Thermal conductivity of the degenerate one-dimensional Fermi gas

TL;DR

This paper develops a microscopic theory for heat transport in a one-dimensional degenerate Fermi gas, revealing two distinct relaxation processes affecting thermal conductivity at low temperatures.

Contribution

It introduces the concept of thermal conductivity of elementary excitations and analyzes their role in heat dissipation in weakly interacting 1D Fermi gases.

Findings

Identification of two relaxation processes with different rates

Introduction of thermal conductivity of elementary excitations

Quantitative description of heat transport at low temperatures

Abstract

We study heat transport in a gas of one-dimensional fermions in the presence of a small temperature gradient. At temperatures well below the Fermi energy there are two types of relaxation processes in this system, with dramatically different relaxation rates. As a result, in addition to the usual thermal conductivity, one can introduce the thermal conductivity of the gas of elementary excitations, which quantifies the dissipation in the system in the broad range of frequencies between the two relaxation rates. We develop a microscopic theory of these transport coefficients in the limit of weak interactions between the fermions.

Equations

$$\kappa =\frac{\pi Tv\tau }{3\hslash }.$$

$${\epsilon }_p=\frac{p^2}{2m},$$

$${\partial }_t{n}_p+\frac{p}{m}{\partial }_x{n}_p=I[n_p].$$

$$n_p^{(0)}=\frac{1}{e^{\beta {\epsilon }_p-\gamma p-\alpha }+1},$$

$$n_p=n_p^{(0)}+\delta n_p,$$

$$\int \delta n_{p}dp=0,\int {\epsilon }_p\delta n_{p}dp=0,\int p\delta n_{p}dp=0$$

$$j_Q=-\kappa {\partial }_{x}T.$$

$$j_Q=\int \frac{dp}{2\pi \hslash }{\epsilon }_p\frac{p}{m}\delta n_p$$

$${\dot{n}}_p=I[n_p^{(0)}+\delta n_p],$$

$${\dot{n}}_p={\partial }_t{n}_p^{(0)}+\frac{p}{m}{\partial }_x{n}_p^{(0)}.$$

$$\alpha =\alpha (x),\beta =\beta (x),\gamma =0.$$

$${\dot{n}}_p=-g_p^2[({\partial }_x\beta ){\epsilon }_p-{\partial }_x\alpha ]\frac{p}{m},$$

$$g_{p}=\sqrt{n_{p}^{(0)}\Big{(}1-n_{p}^{(0)}\Big{)}}=\frac{1}{2\cosh\frac{\beta\varepsilon_{p}-\alpha}{2}}.$$

$$\int \frac{dp}{2\pi \hslash }p{\dot{n}}_p=0.$$

$${\dot{n}}_p=-g_p^2\frac{{\partial }_x\beta }{m}[{\epsilon }_p-\tilde{\mu }]p,\tilde{\mu }=\frac{1}{2m}\frac{I_4}{I_2},$$

$$I_{\lambda }={\int }_0^{+\infty }dp{g}_p^2{p}^{\lambda }.$$

$$\delta n_p=g_p{\varphi }_p.$$

$$\int \frac{d{p}^{\prime }}{2\pi \hslash }K(p,p^{\prime }){\varphi }_{p^{\prime }}=\frac{{\dot{n}}_p}{g_p}$$

$$\int \frac{d{p}^{\prime }}{2\pi \hslash }K(p,p^{\prime }){\varphi }_{p^{\prime }}^{(l)}=-\frac{1}{{\tau }_l}{\varphi }_p^{(l)},⟨{\varphi }_p^{(l)}|{\varphi }_p^{(l^{\prime })}⟩={\delta }_{l,l^{\prime }},$$

$$⟨{\varphi }_p|{\psi }_p⟩=\int \frac{dp}{2\pi \hslash }{\varphi }_p{\psi }_p.$$

$${\varphi }_p^{(N)}=g_p,{\varphi }_p^{(E)}=\left({\epsilon }_p-\frac{I_2}{2m{I}_0}\right)g_p,{\varphi }_p^{(P)}=p{g}_p,$$

$${\varphi }_p=-\frac{{\partial }_{x}T}{m{T}^2}\sum _{l\ne N,E,P}{\tau }_l⟨{\varphi }_p^{(l)}|g_p({\epsilon }_p-\tilde{\mu })p⟩{\varphi }_p^{(l)},$$

$$j_Q=\frac{1}{m}⟨g_p({\epsilon }_p-\tilde{\mu })p|{\varphi }_p⟩.$$

$$\kappa =\frac{1}{m^2{T}^2}\sum _{l\ne N,E,P}{\tau }_l⟨{\varphi }_p^{(l)}|g_p({\epsilon }_p-\tilde{\mu })p{⟩}^2.$$

$$n_p=\frac{\theta (p)}{e^{\beta {\epsilon }_p-\gamma p-\alpha -\delta \alpha }+1}+\frac{\theta (-p)}{e^{\beta {\epsilon }_p-\gamma p-\alpha +\delta \alpha }+1}.$$

$$\delta n_p=\delta \alpha g_p^2\left(\mathrm{sgn}p-\frac{I_1}{I_2}p\right).$$

$${\varphi }_p^{(J)}=C_J{g}_p\left(\mathrm{sgn}p-\frac{I_1}{I_2}p\right)$$

$$C_J=\sqrt{\frac{\pi \hslash I_2}{I_2{I}_0-I_1^2}}$$

$$⟨{\varphi }_p^{(J)}|g_p({\epsilon }_p-\tilde{\mu })p⟩=\frac{C_J}{2\pi \hslash m}\frac{I_3{I}_2-I_4{I}_1}{I_2}.$$

$$I_{\lambda }$$

Full text

I Introduction

Relaxation of one-dimensional systems toward equilibrium has a number of special features. The two-particle scattering processes, which control relaxation in higher dimensions, are strongly restricted in one dimension by the conservation laws, and do not lead to effective relaxation of the system. As a result, the relaxation is dominated by three-particle processes. In a quantum system at low temperature $T$ these scattering processes are strongly suppressed, resulting in a slow relaxation toward equilibrium Imambekov et al. (2012). This leads to a different temperature dependence of the transport coefficients at low temperatures. For example, while the bulk viscosity of the three-dimensional Fermi liquid vanishes at $T\to 0$ as $\zeta\propto T^{2}$ Sykes and Brooker (1970), in one dimension it grows as $\zeta\propto T^{-3}$ Matveev and Pustilnik (2017).

Another important feature of one-dimensional systems is that each particle moves in one of only two directions. As a result, at low temperatures the dominant scattering processes with small momentum transfer are very inefficient at changing the direction of motion. This effect is best illustrated in the case of a one-dimensional Fermi gas. The most efficient three-particle process that changes the relative number of the right- and left-moving particles is shown in Fig. 1(a). In order for an electron to change the direction of motion, this process must involve a hole near the bottom of the band. Thus the rate of such processes is exponentially small, $\tau^{-1}\propto\exp(-\mu/T)$ Lunde et al. (2007); Micklitz et al. (2010); Matveev and Andreev (2012a, b), where $\mu$ is the chemical potential. On the other hand, the scattering processes shown in Fig. 1(b) and (c) do not change the numbers of the right- and left-moving particles, but rearrange excitations near the two Fermi points. The corresponding rate of scattering of thermal excitations $\tau_{\rm ex}^{-1}$ scales as a power of temperature. In spinless one-dimensional systems $\tau_{\rm ex}^{-1}\propto T^{7}$ Imambekov et al. (2012); Arzamasovs et al. (2014); Protopopov et al. (2014), while for weakly interacting spin-$\frac{1}{2}$ fermions $\tau_{\rm ex}^{-1}\propto T$ Karzig et al. (2010).

The presence of exponentially slow relaxation processes in the system results in a very large thermal conductivity $\kappa$ in one dimension. Phenomenological treatment DeGottardi and Matveev (2015) of the transport in a spinless one-dimensional quantum liquid based on the Luttinger liquid theory Haldane (1981) yields

[TABLE]

Here $v$ is the velocity of the bosonic excitations in the Luttinger model and $\hbar$ is the Planck’s constant. It is important to note that the above result applies to thermal conductivity measured at low frequencies $\omega\ll\tau^{-1}$. At higher frequencies the exponentially slow relaxation processes of Fig. 1(a) can be neglected. In this case one can assume that the numbers of the right- and left-moving fermions are conserved, and the relaxation in the system is due to the processes of Fig. 1(b) and (c). A small temperature gradient $\partial_{x}T$ still results in a dissipative contribution to the energy current $-\kappa_{\rm ex}\partial_{x}T$ proportional to it, but with a different thermal conductivity $\kappa_{\rm ex}$. The transport coefficient $\kappa_{\rm ex}$ was recently introduced in the two-fluid hydrodynamic theory of one-dimensional quantum liquids Matveev and Andreev (2018a). It describes the thermal conductivity of the gas of elementary excitations of the quantum liquid and appears in the expressions for damping of the sound modes in this system.

In this paper we develop a microscopic theory of thermal conductivity of a one-dimensional Fermi gas with weak interactions between the particles. Our main focus is on the case of spinless fermions, for which the relaxation processes have been studied in considerable detail Khodas et al. (2007); Matveev and Furusaki (2013); Matveev and Andreev (2012a); Ristivojevic and Matveev (2013). At $T\ll\mu$ our result for the thermal conductivity $\kappa$ is consistent with the phenomenological expression (1), while also providing an expression for the relaxation time $\tau$ in terms of the microscopic interaction potential. More importantly, our approach enables us to obtain the thermal conductivity of the gas of excitations $\kappa_{\rm ex}$, for which no phenomenological theory is available. Because the relaxation processes are sensitive to the form of interaction between fermions Khodas et al. (2007); Ristivojevic and Matveev (2013), we find very different temperature dependence of $\kappa_{\rm ex}$ for the short-range and Coulomb interactions.

The paper is organized as follows. In Sec. II we use Boltzmann equation approach to obtain a microscopic expression for the thermal conductivity $\kappa$ of the degenerate one-dimensional Fermi gas. The same technique is applied to the calculation of the thermal conductivity of the gas of excitations in Sec. III, where a general expression and the order of magnitude extimate of $\kappa_{\rm ex}$ are obtained. A careful evaluation of $\kappa_{\rm ex}$ involves a detailed treatment of the relaxation processes shown in Fig. 1(b) and (c), which is presented in Sec. IV. We discuss our results in Sec. V.

II Thermal conductivity of the Fermi gas

II.1 Boltzmann equation approach

We start by evaluating the thermal conductivity of the one-dimensional gas of spinless fermions with the energy spectrum

[TABLE]

where $p$ is the momentum of the fermion and $m$ is its mass. We will subject the system to an infinitesimal temperature gradient $\partial_{x}T$ and obtain the occupation numbers of the fermionic states $n_{p}$ from the Boltzmann equation Lifshitz and Pitaevskii (1981)

[TABLE]

Weak interactions between fermions give rise to the scattering processes accounted for by the collision integral $I[n_{p}]$. In the left-hand side of the Boltzmann equation interactions will be neglected. In this approximation the energy of the fermion (2) does not depend on its position $x$, which enabled us to omit an additional term $-(\partial_{x}\varepsilon_{p})\partial_{p}n_{p}$ in the left-hand side of Eq. (3).

We are considering a translation-invariant system, in which collisions between the fermions conserve not only the total number of particles and energy of the system, but also its momentum. In this case even in thermodynamic equilibrium the system can move with respect to the lab frame with some velocity $u$, and the equilibrium occupation numbers of the fermionic states are given by

[TABLE]

where $\beta=1/T$, $\alpha=\beta\mu$, and $\gamma=\beta u$. In the presence of the temperature gradient $\partial_{x}T$ the occupation numbers deviate from the equilibrium form (4),

[TABLE]

where the small non-equilibrium correction $\delta n_{p}\propto\partial_{x}T$. The distribution function $n_{p}$ depends on the spatial coordinate $x$. We assume that the parameters $\alpha(x)$, $\beta(x)$, and $\gamma(x)$ of the equilibrium part $n_{p}^{(0)}$ of the distribution function (5) are chosen in such a way that the particle, energy, and momentum densities of the system can be evaluated by substituting $n_{p}^{(0)}$ for $n_{p}$. In other words, we impose the conditions

[TABLE]

upon $\delta n_{p}$.

Our immediate goal is to obtain the thermal conductivity $\kappa$, defined by the relation

[TABLE]

Here

[TABLE]

is the dissipative part of the energy current Lifshitz and Pitaevskii (1981).

We will obtain the correction $\delta n_{p}$ to the equilibrium distribution function by solving the Boltzmann equation (3) written in the form

[TABLE]

where $\dot{n}_{p}$ is given by the left-hand side of Eq. (3). Since the collision integral evaluated for the equilibrium distribution $I[n_{p}^{(0)}]=0$, it is important to keep the infinitesimal correction $\delta n_{p}$ in the right-hand side of Eq. (9). On the other hand, $\dot{n}_{p}$ should be evaluated for $\delta n_{p}=0$, i.e.,

[TABLE]

The thermal conductivity is defined in the steady-state regime, in which the parameters $\alpha$, $\beta$, and $\gamma$ of the equilibrium distribution function (4) do not depend on time. In addition, the system is assumed to be stationary, i.e., $\gamma=0$. The latter condition is satisfied due to the gradient of the chemical potential emerging in the system, which compensates for the force acting on the system as a result of the applied temperature gradient. Thus we substitute into Eq. (9) the equilibrium distribution (4) with the parameters

[TABLE]

This results in

[TABLE]

where we have introduced

[TABLE]

The value of $\partial_{x}\alpha$ in Eq. (12) can be found by noticing that the scattering processes accounted for by the collision integral $I[n_{p}]$ conserve the total momentum of the system, i.e.,

[TABLE]

Imposing this condition on Eq. (12), we obtain

[TABLE]

where

[TABLE]

In the zero temperature limit $\widetilde{\mu}\to\mu$.

II.2 Linearized collision integral

Our next step is to obtain $\delta n_{p}$ by solving Eq. (9). Since we are interested in the linear response to an infinitesimal temperature gradient, we can linearize the collision integral in the right-hand side of Eq. (9). In addition, it is convenient to write the resulting integral equation in terms of function $\phi_{p}$ defined by

[TABLE]

As a result, Eq. (9) becomes a linear integral equation

[TABLE]

with a real symmetric kernel $K(p,p^{\prime})$. The latter property means that one can, in principle, obtain an orthonormal set of eigenfunctions of this integral operator,

[TABLE]

with real eigenvalues $-1/\tau_{l}$. Here we defined the inner product by

[TABLE]

Because the collisions result in the evolution of the distribution function toward equilibrium, the relaxation rates $\tau_{l}^{-1}$ are non-negative.

The full set of eigenfunctions $\phi_{p}^{(l)}$ includes three modes with zero eigenvalues. The existence of such zero modes is due to the conservation of the total number of particles $N$, energy $E$, and momentum $P$ of the system. They are obtained by small variations of parameters $\alpha$, $\beta$, and $\gamma$ in the expression for the equilibrium distribution (4),

[TABLE]

where we orthogonalized the modes, but omitted the normalization constants. Indeed, any such variation transforms $n_{p}^{(0)}$ to another equilibrium distribution. Collisions do not modify equilibrium distributions, resulting in vanishing eigenvalues in Eq. (19).

We can now expand $\phi_{p}$ in the basis of the eigenfunctions $\phi_{p}^{(l)}$ and obtain a formal solution of the integral equation (18),

[TABLE]

where we used Eq. (15) for $\dot{n}_{p}$. In the sum in Eq. (22) we excluded the zero modes (21) for which the corresponding $\tau_{l}$ is infinite. This is due to the fact that the overlaps of $g_{p}(\varepsilon_{p}-\widetilde{\mu})p$ with $\phi_{p}^{(N)}$ and $\phi_{p}^{(E)}$ vanish due to opposite symmetries with respect to $p\to-p$, while the overlap with $\phi_{p}^{(P)}$ vanishes because of the momentum conservation condition (14). As a result, the conditions (6) for $\delta n_{p}$ are satisfied.

Next, we notice that due to the last of the conditions (6) one can replace $\varepsilon_{p}\to\varepsilon_{p}-\widetilde{\mu}$ in the expression for the dissipative contribution (8) to the energy current, after which the latter becomes

[TABLE]

Substitution of Eq. (22) and comparison with Eq. (7) yield

[TABLE]

This expression gives the thermal conductivity of the one-dimensional spinless Fermi gas at any temperature. Significant further progress in understanding thermal conductivity can be made in the regime of low temperature, $T\ll\mu$.

II.3 Thermal conductivity at low temperatures

At low temperatures the relaxation of the one-dimensional Fermi gas is dominated by the processes shown in Fig. 1. The process of Fig. 1(a) is exponentially suppressed, whereas relaxation rate $\tau_{\rm ex}^{-1}$ associated with the processes of Fig. 1(b) and (c) has a power-law temperature dependence. It is important to note, however, that understanding the full relaxation of the system to equilibrium requires accounting for the effect of the processes of Fig. 1(a), because the remaining processes do not change the numbers of the left- and right-moving particles in the system.

Let us consider a Fermi gas with zero total momentum, which relaxes to the equilibrium distribution (4) with $\gamma=0$. Keeping in mind the presence of two very different relaxation times $\tau\gg\tau_{\rm ex}$, one concludes that the relaxation of the degenerate one-dimensional Fermi gas proceeds in two steps. First, at the time scales of the order of $\tau_{\rm ex}$ the particle-hole excitations come to equilibrium with each other, but the chemical potentials of the left- and right-moving particles remain different. This means that the distribution function takes the form

[TABLE]

The presence of the new parameter $\delta\alpha$ in the distribution function accounts for the fact that in addition to the total number of particles, energy, and momentum of the system, the difference $J$ of the numbers of the right- and left-moving particles is also conserved at time scales $t\ll\tau$.

To linear order in $\delta\alpha$ and $\gamma$, the momentum density of the Fermi gas with the distribution (25) is $(\delta\alpha I_{1}+\gamma I_{2})/\pi\hbar$. Thus the parameter $\gamma$ in Eq. (25) is related to $\delta\alpha$ by $\gamma=-(I_{1}/I_{2})\delta\alpha$, and the deviation of the distribution function (25) from the equilibrium form $(e^{\beta\varepsilon_{p}-\alpha}+1)^{-1}$ is

[TABLE]

At the second stage of the relaxation process the system slowly approaches equilibrium, $\delta\alpha\propto e^{-t/\tau}$, while the momentum dependence of $\delta n_{p}$ retains the form (26). We therefore conclude that the eigenvalue problem (19) for the collision integral has the solution

[TABLE]

with exponentially small eigenvalue $-1/\tau$. The constant

[TABLE]

is obtained from the normalization condition in Eq. (19).

The remaining relaxation modes $\phi_{p}^{(l)}$ describe how the non-equilibrium distribution function approaches the form (25) at the first stage of the relaxation process, dominated by the processes of Fig. 1(b) and (c). The corresponding relaxation times $\tau_{l}\sim\tau_{\rm ex}$ are much smaller than $\tau\propto e^{\mu/T}$. Thus, at low temperatures $T\ll\mu$, the thermal conductivity (24) is dominated by the mode (27). Evaluation of the corresponding matrix element is straightforward,

[TABLE]

At $T\ll\mu$ the integrals (16) can be approximated by the Sommerfeld expansion

[TABLE]

For the dominant contribution to the thermal conductivity (24) we then find

[TABLE]

The speed $v$ of the low-energy elementary excitations in a Fermi gas at $T\ll\mu$ is the Fermi velocity $v_{F}=\sqrt{2\mu/m}$. Therefore the result (31) is consistent with the phenomenological expression (1) for the thermal conductivity of a spinless one-dimensional quantum liquid. We note that both Eqs. (1) and (31) express $\kappa$ in terms of the relaxation time $\tau$. The latter has been studied in some detail phenomenologically in Refs. Matveev and Andreev (2012a, b), where it was expressed in terms of the quasiparticle spectrum of the quantum liquid. In Sec. V we discuss how one can obtain a microscopic expression for $\tau$ in terms of interactions between fermions.

III Thermal conductivity of the gas of elementary excitations

In Sec. II we found that at $T\ll\mu$ the thermal conductivity of the Fermi gas (31) is proportional to the relaxation time $\tau$ and is, therefore, exponentially large. This result holds as long as thermal conductivity is measured at frequencies $\omega\ll\tau^{-1}$. On the other hand, interesting new behavior of one-dimensional systems is expected in the broad range of frequencies

[TABLE]

It was shown recently Matveev and Andreev (2017, 2018b) that in this regime one-dimensional systems behave like superfluids and support two sound modes, in contrast to a single sound mode at $\omega\ll\tau^{-1}$.

In the presence of an ordinary sound wave the temperature of the system depends on position. This results in dissipation, which is proportional to the thermal conductivity $\kappa$ and contributes to the attenuation of sound Landau and Lifshitz (2013). The same physics applies in the two-sound regime (32), but the resulting contribution to the sound attenuation is controlled by a different thermal transport coefficient $\kappa_{\rm ex}$ Matveev and Andreev (2018a), which has the meaning of the dissipative part of the thermal conductivity at frequencies in the range (32).

To evaluate $\kappa_{\rm ex}$ we will assume that the exponentially long relaxation time $\tau\to\infty$ and then take the limit $\omega\to 0$. In other words, we will find the thermal conductivity assuming that the relaxation processes conserve not only the number of particles, energy, and momentum of the system, but also the difference $J$ of the numbers of right- and left-moving particles. We now adapt the evaluation of the thermal conductivity in Sec. II.1 and II.2 to account for the this additional conservation law.

First of all, the equilibrium distribution $n_{p}^{(0)}$ now takes the form (25) and depends on four parameters, $\alpha$, $\beta$, $\gamma$, and $\delta\alpha$. A small temperature gradient $\partial_{x}T$ results in a small deviation of the distribution function from the equilibrium form, see Eq. (5). Because of the fourth conservation law, in addition to Eq. (6) we impose the condition

[TABLE]

to ensure that the value of $J$ is determined by the equilibrium part of $n_{p}$.

We next obtain $\delta n_{p}$ in the expression for the dissipative energy current (8) by solving the Boltzmann equation in the form (9) with $\dot{n}_{p}$ given by Eq. (10). The conditions (11) must be modified to account for the extra conservation law. First, the condition of zero total momentum of the system now takes the form $\gamma=-(I_{1}/I_{2})\delta\alpha$, see Sec. II.3. Second, in the absence of the relaxation processes of Fig. 1(a) one can no longer exclude the possibility of a time-dependent difference of the chemical potentials of the right- and left-moving particles. This results in the following assumptions regarding the parameters of the equilibrium distribution (25)

[TABLE]

Substituting the expression (25) for $n_{p}^{(0)}$ in Eq. (10) we get

[TABLE]

The values of $\partial_{x}\alpha$ and $\partial_{t}\delta\alpha$ are determined by imposing the condition (14) of conservation of momentum in collisions along with the new condition of conservation of $J$,

[TABLE]

This yields

[TABLE]

where

[TABLE]

We note that by imposing the conservation laws (14) and (36) we ensured that the function $\phi_{p}=g_{p}\nu_{p}$ is orthogonal to both the usual zero modes (21) and the additional zero mode (27) corresponding to the conservation of $J$.

The next step is to find the non-equilibrium correction $\delta n_{p}$ to the distribution function by solving the integral equation (18). Writing $\delta n_{p}$ as prescribed in Eq. (17), and expanding $\phi_{p}$ in the basis of the eigenfunctions $\phi_{p}^{(l)}$ of the linearized collision integral with non-zero eigenvalues, we obtain

[TABLE]

Because $\phi_{p}$ is orthogonal to the four zero modes (21) and (27), the conditions (6) and (33) imposed on $\delta n_{p}$ are satisfied.

Using Eq. (33) and the last of the conditions (6), it is convenient to replace $\varepsilon_{p}p\to\nu_{p}/2m$ in the definition (8) of $j_{Q}$, which results in

[TABLE]

Substitution of the expression (39) for $\phi_{p}$ yields $j_{Q}=-\kappa_{\rm ex}\partial_{x}T$ with

[TABLE]

Unlike the similar expression (24) for $\kappa$, the result (41) assumes the low-temperature regime, $T\ll\mu$, because the transport coefficient $\kappa_{\rm ex}$ is defined only at $\tau\gg\tau_{\rm ex}$.

At $T\ll\mu$ the particle and hole excitations are confined to the vicinities of the two Fermi points $p=\pm p_{F}$, where $p_{F}=\sqrt{2m\mu}$. For such values of momentum we can use Eq. (30) to approximate $\nu_{p}$ defined by Eq. (38) as

[TABLE]

For typical values of momentum, $|p|-p_{F}\sim T/v_{F}$, the leading order correction to Eq. (42) scales as $T^{3}$ and can be easily shown to give a subleading contribution to Eq. (41). Thus, the dominant contribution to $\kappa_{\rm ex}$ can be obtained by combining Eqs. (41) and (42).

We now obtain an order of magnitude estimate of the transport coefficient $\kappa_{\rm ex}$ using Eqs. (41) and (42). The typical relaxation time $\tau_{l}\sim\tau_{\rm ex}$. Evaluation of the inner product (20) adds a factor of order $T/\hbar v_{F}$. Thus the normalization of the eigenfunctions prescribed by Eq. (19) gives $\phi_{p}^{(l)}\sim\sqrt{\hbar v_{F}/T}$. Combining these estimates we find

[TABLE]

To evaluate $\tau_{\rm ex}$ and obtain the numerical prefactor in Eq. (43), one has to carefully consider the collision integral of the Boltzmann equation (3). We present this treatment in the next section.

IV Relaxation of the degenerate Fermi gas to equilibrium

We now evaluate the transport coefficient $\kappa_{\rm ex}$ in terms of the two-particle interaction potential $U(x)$ between the fermions. Our prescription (41) requires one to find the full spectrum of the relaxation rates $\tau_{l}^{-1}$ in the system as well as the respective relaxation modes $\phi_{p}^{(l)}$ by solving the eigenvalue problem (19) for the linearized collision integral. We will show that the problem simplifies considerably for the interaction potentials that decay slowly with the distance between particles, such as the Coulomb interaction. In this case one of the relaxation modes coincides with $\nu_{p}$ given by Eq. (42) up to a normalization factor. As a result the sum in Eq. (41) includes just one term, and the evaluation of $\kappa_{\rm ex}$ simplifies considerably. This is not the case for interaction potentials that decay rapidly with the distance between fermions, which will be considered separately.

IV.1 Coulomb and dipole-dipole interactions

In the case of charged particles, their interactions are usually dominated by Coulomb repulsion. For particles with charge $e$ confined to a narrow channel the interaction potential takes the form $U(x)={e^{2}}/{|x|}$ at distances $x$ that are large compared to the the width of the channel $w$. The behavior of $U(x)$ at $x\lesssim w$ is determined by the nature of the confining potential. The study of the relaxation spectrum requires evaluation of the Fourier transform of the interaction potential

[TABLE]

Substitution of $U(x)={e^{2}}/{|x|}$ into Eq. (44) results in a logarithmic singularity. We therefore account properly for the short-distance behavior of $U(x)$ for particles confined to a channel of width $w$, see Appendix A. At low momenta, $|p|\ll\hbar/w$, we find

[TABLE]

A numerical factor in the argument of the logarithm in Eq. (45) depends on the details of the confinement and is not included in the above expression.

Another important special case is dipole-dipole interaction $U(x)={\Upsilon}/{|x|^{3}}$. It can be realized, for example, in a quantum wire in the vicinity of a metal gate parallel to it. In this case $\Upsilon=2e^{2}d^{2}$, where $d$ is the distance between the wire and the gate. The Fourier transform of dipole-dipole interaction is

[TABLE]

Similarly to Eq. (45) for Coulomb interaction, Eq. (46) is written within the logarithmic accuracy and is restricted to small momenta, $|p|\ll\hbar/w$. In Eq. (46) we omitted a large constant term that corresponds to the contact interaction. For spinless fermions, Pauli principle forbids two particles to occupy the same position in space, and thus the contact interaction does not affect this system.

A special feature of slowly decaying potentials, such as Coulomb and dipole-dipole ones, is that the relaxation processes which involve co-propagating particles, shown in Fig. 1(c), occur at a higher rate than the ones that involve counter-propagating particles, see Fig 1(b). The estimate of the corresponding rates can be obtained from the results of Ref. Ristivojevic and Matveev (2013), which studied the relaxation of quasiparticles with energies much greater than $T$. The typical decay rate of thermal quasiparticles is $\tau_{\rm c}^{-1}\propto T^{2}$ for Coulomb interaction (45), and $\tau_{\rm c}^{-1}\propto T^{6}$ for dipole-dipole interaction (46). These decay rates are larger than the ones involving processes depicted in Fig. 1(b), which occur at rates $\tau_{\rm b}^{-1}\propto T^{3}$ and $\tau_{\rm b}^{-1}\propto T^{7}$, respectively. As a result, at time scales longer than $\tau_{\rm c}$ but shorter than $\tau_{\rm b}$, each branch of excitations independently achieves equilibrium characterized by its own parameters $\alpha$, $\beta$, and $\gamma$. Thus, the distribution function takes a partially-equilibrated form Micklitz et al. (2010)

[TABLE]

Further relaxation of the distribution (47) toward the form (25) is controlled by the processes shown in Fig. 1(b), with the relaxation time $\tau_{\rm b}\gg\tau_{\rm c}$.

In the following we will calculate the eigenmodes and the corresponding rates for relaxation of the distribution (47), which will be sufficient to obtain the thermal conductivity (41). We note that this is a much simpler problem than the full solution of the eigenvalue problem (19). Instead of diagonalizing the full collision integral, which has an infinite number of eigenmodes, the problem is reduced to the study of the evolution of only six parameters in Eq. (47).

At small deviations from equilibrium, we expand Eq. (47) as $n_{p}=\left(e^{\beta\varepsilon_{p}-\alpha}+1\right)^{-1}+\delta n_{p}$, where

[TABLE]

The relaxation of $\delta n_{p}$ of Eq. (IV.1) is constrained by the four conditions given by Eqs. (6) and (33). This leads to four equations for the six parameters of Eq. (IV.1). We use them to express four parameters as a function of the two remaining ones, which we select to be $\beta_{R}-\beta_{L}$ and $\gamma_{R}-\gamma_{L}$. The correction to the distribution function (IV.1) then takes the form

[TABLE]

where

[TABLE]

We note that $\eta_{p}$ is an odd function of $p$, whereas $\sigma_{p}$ is an even one. At low temperature we find

[TABLE]

In this regime $\eta_{p}$ and $\sigma_{p}$ are equal in absolute value.

To find the evolution of the distribution function (47), we consider the rate of change of the occupation number of the state $p$ due to three-particle collisions

[TABLE]

Here the scattering matrix element $\mathcal{A}_{p,p_{1},p_{2}}^{p^{\prime},p_{1}^{\prime},p_{2}^{\prime}}$ depends on the details of the two-body interaction potential Imambekov et al. (2012); Ristivojevic and Matveev (2013) and will be discussed below. The factor $1/12$ accounts for $12$ identical configurations that exist due to unrestricted summations over the two initial ($p_{1}$ and $p_{2}$) and three final ($p^{\prime}$, $p_{1}^{\prime}$, and $p_{2}^{\prime}$) states.

For systems close to the thermal equilibrium, we can linearize the occupation factors by using Eqs. (5) and (17). This yields

[TABLE]

where we introduced

[TABLE]

Using the separation of variables we transform Eq. (55) into the eigenvalue problem [cf. Eq. (19)]

[TABLE]

Here the eigenvalue $1/\tau_{l}$ represents the relaxation rate associated with the eigenfunction $\phi_{p}^{(l)}$ of the collision operator. Multiplying both sides in Eq. (57) by $\phi_{p}^{(l)}$ and performing the summation over $p$, we find

[TABLE]

Here we used the symmetries of $W$ to extend the summation over six equivalent terms, thereby conveniently making the right-hand side symmetric. As a result we have obtained an extra 1/6 prefactor and the square of the expression in parentheses in the right-hand side. We notice that Eqs. (54)–(58) do not have restriction on the summation range and therefore apply for all three kinds of processes shown in Fig. 1.

The momentum inversion symmetry of $W$ enables one to classify the eigenfunctions of Eq. (57) with respect to parity. This guarantees that the eigenfunctions in Eq. (57) will be either odd or even in $p$. Thus, the odd and even components of $\phi_{p}$ obtained from Eq. (49) as $\delta n_{p}/g_{p}$, which are proportional to $g_{p}\eta_{p}$ and even $g_{p}\sigma_{p}$, respectively, relax as two independent eigenmodes. We now normalize them and introduce

[TABLE]

where to leading order in $T/\mu$ the normalization constants are equal,

[TABLE]

Next, we calculate the relaxation rates corresponding to $\phi_{p}^{(\eta)}$ and $\phi_{p}^{(\sigma)}$ using Eq. (58). We split the six-fold summation in that expression into $2^{6}=64$ six-fold summations over momenta that are either positive or negative. Out of 64 different terms, 44 describe particle backscattering, i.e., the numbers of particles with positive (negative) momenta are different in the initial and the final states [see, for example, Fig. 1(a)]. At low temperatures, such processes occur at exponentially long time scales as discussed in Sec. II. They determine the thermal conductivity of the system $\kappa$ but must be neglected in evaluating the thermal conductivity of the gas of excitations $\kappa_{\rm ex}$.

Out of the remaining 20 terms, two have all the six momenta of the same sign. The expression in parentheses in Eq. (58) evaluated for such restricted range of momenta, i.e., all positive or all negative, nullifies for the two eigenfunctions (59) and (60) due to the conservation laws of energy and momentum. Therefore, for the processes represented in Fig. 1(c), in addition to four zero modes (21) and (27), Eqs. (59) and (60) define two additional zero modes. This conclusion is consistent with six-parameter partially equilibrated distribution (47).

The remaining 18 terms of the sum in Eq. (58) are equivalent and contain configurations of six momenta with two near one Fermi point, and the remaining four near the opposite one, in both the initial and final states, see Fig. 1(b). Using the notation of the figure where $q$ and $q^{\prime}$ are negative while $k,k_{1},k^{\prime},k_{1}^{\prime}$ positive, the expression in parentheses of Eq. (58) becomes

[TABLE]

where we accounted for the conservation laws of momentum and energy. Since $\sum_{p}(\phi_{p}^{(\eta)})^{2}=\sum_{p}(\phi_{p}^{(\sigma)})^{2}=L$, where $L$ is the system size, we obtain

[TABLE]

At low temperature, the momenta in the summation in Eqs. (64) and (65) are confined near the corresponding Fermi points. We can thus linearize $(q^{\prime 2}-q^{2})^{2}\simeq 4p_{F}^{2}(q^{\prime}-q)^{2}$. Since $C_{\eta}=C_{\sigma}$, the two relaxation rates (64) and (65) are equal at the leading order in small $T/\mu$.

IV.1.1 Evaluation of $\tau_{\eta}^{-1}$

To find $\tau_{\eta}^{-1}$ we need an expression for the three-particle scattering matrix element $\mathcal{A}$ that enters Eq. (64) via $W$ [see Eq. (56)]. The matrix element $\mathcal{A}$ was calculated in Ref. Ristivojevic and Matveev (2013) for an arbitrary configuration of momenta. In the special case of momenta that corresponds to the process shown in Fig. 1(b), the result of Ref. Ristivojevic and Matveev (2013) for Coulomb interaction takes the form

[TABLE]

Similarly, for dipole-dipole interaction we have

[TABLE]

We now proceed to the evaluation of the expression (64). Using the conservation laws of momentum and energy we find

[TABLE]

because for the process shown in Fig. 1(b), $k_{1}^{\prime}-q\simeq 2p_{F}$. Therefore, we substitute

[TABLE]

in Eq. (64). For thermally excited quasiparticles, $|k^{\prime}-k|\sim|k^{\prime}-k_{1}|\sim T/v_{F}$. On the other hand, from Eq. (68) we find $|q^{\prime}-q|\sim T^{2}/\mu v_{F}$. We therefore find $q^{\prime}=q+O(T^{2}/\mu v_{F})$. After the substitution (69), the remainder of the expression (64) does not depend on the difference $q^{\prime}-q$ apart from the delta functions contained in $W$. We therefore approximate

[TABLE]

The integration over $q$ and $q^{\prime}$ is now straightforward, resulting in $\int dqdq^{\prime}\delta(q-q^{\prime})g_{q}g_{q^{\prime}}=T/v_{F}$. Here we linearized the spectrum at low temperature, such that

[TABLE]

The remaining four integrations involve one delta function. For Coulomb interaction (45) we were able to perform analytically one more integration and found the relaxation rate

[TABLE]

Here $a_{B}=\hbar^{2}/me^{2}$ is the Bohr radius and

[TABLE]

For dipole-dipole interaction (46) we have been able to evaluate the numerical prefactor in the relaxation rate analytically,

[TABLE]

Here the momentum-dependent logarithm originating from the scattering matrix element (IV.1.1) is replaced by $\ln\left({\mu}/{T}\right)$, in accordance with the logarithmic accuracy adopted earlier.

IV.1.2 Evaluation of $\kappa_{\rm ex}$

As a result of separation of time scales $\tau_{\rm c}\ll\tau_{\rm b}$, at scales longer than $\tau_{\rm c}$ there exist only two relaxation modes of the distribution function given by Eqs. (59) and (60). Up to a normalization constant, the former one, $\phi_{p}^{(\eta)}$ coincides with $g_{p}\nu_{p}$ of Eq. (42). Therefore the eigenmode (59) actually exhausts the sum in Eq. (41) yielding

[TABLE]

The overlap entering the latter expression can now be easily obtained by a comparison between Eqs. (42) and (52):

[TABLE]

which gives

[TABLE]

This is our final expression for the thermal conductivity of the gas of elementary excitations. It applies to systems with long-range two-body interactions. For the Coulomb and dipole-dipole interactions, the relaxation time $\tau_{\eta}$ is given by Eqs. (72) and (74).

We stress that the simplification (77) corresponding to just one eigenmode of the linearized collision integral contributing to the general expression (41) holds only for the long-range interactions. A different scenario occurs in systems where the interaction potential decays rapidly with the distance. In this case the evaluation of $\kappa_{\rm ex}$ requires more involved study of the relaxation modes of the collision integral, which we turn to next.

IV.2 Short-range interactions

IV.2.1 Scattering matrix element

In the case of short-range interactions, the rates associated with the processes of Figs. 1(b) and (c) scale with the temperature as $\tau_{\rm b}^{-1}\propto T^{7}$ Imambekov et al. (2012); Arzamasovs et al. (2014); Protopopov et al. (2014) and $\tau_{\rm c}^{-1}\propto T^{14}$ Protopopov et al. (2014), respectively. Thus, unlike the cases of Coulomb and dipole-dipole interactions, at low temperatures $\tau_{\rm b}\ll\tau_{\rm c}$. As a result, only the processes of Fig. 1(b) need to be taken into consideration one .

At low temperature $T\ll\mu$, when all the states are close to the respective Fermi points, the scattering matrix element takes the form

[TABLE]

where $L$ is the system size. This expression was obtained in Ref. Matveev and Furusaki (2013) for a spinless quantum liquid with arbitrarily strong interactions. It is consistent with the matrix element used in Ref. Khodas et al. (2007) provided that

[TABLE]

where $V(p)$ is the Fourier transform (44) of the interaction potential. We note that the calculation of Ref. Khodas et al. (2007) assumes that $V(p)$ falls off rapidly away from the peak at $p=0$, such that $V(2p_{F})$ is negligible compared with $V(0)$.

In Ref. Matveev and Furusaki (2013) the parameter $\Lambda$ was expressed in terms of the properties of the quasiparticle spectrum of the spinless quantum liquid. In Appendix B we apply that prescription to the weakly-interacting spinless Fermi gas and find

[TABLE]

This expression recovers Eq. (79) at $V(2p_{F})=0$.

IV.2.2 Linearized collision integral in the low-temperature limit

Collision processes shown in Fig. 1(b) change the occupation number of the state $k$ on the right-moving branch with the rate

[TABLE]

Here we assume that the sums over $k^{\prime}$, $k_{1}$, and $k_{1}^{\prime}$ are limited to the right-moving branch, while those over $q$ and $q^{\prime}$ are limited to the left-moving one, Fig. 1(b). The factor $1/2$ compensates for the double counting in the sum due to the permutation of $k^{\prime}$ and $k_{1}^{\prime}$. Note, that there is an additional contribution to $\dot{n}_{k}$ due to the processes involving one right-moving and two left-moving particles. We will see below that at low temperatures this contribution is negligible.

We now focus on systems close the thermal equilibrium by substituting Eqs. (5) and (17) and linearizing the collision integral in small $\phi_{p}$. This yields

[TABLE]

At low temperature the typical values of momentum of thermally excited quasiparticles measured from the nearest Fermi point are of the order of $T/v_{F}$. On the other hand, as we saw in Sec. IV.1, for the processes of Fig. 1(b) the difference of the momenta on the left branch is $|q-q^{\prime}|\sim T^{2}/\mu v_{F}\ll T/v_{F}$. Thus, when solving Eq. (82) to leading order in $T/\mu\ll 1$, after substituting the expression (78) for the matrix element one can apply the approximation (IV.1.1). This yields

[TABLE]

We note that because Eq. (83) holds only in the low-temperature limit, the same approximation must be used in Eq. (13), resulting in

[TABLE]

Equation (83) shows that to leading order in $T/\mu$ the processes involving two right-moving and one left-moving particles [Fig. 1(b)] affect only the distribution function on the right-moving branch. This justifies our earlier approximation that neglected the contribution to $\dot{n}_{k}$ from the processes involving one right-moving and two left-moving particles.

IV.2.3 Dimensionless form of the collision integral

Let us now bring Eq. (83) to a dimensionless form by introducing dimensionless momentum $\xi$ and time $\theta$,

[TABLE]

where the relaxation time $\tau_{\rm ex}$ is defined by

[TABLE]

and has the expected power-law scaling $\tau_{\rm ex}^{-1}\propto T^{7}$ Imambekov et al. (2012); Arzamasovs et al. (2014); Protopopov et al. (2014). In these units, Eq. (83) takes the form

[TABLE]

where

[TABLE]

Here $\Phi(\xi)=\phi_{k}$ and $G(\xi)=g_{k}$, i.e.,

[TABLE]

After some algebra the integral operator (88) can be rewritten in the form

[TABLE]

where

[TABLE]

As expected, the kernel of the integral operator (90) is symmetric with respect to permutation $\xi\leftrightarrow\xi^{\prime}$. This property of the operator $\widehat{M}$ ensures that the eigenvalue problem

[TABLE]

has an orthonormal set of solutions $\Phi_{l}(\xi)$ with real eigenvalues $\lambda_{l}$.

The eigenvalue problem (94) was derived from Eq. (83), which describes time evolution of the distribution function of fermions near the right Fermi point. Particles near the left Fermi point can be treated in the same way. Therefore, each eigenfunction $\Phi_{l}(\xi)$ gives a solution $\phi_{p}^{(l)}$ of the full eigenvalue problem (19) that is confined to either right- or left-moving part of the quasiparticle spectrum. The corresponding eigenvalue is

[TABLE]

Alternatively, one can symmetrize and antisymmetrize the eigenfunctions, resulting in two sets of solutions $\phi_{p}^{(l)}$ that are either even or odd in $p$ with the same eigenvalues (95). Evaluation of the transport coefficient $\kappa_{\rm ex}$ given by Eqs. (41) and (42) requires odd solutions, which take the form

[TABLE]

where the prefactor assumes that the eigenfunctions $\Phi_{l}(\xi)$ are normalized according to

[TABLE]

Using Eq. (96), we can evaluate the matrix element in our expression (41) for $\kappa_{\mathrm{ex}}$,

[TABLE]

Substituting this matrix element into Eq. (41), we obtain

[TABLE]

where $\tau_{\rm ex}$ is given by Eq. (86) and the numerical coefficient $c_{2}$ is defined as

[TABLE]

Summation in Eq. (100) excludes the zero modes for which $\lambda_{l}=0$.

Because the dimensionless eigenvalue problem (94) describes relaxation of the right-moving particles, which at $T/\mu\to 0$ is decoupled from the relaxation of the left movers, operator $\widehat{M}$ has only two zero modes:

[TABLE]

The modes $\Phi_{0}$ and $\Phi_{1}$ correspond to the conservation of the particle number and momentum, respectively. Their forms are easily verified analytically using Eq. (88). The integral operator (90) can be diagonalized numerically. Excluding the zero modes (101) and (102) from the sum (100), we obtained $c_{2}\approx 0.41088$.

V Discussion of the results

In this paper we have developed a microscopic theory of the thermal transport coefficients of one-dimensional Fermi gas at low temperature $T\ll\mu$. A special feature of one-dimensional quantum systems is that in addition to the usual thermal conductivity of the system $\kappa$, one can introduce thermal conductivity of the gas of elementary excitations $\kappa_{\rm ex}$. This is a consequence of separation of scales of the rates of various processes responsible for the relaxation of the system to equilibrium. Specifically, the rates of the processes of Fig. 1(a), which are responsible for the equilibration of the chemical potentials of the left- and right-moving particles, are exponentially small, $\tau^{-1}\propto e^{-\mu/T}$. On the other hand, the remaining scattering processes illustrated in Fig. 1(b) and (c) occur at rates $\tau_{\rm ex}^{-1}$ that scale as a power law of $T/\mu$ and are therefore much faster.

In general, transport coefficients are proportional to the relevant relaxation times. As a result, the thermal conductivity $\kappa\propto\tau$ is exponentially large. Our microscopic theory gives the result (31), which is consistent with the phenomenological expression (1) obtained within the Luttinger liquid theory. These results relate $\kappa$ to the relaxation time $\tau$, which requires special evaluation. A phenomenological theory Matveev and Andreev (2012a, b) expresses $\tau$ in terms of the properties of the excitation spectrum of the quantum liquid. For the system of weakly interacting fermions further progress can be made. In Appendix C we obtained an expression for $\tau$ in terms of the interaction potential. For example, in the case of spinless fermions with dipole-dipole interaction (46) we found

[TABLE]

A more general expression given by Eqs. (130) and (131) applies to any interaction $U(x)$ that falls off at large distances faster than $1/x$.

In one-dimensional systems the transport coefficient $\kappa$ describes the thermal conductivity only at low frequencies $\omega\ll\tau^{-1}$. In particular, it gives the dominant contribution to the attenuation of sound at low frequencies Matveev and Andreev (2018a). On the other hand, at $\omega\gg\tau^{-1}$ the exponentially slow relaxation processes can be neglected, which leads to a very different behavior of the system. Instead of the conventional sound, the system now supports two sound modes Matveev and Andreev (2017, 2018b), whose attenuation is no longer exponentially strong. Instead, the thermal transport coefficient that enters the expression for sound attenuation is $\kappa_{\rm ex}$ Matveev and Andreev (2018a), which can be thought of as the thermal conductivity at frequencies $\omega\gg\tau^{-1}$.

Our microscopic theory relates $\kappa_{\rm ex}$ to the relaxation modes of the system, see Eqs. (41) and (42). This enables one to find the order of magnitude estimate (43), which expresses $\kappa_{\rm ex}$ in terms of the relaxation time $\tau_{\rm ex}$. To obtain a full microscopic expression for $\kappa_{\rm ex}$, a more detailed treatment of the relaxation processes is required. We performed such a treatment for two kinds of interactions between fermions. For Coulomb and dipole-dipole interactions, the long range of the interaction potential results in processes of the type shown in Fig. 1(c) having a higher rate than those of Fig. 1(b). The relaxation rate $\tau_{\eta}^{-1}$ that controls the thermal conductivity of the gas of excitations is due to the latter type of processes and is given by Eqs. (72) and (74). Our microscopic result for $\kappa_{\rm ex}$ is given by Eq. (77). It is consistent with the earlier estimate (43) provided that the rate $\tau_{\rm ex}^{-1}$ of relaxation of the gas of excitations is identified with $\tau_{\eta}^{-1}$. On the other hand, in the case of short-range interactions, the processes shown in Fig. 1(c) are negligible. The resulting $\kappa_{\rm ex}$ is expressed in terms of the interaction potential using Eqs. (80), (86) and (99).

In our treatment we considered the model of spinless fermions. Spins can be added to the Boltzmann equation treatment of thermal transport in a straightforward way, but they may strongly affect the relaxation processes. The processes of Fig. 1(a) remain exponentially suppressed in the presence of spins, and we again expect $\kappa\propto\tau\propto e^{\mu/T}$. The rate of processes shown in Fig. 1(b) was studied in Ref. Karzig et al. (2010), where $\tau_{\rm ex}^{-1}\propto T$ was obtained, while the processes of Fig. 1(c) have not been explored. In the case of spinless fermions the relaxation rate relevant for the evaluation of $\kappa_{\rm ex}$ was that of the processes shown in Fig. 1(b), and this should hold for fermions with spin. Thus, using the estimate (43) we expect that for weakly interacting spin-$\frac{1}{2}$ fermions $\kappa_{\rm ex}\propto T^{2}$. We leave more careful microscopic treatment of this problem for future study.

Finally, we note that our treatment of the thermal conductivity assumes that at every point in space the Fermi gas is near local thermal equilibrium. This assumption is violated if one takes into account the sound modes in the system, which are weakly damped at low frequencies Andreev (1971). This effect is particularly strong in one dimension, where it leads to power-law scaling of sound absorption Andreev (1980). Violation of the assumption of local equilibrium gives rise to power-law scaling of thermal conductance of one-dimensional classical systems with the system size, see, e.g., Ref. Lepri et al. (2003). This effect should also manifest itself in the low-frequency behavior of thermal conductivity of one-dimensional Fermi gas at low temperatures. On the other hand, because the local contribution (1) to the thermal conductivity is exponentially amplified by the long relaxation time $\tau$, one should expect the non-local effects to become significant only at frequencies that are exponentially smaller than $\tau^{-1}$. Frequency dependence of thermal conductivity of one-dimensional electronic fluid was recently studied by R. Samanta et al. Samanta et al. . In addition to the crossover from the result (1) to a power-law scaling at $\omega\to 0$, they obtained several additional parametric frequency regions, including one in which $\kappa(\omega)$ is consistent with our result for $\kappa_{\rm ex}$.

Acknowledgements.

The authors are grateful to A. V. Andreev for stimulating discussions. Work at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division. Work at Laboratoire de Physique Théorique was supported in part by the EUR grant NanoX ANR-17-EURE-0009 in the framework of the “Programme des Investissements d’Avenir.”

Appendix A Fourier transform of the Coulomb interaction potential

Let us consider two three-dimensional particles with charge $e$ confined to a one-dimensional channel. The position of each particle will be described by its coordinate along the channel and the vector $\bm{\rho}$ in the transverse direction. Coulomb interaction between the two particles at the distance $x$ along the channel is given by

[TABLE]

where $\Psi(\bm{\rho})$ is the normalized wave function of the transverse motion of a particle in the channel. Performing the Fourier transform (44), we find

[TABLE]

where $K_{0}(z)$ is the Macdonald function. Its asymptotic behavior at small $z$ is given by

[TABLE]

where $\bm{C}\approx 0.5772$ is the Euler’s constant. Substituting Eq. (106) into Eq. (105) and omitting coefficients of order unity in the argument of the logarithms, we obtain the result (45), provided that the channel width $w$ is defined by

[TABLE]

Here the averaging

[TABLE]

is performed over the distribution of the particle density in the transverse direction.

Appendix B Evaluation of $\Lambda$ in the limit of weak interactions

In this Appendix we evaluate the parameter $\Lambda$ in the expression for the scattering matrix element (78) characterizing the process shown in Fig. 1(b). We will express $\Lambda$ in terms of the Fourier components $V(p)$ of the interaction potential and obtain the result (80). Our calculation is based on the approach suggested in Ref. Matveev and Furusaki (2013) where the matrix element (78) was used to study the decay rate of quasiparticle excitations of a spinless quantum liquid at zero temperature. The prescription of Ref. Matveev and Furusaki (2013) is

[TABLE]

where $Y_{p}$ is expressed in terms of the quasiparticle energies $\epsilon_{p}$ as follows

[TABLE]

Here the quasiparticle velocity is defined as a derivative of quasiparticle energy, $v_{p}=\epsilon_{p}^{\prime}$, the effective mass $m^{*}_{p}=1/\epsilon_{p}^{\prime\prime}$, the speed of the low energy excitations $v=v_{p_{F}}$, and for Galilean invariant systems the Luttinger liquid parameter $K=v_{F}/v$. In an interacting system the quasiparticle energies depend on the density of particles $n$ and momentum per particle $\chi$. The partial derivatives in Eq. (110) are defined by

[TABLE]

Finally, since the sign of $\Lambda$ has no physical significance, for convenience, in Eq. (109) we changed the sign of the expression used in Ref. Matveev and Furusaki (2013).

A quasiparticle can be added to the ground state of the system in two ways. First, one can move a fermion from a Fermi point $p_{F}$ to a state with momentum $p>p_{F}$. In this approach, the quasiparticle is essentially a particle-hole pair, with the hole remaining at the Fermi point. Alternatively, an additional fermion with momentum $p>p_{F}$ can be added to the system. The former approach was used in Ref. Matveev and Furusaki (2013), but the prescription (109) and (110) applies in both cases. In this Appendix we will use the second approach. In particular, the energy of the excitation in the system of non-interacting fermions is

[TABLE]

This energy, of course, depends on neither the density $n$ nor the momentum per particle $\chi$ of the system, and therefore yields $\Lambda=0$. In a weakly interacting system, the energies of both the ground state and the state with the additional particle with momentum $p$ change. In this case the quasiparticle energy should be understood as the difference of the energies of those many-body states. The energy $\epsilon_{p}$ defined this way does depend on $n$ and $\chi$, resulting in a nonvanishing $\Lambda$.

Instead of treating quasiparticle energies as functions of $n$ and $\chi$, it will be convenient to think of the ground state of a moving system in terms of the positions of the left and right Fermi points, $p_{L}$ and $p_{R}$. The two sets of variables are related by

[TABLE]

Then the derivatives (111) can be written as

[TABLE]

In the following, we evaluate the quasiparticle energy $\epsilon_{p}(p_{L},p_{R})$ up to terms quadratic in interactions, substitute the resulting expressions into Eq. (110), take the limits $p_{L}\to-p_{F}$ and $p_{R}\to p_{F}$, and obtain $\Lambda$ from Eq. (109).

We start by evaluating the first order correction $\delta\epsilon_{k}^{(1)}$ to the quasiparticle energy (112). The interaction between fermions is given by

[TABLE]

where $V(q)$ is defined by Eq. (44). The first-order correction to the energy of a many-body state

[TABLE]

where $n_{p}=\langle a_{p}^{\dagger}a_{p}\rangle$ is the occupation number of the state with momentum $p$. In a moving ground state

[TABLE]

where $\theta(x)$ is the unit step function, and we assumed $p_{R}>p_{L}$. The extra contribution to $\delta E^{(1)}$ from an additional fermion at state $p$ is given by

[TABLE]

Differentiating this expression with respect to $p$, we find the first order correction to quasiparticle velocity $v_{p}^{(0)}=p/m$ in the form

[TABLE]

In a stationary system, $p_{R}=-p_{L}=p_{F}$, the velocity at the Fermi point is

[TABLE]

As a result, to first order in interaction the Luttinger liquid parameter $K=v_{F}/v$ given by

[TABLE]

Because we have $K=1$ in the absence of interactions, the derivative of $K$ with respect to the particle density appears in the first order in interactions,

[TABLE]

Similarly, the derivatives of $\epsilon_{p}$ and $v_{p}$ in Eq. (110) appear only in the first order. These leading contributions are found by using Eq. (114), (118), and (119),

[TABLE]

On the other hand, the second derivative $\partial^{2}_{LR}\epsilon_{p}^{(1)}$ appears only in the second order in interactions

[TABLE]

Substituting Eqs. (122)–(B) into Eq. (110), we conclude that to first order in interactions $Y_{p}$ and, therefore, $\Lambda$ vanish. This is an expected outcome because the three-particle scattering matrix element (78) cannot be generated from the two-particle interaction (115) in the first order.

To obtain $Y_{p}$ in the second order in interaction strength, in addition to Eqs. (122)–(B) we need to find the second-order correction $\delta\epsilon_{p}^{(2)}$ to the quasiparticle energy, which will contribute to the first term in the right-hand side of Eq. (110). Applying the standard second-order perturbation theory, we obtain

[TABLE]

The first term in the integral accounts for the second-order contribution to the many-body state due to an additional fermion in state $p$, assuming $p>p_{F}$. The second term subtracts the contribution to the ground state energy due to processes involving state $p$, which are not allowed in the presence of the additional fermion. The two contributions are illustrated in Fig. 2.

The second-order correction (125) contributes to the first term in Eq. (110) via $\partial^{2}_{LR}\epsilon_{p}$. To leading order in interactions, $\partial^{2}_{LR}\delta\epsilon_{p}^{(2)}=-(2\pi\hbar)^{2}\partial^{2}\epsilon_{p}^{(2)}/\partial_{p_{R}}\partial_{p_{L}}$, see Eq. (114c). The positions $p_{L}$ and $p_{R}$ of the two Fermi points enter Eq. (125) via the occupation numbers (117).

Differentiation of Eq. (125) with respect to $p_{L}$ and $p_{R}$ yields

[TABLE]

Here the first term originates from the first term in Eq. (125), whereas the second and third ones originate from the second term in (125).

We are now in a position to evaluate $Y_{p}$ in the second order in interaction strength. To this end we substitute Eqs. (122) and (123) for the corresponding derivatives in the first order and add expressions (B) and (126) evaluated in the second order. (We replace $p_{L}\to-p_{F}$ and $p_{R}\to p_{F}$.) The remaining parameters need not account for interactions, i.e., we substitute $m^{*}_{p}=m$, $v=v_{F}$, $v_{p}=p/m$, and $K=1$. The result has the form

[TABLE]

Substitution of the above result into Eq. (109) yields Eq. (80).

Appendix C Relaxation rate $\tau^{-1}$ in a system of weakly interacting spinless fermions

A phenomenological expression for the relaxation rate of in a spinless quantum liquid at low temperatures was obtained in Refs. Matveev and Andreev (2012a, b). In this Appendix we apply that result to find the rate $\tau^{-1}$ for the special case of weakly interacting spinless fermions. To this end it is convenient to express the relaxation rate as Matveev (2013)

[TABLE]

Here $\Delta$ is the maximum energy of a hole-like excitation in the quantum liquid, $m^{*}$ is the effective mass of the hole at the maximum of energy, and $v$ is the velocity of the low-energy excitations in the system. In a weakly interacting Fermi gas $\Delta=\mu$, $m^{*}=m$, and $v=v_{F}$.

The quantity $B$ was expressed in Ref. Matveev and Andreev (2012a) in terms of $\Delta$, $m^{*}$, and $v$ as functions of the particle density. An alternative expression

[TABLE]

was obtained in Refs. Matveev and Andreev (2012b); Matveev (2013). Here $Y_{0}$ is a function of the spectrum of holes in the quantum liquid, which is analogous to the $Y_{p}$ for particle-like excitations given by Eq. (110). Because of the difference in the type of excitations, the effective mass $m^{*}$ in Eq. (110) should be replaced with $-m^{*}$. The momentum $p$ in the resulting $Y_{p}$ should correspond to the maximum of the energy of the hole. If the hole is formed by moving a fermion from a state below the Fermi level to the right Fermi point, the maximum of energy corresponds to $p=p_{F}$. Alternatively, one may create a hole by removing a particle from the system, in which case the maximum of energy corresponds to $p=0$. Here we adopt the latter approach.

Evaluation of $Y_{p}$ for a hole-like excitation can be performed by retracing the steps leading from Eq. (110) to Eq. (127). We find that for the hole $Y_{p}$ is given by Eq. (127) with the opposite sign. Thus one can substitute for $Y_{0}$ into Eq. (129) the result (127) taken at $p=0$. This yields

[TABLE]

where

[TABLE]

The derivation of the expression (128) in Refs. Matveev and Andreev (2012a, b) assumed that the interactions between the particles fall off sufficiently fast at the long distances for the velocity $v$ of the elementary excitations to be well defined. In practice this means that the interaction potential $U(x)$ falls off at $x\to\infty$ faster than $1/x$. In particular, the result (130) does not apply in the case of Coulomb interactions, for which $V(0)$ in Eq. (131) is ill-defined. On the other hand, the dipole-dipole interaction with the short-distance cutoff $w$ does have a well-defined $V(0)$. Substituting Eq. (46) into Eq. (131), we obtain

[TABLE]

This expression is obtained for $p_{F}w/\hbar\ll 1$ within logarithmic accuracy.