wave function dictates that its orbital component

fer qi ∼ T/vF allowed by thermal smearing of states

should be odd and therefore relevant exchange ampli-

near the Fermi energy. As RRL relaxation is controlled

tudes are suppressed by Pauli exclusion. Mathemati-

by the momentum transfer between the branches, one

cally, one sees this in a cancellation of various terms

needs to compare phase spaces available to left movers.

that lead to Eq. (18). In contrast, for spinful elect-

Since at zero temperature q₃ ∼ ε²/v2F p_F , one deduces

rons singular parts of the amplitude do not cancel.

from comparison to q₃ ∼ T/v_F the crossover scale ε_T ∼

782

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

∼

√ε_F T . Technically, this argument can be also made

where ε_d = v_F /d. In fact, ∝ T⁷ is a generic property

clear by observing that

for any non-integrable finite-range interaction model

with a sufficient degree of analyticity at small momenta

∑

[

]-1

⁽v_Fq₃)

q3 exp

-1

[32, 33]. Lastly, in the case of spin-1/2 chiral electrons

f-k3 (1 - f-k3-q3 ) =

2π

one estimates the decay rate to be of the form

and reducing to LT/2πv_F as q₃ → 0. These consider-

Tε6T

⁽d⁾

τ-1qp = c₉g⁴

ln⁴

(27)

ations suggest that Eqs. (16), (19), and (21) are valid

ε²ε4d

for T ≪ ε²/ε_F . Above this threshold one finds

In addition to relaxation of particles with the same

(

)₂

chirality, thermal broadening allows for the relaxation

τ-1qp = c₄ε_F g⁴λ²¹(p_F w)

(22)

ε_F

of hot quasiholes, a process kinematically forbidden at

zero temperature. The derivation of the corresponding

instead of Eq. (16) for the spinless Coulomb case. Si-

decay rate τ-1qh proceeds in close analogy to that for the

milarly,

RRL-process. Crucial modifications are (i) the sign of

(

)₆

q₃, (ii) a smaller phase space volume, now suppressed

)

⁽ε_F

τ-1qp = c₅ε_F g⁴λ²²(p_F d)

ln²

(23)

by an additional factor ∼ T/(ε²/ε_F ), and (iii) that it

ε_F

takes ∼ (ε/ε_T )² steps to relax the excess energy. As a

result, the quasihole relaxation rate e. g. for the spin-

instead of Eq. (19) for the spinless screened case, and

1/2 model,

finally

)

⁽ε_F

⁽T)2

⁽ε_F )

τ-1qp = c₆ε_F g⁴λ²³(p_F w)

ln²

(24)

τ-1qh = c₁₀ε_F g⁴λ²³(p_F w)

ln²

(28)

ε_F

instead of Eq. (21) for the spin-1/2 Coulomb case. The

is by a factor (ε_T /ε)⁴ smaller than τ-1qh defined in

set of coefficients c4,5,6 can be determined from numer-

Eq. (21) when taken at the same energy. This pro-

ical integrations, however, their specific values are of

nounced asymmetry in the relaxation rates of elect-

no particular significance here.

ron-like and hole-like excitations is a direct consequence

At elevated temperatures the above mechanism of

of the 1D kinematics of three-particle scattering with

relaxation competes with another process involving

nonlinear spectrum. This feature marks a sharp dis-

only particles of the same chirality. As indicated ear-

tinction between the quantum 1D Fermi-Luttinger liq-

lier, this RRR- (or equivalently LLL-) process is kine-

uids and higher dimensional Fermi liquids.

matically possible only at finite energies. It follows

We summarize in Table the discussed quasiparticle

from the same amplitude Eq. (6), but admits differ-

relaxation rates in the different regimes.

ent conditions on the involved momenta. In this pro-

cess, a high-energy particle with excess energy ε can

2.3. Distribution imbalance rates. Another

relax on two other comoving particles, which during

common technique in kinetic theory applied to the de-

the collision are scattered in opposite directions in en-

termination of relaxation rates is to project the collision

ergy. Namely, one is drifting slightly upwards in energy,

integral onto specific modes of interest, to infer their

whereas the other float downwards, closer to the Fermi

corresponding decay times. For instance, in the context

energy. A detailed calculation in the spinless Coulomb

of the present problem, one can look at the thermal im-

model shows that the corresponding relaxation rate is

balance relaxation between left- and right-movers. This

given by

amounts to projecting the collision term onto the ener-

, which is even

gy mode of the distribution function n_p

⁽ε_w)

in momentum.

τ-1qp = c₇g⁴(p_F w)⁴

ln²

(25)

εε_F

To see the practical implementation of this method,

consider a situation in which right-movers are hotter

where ε_w = v_F /w. This rate exceeds that given in

than left-movers. The goal is then to derive an equa-

Eq. (22), provided that temperature is higher than

√

tion which describes the relaxation of the difference in

∼ ε

ε/ε_F . In the case of screened Coulomb interac-

temperatures ΔT = T^R - T^L of left- and right-moving

tion, the same mechanism is more strongly suppressed

electrons. It should be noted that the physical set-

⁽ε_d)

⁽ε⁾

ting with imbalanced temperature is justified in 1D:

τ-1qp = c₈g⁴(p_F d)⁸

ln²

(26)

εε5F

while three-particle collisions generate both right- and

783

A. Levchenko, T. Micklitz

ЖЭТФ, том 159, вып. 4, 2021

Table. Energy and temperature dependencies of quasi-

The next step is to also linearize the right-hand-si-

particle relaxation rates (only the leading parametric

de of Eq. (29) in ΔT . To accomplish this task we

behavior is indicated and logarithmic terms are omit-

parametrize n_p = f_p+f_p(1-f_p)ψ_p, which allows to con-

ted for brevity). First two rows summarize results for

veniently take advantage of the detailed balance con-

spinless electrons interacting via Coulomb and screened

dition in the collision integral St{n}. For the thermal

short-range interaction models, respectively, and the

imbalance ψ_p = (ε_p - ε_F )ΔT/T², and one finds upon

last row gives the result for the spin-1/2 model. The

expansion in ΔT

first two columns describe processes involving particles

∑

of both chiralities (e. g. the RRL-process), and the

(εp1 - ε_F )St{n} =

last column describes the relaxation of comoving par-

p1>0

ticles with only same chirality (e. g. the RRR-process).

∑

√

ΔT

(v_F k₁)(v_F q₃)W.

(32)

In all cases T1 ∼ ε²/εF , while T2 ∼ ε

ε/εF in the

√

T²

{k,q,σ}

Coulomb model, T2 ∼ ε⁶

ε/εF in the screened model,

√

and T2 ∼ ε(εd/εF )³

ε_d/ε in the spinful model

Here

τ-1qp

T <T₁

T₁ < T < T₂ T₂ < T < ε

W = Wfk1(1 - fk1+q1)fk2(1 - fk2+q2)×

Coulomb ε⁴/ε3F

Tε²/ε2F

T³/εε_F

× f-k3(1 - f-k3-q3),

(33)

Screened ε⁸/ε7F

Tε⁶/ε6F

T⁷/εε5F

and at intermediate steps we made use of the energy

conservation implicit in W , and approximated εp1 -

Spin-1/2

ε²/ε_F

Tε6T/ε²ε4d

- ε_F ≈ v_Fk₁ and ε_p′

- εp3 ≈ -v_Fq₃. It is now evident

that Eq. (29) can be cast in form of the usual relaxation

time approximation,

left-moving particle-hole pairs the intrabranch relax-

ation induced by these processes is faster, while inter-

∂_tΔT = -ΔT/τ_th,

(34)

branch is a slow.

We start from the Boltzmann equation, multiply

where we introduced the corresponding thermalization

both sides by εp1 - ε_F , and sum over p₁ > 0

time. For the kinematics of the RRL-process, the latter

∑

evaluates to

(εp1 - ε_F )∂_tnp1 =

(εp1 - ε_F )St{n},

(29)

)₂

p1>0

⁽T

⁽ε_F )

-1

τ_t

= c₁₁ε_Fg⁴λ²³(p_Fw)

ln²

(35)

where, as above, momentum p₁ is that of a right-

ε_F

moving particle. We then assume np1 to be of Fer-

In a similar fashion one can find the relaxation rate for

mi-Dirac form with nonequilibrium temperature T^R =

the odd part of the imbalanced distribution. For this

= T +ΔT of right-moving excitations, and linearize in

purposes one may consider a boosted frame of referen-

the left-hand-side with respect to ΔT ,

ce, ε_p - pu, and derive the relaxation equation for u

(εp1 - ε_F )∂_tΔT

by projecting the collision integral onto the momentum

∂_tnp1 = ∂_T np1 ∂_tΔT =

(30)

⁽εp1 -ε_F )^.

mode. Kinematics of the respective collision is different

4T² ch²

though, and will be considered in the next section.

To get an idea of the order of magnitude of the

When computing integral over p₁ it is convenient to

different timescales, it is instructive to consider the fol-

shift momentum to the respective Fermi point, p₁ =

lowing estimates for GaAs quantum wires using exper-

= p_F + k₁. Linearizing further the dispersion relation

imental parameters of Ref. [52]. For v_F ∼ 2 · 10⁵ m/s

in k₁, εp1 - ε_F ≈ v_F k₁, one may use that the integral

and κ ∼ 10, the interaction parameter is just within

is peaked at p_F and rapidly converging. Noting that

the applicability criterion of the perturbative expres-

∫

sions g ∼ 1. For the typical electron density we use

z²dz/ ch²(z) = π²/6,

p_F ∼ 10⁸ m-1, w ∼ 10 nm, and ε_F ∼ 1 meV. Then for

-∞

ε ∼ ε_F/4, which is a typical excess energy of injected

particles in tunneling experiments, and T ∼ 0.25 K one

one readily finds

is securely in the regime T ≪ ε²/ε_F . For this set of

∑

πLT

parameters τ-1qp ∼ 10¹¹ s-1, τ-1qh ∼ 10⁹ s-1, and τ-1th ∼

(εp1 - ε_F )∂_tnp1 =

∂_tΔT.

(31)

6v_F

∼ 10⁶ s-1.

p1>0

784

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

2.4. Backscattering hole mobility rates. Rela-

change in each scattering event is small, and holes ef-

xation processes of low-energy excitations leading to

fectively drift through the bottom of the band. Thus

the decay of quasiparticles near the Fermi energy do not

relaxation occurs in multiple steps and the underlying

change the numbers of right- and left-moving particles.

dynamics is momentum space diffusion. Under these

Thus they are chirality conserving. It turns out that it

conditions, the mobile impurity falls into the universal

is also possible to have backscattering processes. The

class of problems described by a Fokker-Planck equa-

kinematics of these collisions involves states deep in the

tion [72]. The collision integral Eq. (36) can then be

Fermi sea, and for this reason it is useful to consider

simplified by expanding in the small momentum step

the mobility of holes at the bottom of the band. These

q₁ ≪ p₁, and maps to the differential operator

processes are commonly considered from the perspec-

tive of mobile impurities in a Luttinger liquid [63-69].

St{hp1 } ≈ -∂p1 [A(p₁)hp1 ] +

∂2p

[B(p₁)hp1 ] .

(38)

Here we will continue using the kinetic equation ap-

proach for their description. The idea is then to single

Here we introduced

out hole states at the bottom of the band with small

∑

momenta, and to derive an effective kinetic equation

A(p₁) = - q₁Pq1(p₁),

capturing their dynamics and allowing the calculation

∑

(39)

of corresponding backscattering rates [70,71].

B(p₁) =

q²¹Pq1 (p₁),

For this purpose, let p₁ and p′1 be momenta near the

band bottom, p₂ and p′2 lie near the right Fermi point

(+p_F ), and p₃ and p′3 be taken near the left Fermi point

and used the short-hand notation Pq1 (p₁) = P(p′1, p₁).

(-p_F ). As before, the unprimed momenta correspond

The diffusion coefficient in momentum space B(p₁) is a

to incoming states whereas primed ones are associated

function of the hole-momentum p₁ varying on a scale

with outgoing states. With these conventions, we intro-

set by p_F . For holes at the bottom of the band, one

duce the hole distribution function, hp1 = 1 - np1 , and

may thus approximate B(p₁) by its value at p₁ = 0,

the collision integral for holes, St{hp1 } = -St{np1 }.

in the following simply denoted by B without argu-

Starting from Eq. (11), the latter can be cast in the

ment. Furthermore, the drift coefficient A(p₁) is read-

form

ily obtained from noting that the collision integral (38)

∑[

]

has to vanish for hole distributions of an equilibrium

St{hp1 } =

P (p₁, p′1)h_p′

- P(p′1,p₁)hp1

(36)

Boltzmann form. This condition leads to the relation

p^′

A(p) = pB/2m^∗T .

where

The rest of the calculation depends on the struc-

∑

P (p₁, p′1) = 12

Wfp2(1-f_p′

)fp3 (1 - f_p′

)

(37)

ture of the amplitude for the given kinematics of the

{σ}

p2p3

three-particle process. In calculating A from Eq. (6)

p′2p′3

for the momentum configuration under consideration,

is the rate for a transition in which a hole scatters from

and up to small corrections in T/ε_F ≪ 1, it is sufficient

some state p′1 into p₁, while P(p′1, p₁) denotes the rate

to approximate p₁ ≈ 0, p₂ ≈ +p_F and p₃ ≈ -p_F . Mo-

for the inverse process. In the above sums, all momenta

mentum and energy conservations provide additional

have been restricted to the discussed ranges, which ex-

restrictions on the transferred momenta, enforcing that

plains the combinatorial overall factor of 12. Since both

q2 ≈ q3 ≈ -q1/2, again up to small corrections in

p₁ and p′1 lie near the bottom of the band, the distri-

T/ε_F

≪ 1. As a result, the amplitude A can be

bution functions hp1 and h_p′ are exponentially small

parametrized only by a single momentum q₁. Expan-

∝ e-εF/T due to Pauli exclusion, and so is the collision

ding Eq. (6) and summing over spins one then finds

integral of holes St{h_p}. It is therefore unnecessary

∑

p2F

to account for additional exponentially small contribu-

|A|² =

V2p

(VpF - V2pF )²

(40)

tions in the transition rates P(p₁, p′1) and P(p′1, p₁),

ε2F L⁴

q²

{σ}

and this is why we replaced fp1 ≃ 1 and f_p′

≃ 1 in

both. As in the case of the forward scattering process,

The singularity of A at small momenta is cancelled in

the typical scale for momentum change of all three par-

the spinless case. Specifically, for the long-range inter-

ticles in a hole backscattering is set by temperature,

action model with Eq. (10) one finds

qi = p′i - pi ∼ T/vF. At the same time, the typical

√

momentum of a hole is p₁ ∼

m^∗T so that q₁/p₁ ∼

⁽2e² )⁴

⁽p_F )

√

A≈

λ²¹(p_F w)ln²

(41)

∼

T/ε_F ≪ 1. This means that the net momentum

16ε2FL4

|q₁|

785

ЖЭТФ, вып. 4

A. Levchenko, T. Micklitz

ЖЭТФ, том 159, вып. 4, 2021

whereas for the screened model

energy. Thus for Δp² ≃ m^∗T the result is (omitting

numerical factor for brevity)

9(ln 4 - 1)

⁽2e² )⁴

A≈

λ²²(p_F d).

(42)

)₂

ε2FL4

-1

⁽ T

τ_d

≃ g⁴λ²³(p_Fw)

(47)

ε_F

In order to perform remaining momentum integrations

implicit in the definition of B, one can approximate

Finally we recall that the mobility of particles μ is re-

delta functions in the scattering probability by

lated to the diffusion constant by the simple kinetic

formula μ = T/B, and therefore μ ∝ 1/T².

The result is different in the spinless case. From

δ_P,P′ δ(E - E^′) ≈

δ(q₂ - q₃)δ_q

2,-q1/2.

v_F

Eqs. (41), (42) and (45) one finds B ∝ T⁵ in both cases,

modulo a logarithmic factor ln² T in the Coulomb case,

This removes two integrations out of five, and gives

. The results dis-

and thus τ-1dh ∝ T⁴ and μ ∝ 1/T⁴

∑

cussed in this section are again perturbative in the in-

12L

q²

|A|²f_k

2-q1/2(1-fk2)×

teraction. The power laws in the temperature depen-

v_F

q1k2k3

{σ}

dence of relaxation rates are, however, generic and also

apply to the strongly interacting regime, as we further

(43)

× fk3+q1/2⁽¹- fk3),

elaborate below, see also Refs. [67, 68].

2.5. Electron-phonon relaxation rates. Apart

where we shifted momenta p2,3 to the respective Fermi

from the purely electronic mechanisms of relaxation

points, ±p_F + k2,3, and linearized the dispersion rela-

electrons may scatter on phonons, disorder, and sam-

tion in all Fermi occupation functions. Finally, using

ple imperfections thus relaxing their energy and mo-

the tabulated integral

mentum. At extremely low temperatures phonons are

∑

not expected to be efficient at cooling the electronic

fk+q(1 - f_k) =

qb_q, b_q =

(44)

sub-system. On the other hand, electron-phonon scat-

2π

e^vF^q/T - 1

tering has no such severe phase space restrictions like

where b_q is the equilibrium Bose distribution, we arrive

the three-particle collisions considered above. It is thus

at the general expression

instructive to estimate the temperature dependence for

the corresponding relaxation rate. Unlike the previous

)₃ ∑

∑

6π

⁽L

studies of electron-phonon relaxation in multichannel

q⁴

|A|²b_q

1/2(1+bq1/2).

(45)

v_F

2π

quantum wires [73, 74], and phonon-induced backscat-

{σ}

tering relaxation [75, 76], we focus on the complemen-

A notable feature of this expression is that it is entirely

tary effect of soft collisions in a single-channel geometry

expressed in terms of bosonic modes. In essence, this is

of strictly 1D electrons and 3D phonons.

a manifestation of bosonization at the level of fermionic

The coupling of electrons and phonons is described

kinetic theory, as the occupation of an electron-hole

by the collision integral [77]

pair near one of the Fermi points integrated over the

St{n_p, N_q} =

center of mass momentum is equivalent to a collective

boson emitted/absorbed in a course of hole diffusion.

∑

It will be shown in the subsequent section that struc-

= W_-[n_p′(1 - n_p)N_q-n_p(1-n_p′)(1+N_q)] +

turally the same expression for B can be obtained from

p^′q

a purely bosonic formulation of the problem. Finally,

∑

inserting Eq. (40) into Eq. (45) one finds the momen-

+ W₊[n_p′(1-n_p)(1+N_q)-n_p(1-n_p′)N_q] ,

(48)

tum space diffusion coefficient of spin-1/2 holes

p^′q

)₃

where the scattering rate

768 ln²(2)

⁽ T

g⁴λ²³(p_F w)

p2F ε_F .

(46)

ε_F

(49)

W±(p, p^′, q) = (2π)|A(q)|²δ(εp-εp′ ±ωq)δp=p′ ±qx

The corresponding backscattering relaxation rate can

describes phonon emission and absorption processes

be found from Einstein relation adopted to diffusion in

with an amplitude

momentum space, Δp² = Bτ_dh. The notation τ_dh is

√

meant to emphasize kinetics of a deep hole as opposed

A(q) =

(D|q| + iΛ).

to earlier notation τ_qh

describing quasiholes near Fermi

2ϱVω_q

786

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

Here we took into account that at the level of the lead-

of Fermi and plasmon velocities. In the TL model u =

√

ing Born approximation, the probabilities of scattering

=v_F

1 + V₀/πv_F.

for direct and reverse processes are the same. In the

2.6. Spin-charge scattering rates. The applica-

amplitude we include both deformation (D) and piezo-

bility of the Born approximation, used to construct the

electric (Λ) couplings, ϱ is the mass density, q_x the

quantum amplitude for triple particle processes cap-

phonon wave-vector along the wire, and V is the sys-

tured by Eq. (6), requires that incoming spin-1/2 quasi-

tem volume. For simplicity we assume only a single

particles have sufficiently high energy compared to the

acoustic branch ω_q = s|q|, with sound velocity s.

typical scale of interparticle interaction ε ≫ m^∗v_F V₀.

For equilibrium Fermi and Bose distribution func-

In the generic interacting environment of a 1D

tions of electrons and phonons respectively, n_p → f_p

quantum fluid, quasiparticle excitations break down

and N_q → b_q, the collision integral in Eq. (48) vanishes

into spin and charge modes. At the level of linear Lut-

due to detailed balance condition. As in the above ex-

tinger liquid theory, spin-charge separation is an ex-

ample of the distribution imbalance relaxation, we then

act property of the model [14]. At weak coupling, the

assume that electrons are hot, that is, at an excess tem-

perature T + ΔT with respect to the temperature T of

splitting between velocities of collective spin (v_σ) and

charge (v_ρ) density waves is related to the forward scat-

lattice phonons. Electron-phonon collisions tend to re-

tering component of the interaction v_ρ -v_σ ∼ V₀ (recall

lax ΔT , and the corresponding rate for relaxation can

that for repulsive interactions v_ρ > v_σ). Assuming then

be found by projecting the collision integral onto the

thermal excitations with ε ∼ T , the Born condition can

energy mode,

be equivalently formulated as T/(m^∗v_F ) ≫ v_ρ - v_σ.

∑

ϵ_p n_p = -

ϵ_pSt{n_p, N_q},

In other words, for fermionic quasiparticles to preserve

their integrity the excitation energy (or temperature)

should be bigger than the energy scale of spin-charge

with ϵ_p = ε_p - ε_F . To linear order in ΔT one finds

separation.

from the phonon emission processes of hot electrons,

The interplay of spectrum nonlinearities and in-

∂_tΔT = -ΔT/τ_ep, where

teractions leads to spin-charge coupling [78, 79]. Al-

though irrelevant in the renormalization group sense,

6v_F

τ-1ep = -

the newly emerging higher order operators capture

πT³L

the attenuation of quasiparticles. The kinetic prop-

∑

erties of 1D quantum liquids with spin-charge cou-

× Wϵ_pω_qfεp(1 - fεp)(fεp+ωq + bωq).

(50)

pling are not fully understood. There are basically

pp^′q

two possible approaches one may pursue. The first is

Upon completion of the remaining momentum integra-

to refermionize the nonlinear bosonic theory to obtain

tions, we then find to leading order in T

an effective description in terms of dressed quasiparti-

cles: holons and spinons. Holon relaxation was conside-

9ζ(3)

τ-1ep =

T (Λ²/s²v_F ϱ).

(51)

red in Refs. [36,37] based on non-Abelian bosonization

8π³

[80]. The advantage of this complex theory is that, in

principle, it allows to go beyond the weakly interact-

The scattering rate due to the deformation potential

is parametrically weaker, scaling as τ-1ep ∝ T³. The

ing limit for spinful fermions. Alternatively, one may

backscattering mechanism results in an activated tem-

choose to continue working in bosonic language. In the

perature dependence ∝ e-TA/T with T_A = 2sp_F . It

limit of weak backscattering one can then account for

is straightforward to generalize Eq. (51) to the case

spin-charge interaction perturbatively in the basis of

when electronic relaxation occurs via several acoustic

well-defined spin and charge modes. This second pro-

branches. Notice also that the piezoelectric potential

cedure is limited to weak interactions V2pF ≪ V₀ ≪ v_F .

may have complicated angular dependence in case of

To complement previous studies, we follow in this sec-

wires oriented arbitrarily with respect to the crystallo-

tion the second path. In part this will enable us to ex-

plore the fermion-boson duality. We delegate technical

graphical axis of the sample. A proper angular aver-

aging would change then numerical factors in Eq. (51)

details of bosonization to the Appendix (see full text)

where we took the simplest geometry. Luttinger liquid

and elucidate here the impact of spin-charge scattering

effects lead to renormalization of the linear-T behavior

on various decay rates.

and transform it into a power-law with interaction de-

The lowest order nonlinearity, compatible with

pendent exponent ∝ T^K, where K = v_F /u is the ratio

SU(2) symmetry of the problem is cubic. It contains

787

14*

A. Levchenko, T. Micklitz

ЖЭТФ, том 159, вып. 4, 2021

one charge and two spin operators. Treating this term

which after momentum integration gives a factor of

in a perturbative expansion generates a collision ker-

(πLT/6v_ρ)∂_tΔT . The right-hand-side can be linearized

nel that describes the decay of a plasmon into two spin

with the usual substitution Nq = bq + bq(1 + bq)φq,

modes ρ → σσ. It reads

where φ_q = ω_qΔT/T² for the case of a thermal imbal-

∑

[

ance. After some algebra one finds

St{N^ρ, N^σ} = - W

N^ρq(1+N^σq

)(1+N^σ ) -_q

∑

q1q2

ω^ρqSt{N^ρ, N^σ} =

]

q>0

- (1 + N^ρq)N^σq

N^σ

(52)

∑

where N^ρ/σ are the bosonic occupations of charge (ρ)

= -ΔT (ω^ρq/T)²W(1 + b^ρq)b^σq

b^σ ,

(57)

and spin (σ) excitations. The scattering probability

qq1q2

W = 2π|A|²δq=q1+q2δ(ω^q - ωq1 - ωq )

(53)

where we repeatedly used energy conservation and the

contains an amplitude scaling cubically with momenta

detailed balance condition. Performing the final inte-

of the bosons |A|² = (π³/8L)|q||q₁||q₂|Γ2ρσσ. The per-

grations, we then arrive at

turbative result for the coupling constant is Γρσσ =

√

3π

=V^′

2π², where the prime denotes the derivative

τ-1ρ =

Γ2ρσσ(T³/v4σ)F(v_σ/v_ρ),

(58)

2pF

with respect to p_F . Note that it thus vanishes for the

integrable case of constant interaction. At smallest

where the dimensionless function reads

momenta the dispersion relations are linear ωρ/σ =

∞

∫

= v_ρ/σ|q|. The kinematics of this process uniquely

F (κ) = κ(1 - κ²) dz z⁵(1 + b_z)bz+bz-

(59)

fixes momenta in the final state. Indeed, for concrete-

ness let q > 0, then q1 = q(vρ + vσ)/2vσ and q2 =

= -q(v_ρ - v_σ)/2v_σ, which means that spin waves are

with b_z = (e^z - 1)-1 and z_± = z(1 ± κ)/2. One can

counterpropagating. From dimensional analysis it be-

readily check that F → 32π⁴/15 in the limit κ → 1.

comes apparent that St{N^ρ, N^σ} defines the decay rate

The same scattering process can be alternatively

of a plasmon, and one can introduce the characteristic

viewed as a mutual spin-charge friction. Physically,

rate

∑

v2ρ - v2σ

this is analogous to the electron-phonon drag effect,

τ-1ρ =

W ≃ q³(V ′2p

)²

typically studied in the context of thermoelectricity, or

v³

q1q2

Coulomb drag in double-layers [81] and spin Coulomb

For the sake of an estimate, one may now take V^′

∼

2pF

drag [82]. In each of these examples momentum trans-

∼ V2pF/p_F and replace v_ρ/σ ∼ v_F, except in their dif-

fer between interactively coupled systems leads to drag-

ference where v_ρ - v_σ ∼ V₀, and finds the life-time

ging of one sub-system by the flow of the other. For

)₃

instance, in the context of spin physics in Luttinger liq-

V₀

⁽V2pF )²( q

τ-1

∼ε_F

(54)

uids, generation of spin current is possible by Coulomb

v_F v_F

p_F

drag [83]. To estimate the spin-charge drag rate, one

Notice the nonanalytic dependence of interaction ∝ V³.

can consider a boosted frame of reference for spin and

For thermal plasmons the relaxation rate can be cal-

charge excitations with mismatched boost velocities

culated from Eq. (52) by a projection onto an energy

u_ρ/σ. The scattering leads to momentum exchange be-

mode. We observe that as |q₂| ≪ q the relaxation

tween spins and charge and, as a result, to relaxation

occurs by small energy transfer from right-movers to

∂_tu_ρ = -(u_ρ - u_σ)/τ_ρσ. To capture this effect, we lin-

left-movers (or vise versa) so that interbrach processes

earize the collision integral for N(ω_q - uq) with respect

are slow. Assuming that right-moving excitations are

to u, for both spin and charge occupations, and then

hotter by ΔT , and in complete analogy to the fermionic

project onto the momentum mode to calculate the rate

case, we find

of momentum loss by (e. g.) charge modes

∑

ω^ρq∂_tN^ρq =

ω^ρqSt{N^ρ, N^σ}.

(55)

∑

q>0

∂_tP_ρ =

qSt{N^ρ, N^σ} =

The left-hand-side is straightforward to evaluate fur-

ther, noting that

∑

Δu

ω^ρq∂_t

ΔT

q²W(1 + b^ρq)b^σq

b^σ .

(60)

∂_tN^ρq = ∂_T N^ρq∂_tΔT =

(56)

qq1 q2

4T² sh²(ω^q/2T)

788

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

When we compare this to ∂_tP_ρ = (πLT²/3v3ρ)∂_tu_ρ, we

for momenta q ∼ p_F the occupation is small, N^σq ∝

find that thus defined drag relaxation rate τ-1ρσ coin-

∝ e-εσ/T ≪ 1, and correspondingly 1+N^σ ≈ 1, where

cides with Eq. (58) up to a constant factor. It is per-

ε_σ is the band width of spin excitations. The latter is

haps useful to note that τ-1ρσ ∝ T³ is consistent with

parametrically of the order of the spin exchange cou-

the expectation that Coulomb drag transresistivity be-

pling. Viewing Eq. (61) as the collision integral for

tween double quantum wires due to interwire momen-

spins, we thus write

tum transfer from spin-charge coupling at zero mag-

∑

St{N^σ } =_q

[P(q₁, q′1)N^σq′

- P(q′1,q₁)N^σ ],

(63)

netic field scales as ρ_D ∝ T⁵ [37]. Indeed, this rate is

q^′

accompanied by two thermal phase space factors ∼ T

per wire, thus leading to T⁵. In the drag problem,

where

∑

the factor q³ results from the width of the dynamic

P (q′1, q₁) =

WN^ρq′

(1 + N^ρ )

(64)

charge structure factor and the underlying scattering

′

q2q

that gives rise to q³ is precisely the decay of a charge

is the transition rate for spin scattering processes. More

boson into two spin bosons.

specifically, it describes a collision with momentum

The next in complexity is a quartic nonlinearity in

transfer δq, in which a spin is scattered out of the ini-

spin-charge coupling which leads to two-boson scatte-

tial state q₁. It can thus be rewritten as P(q′1, q₁) =

ring ρσ → ρσ. In particular, we consider backscat-

= P_δq(q₁), and following the same prescription, the

tering of spin excitations on plasmons. Such scattering

transition rate for the inverse process reads P(q₁, q′1) =

processes correspond to the diffusion of spin excitations

= P_-δq(q₁ + δq). Performing then a small-momentum

near the spectral edge, and the goal is to calculate the

expansion,

corresponding diffusion constant. As alluded to earlier,

the discussion parallels the previous calculation of the

P (q₁, q′1)N^σq′

≈ P_-δq(q₁)N^σ +_q

backscattering of a deep hole in the fermionic language.

The corresponding collision integral reads

δq²

+ δq∂q1[P_-δq(q₁)N

∂2q

[P_-δq(q₁)N^σ ],

(65)

St{N^ρ, N^σ} =

the collision integral of spin excitations takes the sim-

∑

[

plified form

W N^σq

(1 + N^σq′)Nρq

(1 + N^ρq′ ) -

q2q¹q^′

St{N^σq

} = -∂q1[A_ρσ(q₁)N^σ ]+_q

]

- N^σq′

(1 + N^σq

)N^ρq′

(1 + N^ρ ) .

(61)

∂2q

[B_ρσ(q₁)N

^σ ],

(66)

The scattering rate for this process is given by

where

W = 2π|A|²δ_Q,Q′δ(E - E^′)

(62)

∑

A_ρσ = -

δqP_δq(q₁), B_ρσ =

δq²P_δq(q₁).

(67)

with the amplitude |A|² = (Γ_ρσ/8L)²|q₁q′1q₂q′2|, where

δq

the coupling constant at the perturbative level reads

At this stage we focus on the derivation of P_δq(q₁). The

Γ_ρσ = V^′′

. The notations for momentum and energy

2pF

momentum conservation implicit in W removes the q′2

conservation here are Q = q₁ + q₂ and E = ω^σq

+ω^ρ ._q

integration. We then notice that distribution functions

Let momenta q₁ and q′1 correspond to the initial and fi-

limit the typical momentum transfer and momenta of

nal states of the spin excitation near the spectral edge.

plasmons to q₂ ∼ δq ∼ T/v_F . At the same time, the

Kinematically each momentum is of the order of the

typical momentum of spins at the spectral edge is q₁ ∼

Fermi momentum, q₁ ∼ q′1 ∼ p_F , while their difference,

∼ p_F and it is sufficient to calculate P_δq(p_F). With

q′1 - q₁ ∼ T/v_F , is small. This corresponds to a small

these observations at hand, we can now approximate

momentum change in each collision, which is accom-

energy conservation by

panied by the excitation of plasmons at low momenta

q₂ ∼ q′2 ∼ T/v_F . For this reason the low-energy de-

δ(E - E^′) ≈

δ(q₂ - δq/2).

scription based on Eq. (61) is sufficient to capture this

v_ρ

physics. Under the specified conditions and n complete

This removes the q₂ integral, and we thus arrive at

analogy with the fermionic case, we can convert the

collision integral into a Fokker-Planck differential op-

V2ρσ

(δq/p_F )²

P_δq =

(68)

erator, thus describing the diffusion of spins. Indeed,

1024Lv_ρ sh²(v_ρδq/4T)

789

A. Levchenko, T. Micklitz

ЖЭТФ, том 159, вып. 4, 2021

with the notation V_ρσ = p2F Γ_ρσ. Finally, this defines

of the injected boson is due to scattering with inter-

the diffusion coefficient of spins in momentum space

branch momentum transfer. Indeed, for q₁, q₂, q′1 > 0

associated to ρσ → ρσ scattering channel

momentum conservation implies that q′2 is order q³,

since energy conservation fixes q′2 ≈ -(3ξ2/2)q1q2q′1

)₅

⁽V_ρσ)²( T

Curiously, even though a finite ξ is crucial to resolve

B_ρσ =

p3F v_ρ.

(69)

v_ρ

p_F v_ρ

the kinematic constraints it drops out from the corre-

√

sponding rate provided that q₁ ≪

T/v_ρξ². In this

In addition to spin-charge scattering, nonlinearities

regime v_ρ|q′2| ≪ T, implying that N^ρq′

≈ T/ω^ρq′

and q′2

also allow for spin-spin scattering. Importantly for the

cancels out from W. The decay rate then scales para-

momentum space diffusion, scattering processes with

metrically as τ-1ρ ∝ T q⁴. This estimate is applicable

√

spin-flips are enhanced. They are thus described by a

as long as T/v_ρ ≲ q ≪³

T/v_ρξ².

different scaling of the probability with momentum as

For thermal plasmons, this rate can be estimated

compared to Eq. (68). That is,

more accurately by projecting the collision integral

onto the energy mode. Assuming that the boson with

V2σσ

P_δq =

(70)

momentum q₁ is “hotter” by a temperature difference

8π²Lv_σ sh²(v_σδq/4T)

ΔT , one finds upon repeating the steps from the pre-

vious similar calculations

and this crucial detail is technically speaking traced

back to the non-commutativity of spin operators when

∑

6v_ρ

ωq1 ωq2

calculating the corresponding amplitude. The impor-

τ-1ρ =

T²

tance of spin flips is also apparent at the level of

πLTq1q2

q′1q^′

fermions. Indeed, the ratio of scattering rates between

spinless and spinful cases has exactly the same param-

×WN^ρq

N^ρq

(1 + N^ρq′

)(1 + N^ρq′ ).

(72)

eter (q/p_F )² ≪ 1 as the ratio between probabilities in

For the kinematics of the process specified above, one

Eqs. (68) and (70). The resulting diffusion constant in

sum is removed by momentum conservation setting

the spin-spin channel is then

q^′

= q₁+q₂. Energy conservation removes another inte-

)₃

4π

⁽V_σσ)²( T

gral, setting q′2 = -(3ξ²/2)q₁q₂(q₁+q₂). The remaining

B_σσ =

p3F v_σ.

(71)

integrals can, after rescaling of momentum variables in

v_σ

p_F v_σ

units of temperature, be brought to a dimensionless

A microscopic calculation of the respective coupling

double-integral. This results in

constants for the different scattering channels is a chal-

)₂

(

)₄

lenging task. Known approaches include weak coupling

3c₁₂

⁽V_ρρ

τ-1ρ =

(73)

results obtained via mobile impurity model [69], results

(4π)⁴

v_ρ

p_F v_ρ

for Kondo polarons [66], and calculations in the strong

where the coefficient

interaction limit within the non-Abelian bosonization

∞

∫

framework [37], as well as a model departing from the

x²y²(x + y)ex+ydxdy

c₁₂ =

Wigner crystal limit [84].

(e^x - 1)(e^y - 1)(ex+y - 1)

Two-boson processes also contribute to the thermal-

ization rates [33,85,86]. For charge excitations this re-

and V_ρρ = p2F Γ_ρρ.

sults in a subleading correction to Eq. (58). In the

The two-spin scattering can be analyzed in the same

spin sector the situation is, however, different since

way, starting out from Eq. (61) by changing N^ρ → N^σ.

at the cubic level of nonlinearities spins are kinemati-

The crucial difference is in the momentum dependence

cally forbidden to scatter. In both cases nonlinearity of

of the scattering rate, which is enhanced by spin-flip

the bosonic spectrum plays an important role to open

processes. The resulting spin wave thermalization rate

phase space for such collisions. In order to general-

due to two-boson scattering processes reads

ize the present model, consider first the charge sector

and assume a weakly anharmonic dispersion of plas-

)₂

(

)₂

⁽V_σσ

mons, ω^ρq ≈ v_ρ|q|(1 - (ξq)²). Assume now that a right-

τ-1σ ∼

(74)

v_σ

p_F v_σ

moving boson with momentum q₁ ≳ T/v_ρ is injected

into the Luttinger liquid. For this setting the collision

This final estimate exhausts all possible scattering pro-

term from Eq. (61), with replacement N^σ → N^ρ, dic-

cesses emerging from the quartic corrections to the lin-

tates that the dominant process limiting the lifetime

ear Luttinger liquid model.

790

ЖЭТФ, том 159, вып. 4, 2021

Kinetic processes in Fermi-Luttinger liquids

Funding. This work was supported by the U. S.

15.

D. L. Maslov, Lecture Notes for the LXXXI Les

Department of Energy (DOE), Office of Science, Ba-

Houches Summer School “Nanoscopic Quantum Tran-

sic Energy Sciences (BES) Program for Materials and

sport” (2004).

Chemistry Research in Quantum Information Science

16.

S. Tomonaga, Prog. Theor. Phys. (Kyoto) 5, 544

under Award No. DE-SC0020313. T. M. acknowl-

(1950).

edges financial support by Brazilian agencies CNPq

17.

J. M. Luttinger, J. Math. Phys. N. Y. 4, 1154 (1963).

and FAPERJ.

18.

D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304

The full text of this paper is published in the English

(1965); see also Mathematical Physics in One Dimen-

version of JETP.

sion, Acad. Press, New York (1966).

19.

I. E. Dzyaloshinskii and A. I. Larkin, Sov. Phys.

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