Viscous Dissipation in One-Dimensional Quantum Liquids

TL;DR

This paper develops a theory for viscous dissipation in one-dimensional quantum liquids, revealing that bulk viscosity diverges at zero temperature and is infinite in integrable models, indicating a breakdown of hydrodynamics.

Contribution

It provides a universal theoretical framework for understanding viscosity in 1D quantum liquids, including the divergence behavior and special cases of integrable models.

Findings

Bulk viscosity diverges at zero temperature for generic interactions.

In integrable models, viscosity is infinite at all temperatures.

Hydrodynamic description breaks down in integrable cases.

Abstract

We develop a theory of viscous dissipation in one-dimensional single-component quantum liquids at low temperatures. Such liquids are characterized by a single viscosity coefficient, the bulk viscosity. We show that for a generic interaction between the constituent particles this viscosity diverges in the zero-temperature limit. In the special case of integrable models, the viscosity is infinite at any temperature, which can be interpreted as a breakdown of the hydrodynamic description. Our consideration is applicable to all single-component Galilean-invariant one-dimensional quantum liquids, regardless of the statistics of the constituent particles and the interaction strength.

Equations

$$W=\zeta L({\partial }_{x}u{)}^2.$$

$$H=\frac{\hslash v}{4\pi }{\int }_0^{L}dx\left[:({\partial }_x\phi {)}^2:+\frac{\hslash }{3m_{\ast }v}:({\partial }_x\phi {)}^3:+\dots \right],$$

$$v^2=\frac{n}{m}{\partial }_n^2{E}_0,\frac{m_{}}{m_{\ast }}=\frac{{\partial }_n(vn)}{2v\sqrt{K}},$$

$$H_{0}=\!\int_{0}^{L}\!\!dx\mkern 1.0mu\Bigl{[}-\mkern 2.0mui\hbar v\mkern 2.0mu\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\mkern 2.0mu\psi^{\dagger}(x)\mkern 1.0mu\partial_{x}\psi(x)\mkern 1.0mu\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}+\frac{\hbar^{2}}{2m_{\ast}}\mkern 2.0mu\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\mkern 2.0mu(\partial_{x}\psi)^{\dagger}(\partial_{x}\psi)\mkern 1.0mu\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\,\Bigr{]},$$

$$H_{0}=\sum_{l}\varepsilon_{l}^{\phantom{\dagger}}\mkern 1.0mu\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\,\psi^{\dagger}_{l}\psi^{\phantom{\dagger}}_{l}\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\,,\quad\varepsilon_{l}=vp_{l}+\frac{p_{l}^{2}}{\mkern 1.0mu2m_{\ast}}.$$

$$\sum_{l}\mkern 2.0mu\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\,\psi^{\dagger}_{l}\psi^{\phantom{\dagger}}_{l}\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\,=0\mkern 2.0mu.$$

$$f_{0l}=\frac{1}{e^{\beta ({\epsilon }_l-\mu )}+1},$$

$$\dot{n}=-n{\partial }_{x}u,{\dot{p}}_l=-p_l{\partial }_{x}u.$$

$$\sum _l{\dot{f}}_l=0,\sum _l{\epsilon }_l{\dot{f}}_l=0.$$

$$\dot{\beta}=\beta\bigl{[}n\mkern 2.0mu\partial_{n}\mkern-2.0mu\ln(nv)\bigr{]}\partial_{x}u,\quad\dot{\mu}=\mu\mkern-2.0mu\left[n\mkern 2.0mu\partial_{n}\mkern-2.0mu\ln\!\left(\frac{m_{\ast}}{n^{2}}\mkern-2.0mu\right)\mkern-1.0mu\right]\mkern-2.0mu\partial_{x}u$$

$$\dot{f}_{0l}=f_{0l}\bigl{(}1-f_{0l}\bigr{)}\mkern 1.0mu\frac{\mkern 1.0mu3\mkern 1.0muv^{2}p_{l}^{2}-\pi^{2}T^{2}}{6m_{\ast}v^{2}T\,}\mkern 1.0mu\bigl{(}n\mkern 2.0mu\partial_{n}\mkern-2.0mu\ln\nu\bigr{)}\mkern 1.0mu\partial_{x}u$$

$$\nu=\frac{\pi\hbar n}{m_{\ast}v}=\sqrt{K\mkern 1.0mu}\bigl{(}1-n\mkern 2.0mu\partial_{n}\mkern-2.0mu\ln\mkern-2.0mu\sqrt{K\mkern 1.0mu}\mkern 1.0mu\bigr{)},$$

$${\dot{f}}_l=I[f],$$

$$W=-\mkern 1.0muT\sum_{l}\frac{\dot{f}_{0l}\mkern 2.0mu\delta\mkern-1.0muf_{l}}{f_{0l}\bigl{(}1-f_{0l}\bigr{)}}.$$

$$\zeta=\mathcal{A}\mkern 2.0mu\frac{\nu^{2}n\mkern 2.0mu}{\mkern 2.0mu(\hbar nv)^{3}}\mkern 2.0muT^{4}\mkern 1.0mu\tau\mkern 2.0mu,\quad\mathcal{A}=c\mkern 1.0mu\bigl{(}n\mkern 2.0mu\partial_{n}\mkern-2.0mu\ln\nu\bigr{)}^{2},$$

$$\frac{1}{\tau }=\mathcal{B}{\nu }^{2}nv{\left(\frac{T}{\hslash nv}\right)}^7,$$

$$\zeta =\frac{\mathcal{A}}{\mathcal{B}}{\left(\frac{\hslash nv}{T}\right)}^3\hslash n,$$

$$\zeta \sim \frac{\hslash n(n{a}_B{)}^2}{{\ln }^2(m{v}_F^2/T)}{\left(\frac{m{v}_F^2}{T}\right)}^2$$

Full text

Hydrodynamics is a universal description of fluid flow at long times and distances, when the state of the fluid is completely characterized by only a few macroscopic parameters LL_VI . Hydrodynamics ideas have been at the center of both experimental and theoretical attention since the early days of the theory of quantum liquids Khalat_book ; PN , originally developed to describe properties of liquid 4He Landau_SF ; Khalat_book ; PN and 3He Landau_FL ; PN ; AK_review . The search for observable manifestations of viscous hydrodynamic flow of electron liquids in solids also has a long history Gurzhi . Although attaining the hydrodynamic regime in this case is difficult because neither the energy nor the momentum of electrons are conserved due to their interaction with crystal lattice and impurities, signatures of hydrodynamic flow were observed in ultraclean two-dimensional graphene and quasi-one-dimensional wires systems. Viscous hydrodynamic behavior was also observed in three-dimensional cold gas_3D and quasi-one-dimensional cold gas_1D resonantly interacting ultracold Fermi gases, which are free of many complications associated with conventional condensed matter systems.

Higher-dimensional fluids are characterized by two viscosity coefficients, the shear viscosity $\eta$ arising due to friction between adjacent layers of the fluid, and the bulk viscosity $\zeta$ associated with the expansion or compression of the fluid. At low temperatures, the bulk viscosity is often neglected. For instance, in the Fermi-liquid theory the shear viscosity is proportional AK_review ; LP_X to the quasiparticle lifetime, $\eta\propto\tau$, and diverges as $\eta\propto T^{-2}$ at $T\to 0$, whereas $\zeta\propto T^{4}\tau\propto T^{2}$ SB , which vanishes in the same limit. In one dimension the shear viscosity is obviously absent. This puts focus on the bulk viscosity $\zeta$ and its behavior at low temperatures.

One-dimensional quantum systems Giamarchi differ dramatically from their higher-dimensional counterparts. Irrespective of the nature of the constituent particles, gapless excitations can be described in terms of waves of density with acoustic low-energy spectra Giamarchi ; Haldane_LL . Similar to fermionic quasiparticles in the Fermi liquid, these excitations are the key ingredients of the Luttinger-liquid theory Haldane_LL . Although this theory has been extremely successful Giamarchi , it is not free of drawbacks Andreev ; Samokhin ; ISG ; PKKG . In particular, it is known ISG ; Andreev ; Samokhin that the conventional Luttinger-liquid theory is not well equipped to handle phenomena in which relaxation of elementary excitations plays an essential part. Viscous dissipation belongs to this class of phenomena. Below we show that, remarkably, the Fermi-liquid result $\zeta\propto T^{4}\tau$ is applicable to both fermionic and bosonic one-dimensional systems. However, because mechanisms of quasiparticle decay in one dimension differ from those in higher dimensions, $\tau$ is anomalously large, resulting in manifestly non-Fermi-liquid temperature dependence: $\zeta$ increases with the decrease of temperature and diverges at $T\to 0$.

To set the stage, consider a system of $N$ identical particles confined in a one-dimensional interval of length $L$. In the thermodynamic limit when both $N$ and $L$ are large, precise choice of the boundary conditions is irrelevant. For our purposes, it is convenient to view the system as a ring of circumference $L$, so that in equilibrium the particle density $n=N/L$ is uniform. Imagine now that the size of the system $L$ slowly changes with time. During such change, fluid elements at distance $\Delta x$ move with respect to each other with velocity $\Delta u=(\Delta x/\mkern-1.0muL)\mkern 1.0mu\dot{L}$. (Hereinafter the dot denotes the derivative with respect to time.) This gives rise to the position-independent gradient of the fluid velocity $\partial_{x}u=\dot{L}/\mkern-1.0muL$. Changing the size of the system inevitably leads to an irreversible transfer of energy to it, i.e., heating. The bulk viscosity $\zeta$ is the transport coefficient that controls the heating rate $W\mkern-2.0mu$. For small $\partial_{x}u$, this rate is given by

[TABLE]

Below we evaluate the heating rate and then use Eq. (1) to extract the viscosity.

Long-wavelength excitations of gapless one-dimensional systems allow for a continuum description Giamarchi ; hydrodynamics ; Haldane_LL ; ISG , similar to that employed in the theory of superfluidity Landau_SF ; Khalat_book . In this approach, the Hamiltonian density of the system is written as an expansion in derivatives of two bosonic fields describing the right- and left-moving excitations. In the leading orders, the right and left movers decouple; hence, it is sufficient to discuss the right movers only. The corresponding part of the Hamiltonian reads left

[TABLE]

where the field $\varphi$ obeys the commutation relation $[\partial_{x}\varphi(x),\varphi(y)]=2\pi i\mkern 1.0mu\delta(x-y)$ and is subject to the periodic boundary condition $\varphi(x)=\varphi(x+L)$; the colons denote normal ordering. The gradient expansion (2) describes low energy properties of single-component one-dimensional systems with interaction between particles decaying with the distance faster than $1/x^{2}$, see, e.g., Refs. ISG ; Pereira ; PM_KdV ; PM_soliton ; RM-14 . The parameters $v$ and $m_{\ast}$ in this expansion have the units of velocity and mass, respectively. For Galilean-invariant systems Khalat_book ; PN ; ISG ; Pereira ; PM_mass

[TABLE]

where $E_{0}$ is the ground state energy per length, $m$ is the mass of the constituent particles, and $K=\pi\hbar\mkern 1.0mun/mv$ is the Luttinger-liquid parameter ISG ; Giamarchi . Both $v$ and $m_{\ast}$ depend on the density and are finite and positive.

The first term in the expansion (2) describes free bosons with linear spectrum, whereas the second term represents the interaction between these bosons. This term is irrelevant in the renormalization group sense Haldane_LL and can often be neglected, which constitutes the Luttinger-liquid Haldane_LL approximation. It is well-known ISG ; Samokhin , however, that naive attempts to account for the interaction perturbatively lead to a divergent inelastic decay rate of the Luttinger liquid bosons. This difficulty is circumvented Rozhkov by rewriting the bosonic Hamiltonian (2) in terms of effective fermions with the help of the identity Giamarchi ; Haldane_LL ; bosonization $\psi(x)=L^{-1/2}\colon\!\mkern-2.0mue^{i\varphi(x)}\mkern-2.0mu\colon\mkern-2.0mu$. In the fermionic representation, the first two terms in the expansion (2) take the form

[TABLE]

where the symbols

⋮

denote the normal ordering of fermionic operators with respect to the ground state in which single-particle states with positive (negative) momenta are empty (occupied) Haldane_LL ; bosonization . Unlike Eq. (2), the fermionic Hamiltonian (4) is quadratic and easily diagonalizable by Fourier transform in the usual way,

[TABLE]

Here $\psi_{l}=L^{-1/2}\!\int_{0}^{L}\!dx\,\psi(x)\mkern 1.0mue^{-ip_{l}x/\hbar}$ and the single-particle momenta are given by $p_{l}=(2\pi\hbar\mkern 1.0mu/L)\mkern 1.0mul$ with integer $l$. In the fermionic language, the periodic boundary condition on $\varphi$ translates to the constraint \int_{0}^{L}\!dx\,\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\mkern 2.0mu\psi^{\dagger}(x)\psi(x)\mkern 1.0mu\raisebox{-0.32289pt}{\scalebox{0.7}{\bm{\vdots}}}\mkern 2.0mu=0, which can be also written as

[TABLE]

The fermions described by Eqs. (4)-(6) emerged as a result of the exact diagonalization of the first two terms in the gradient expansion (2). These fermions do not represent exact eigenstates of the full Hamiltonian, but should be viewed instead as fermionic quasiparticles, akin to those in the Fermi-liquid theory PN ; AK_review ; Landau_FL . Interaction between the quasiparticles originates in higher-order contributions in the gradient expansion (2). Whereas in the conventional Fermi-liquid theory PN ; Landau_FL ; AK_review such interaction affects the excitation spectrum, in our case the interaction leaves the first two terms in the expansion of $\varepsilon_{l}$ given in Eq. (5) intact left ; Rozhkov ; PM_KdV ; PM_mass . It is crucial, however, that in the absence of integrability Sutherland_book the quasiparticles acquire a finite decay rate ISG ; PM_soliton ; LFG-07 ; KPKG-07 ; PWA-09 ; MF-13 ; RM-13 ; ABG-14 ; RM-14 ; PGM-14 .

Similar to the Fermi-liquid theory, the system can be described by the distribution function $f_{l}=\langle\psi^{\dagger}_{l}\psi^{\phantom{\dagger}}_{l}\rangle$. Its equilibrium form $f_{0l}$ maximizes the entropy S\mkern 1.0mu[f]=-\mkern 2.0mu\sum_{l}\bigl{[}f_{l}\ln\mkern-1.0muf_{l}+(1-f_{l})\ln\mkern 1.0mu(1-f_{l})\bigr{]} at constant energy and number of quasiparticles. This immediately leads to the Fermi-Dirac distribution

[TABLE]

where $\beta=1/T$ and $\mu$ is the chemical potential determined by the condition \sum_{l}\bigl{[}\mkern 1.0muf_{0l}-f_{0l}\bigr{|}_{T=0}\mkern 1.0mu\bigr{]}=0 that follows from Eq. (6). In the leading order in temperature this condition yields $\mu=(\pi^{2}\mkern-2.0mu/6)\mkern 1.0mu(T^{2}\mkern-2.0mu/m_{\ast}v^{2})$.

If the system size depends on time, $L=L(t)$, both $\beta$ and $\mu$ in Eq. (7) are time dependent, as is the excitation spectrum $\varepsilon_{l}$. Indeed, the velocity $v$ and the mass $m_{\ast}$ in Eq. (5) depend on the density $n$ that scales with the size as $1/L$, and the momenta $p_{l}$ scale the same way. Taking into account the relation $\partial_{x}u=\dot{L}/\mkern-1.0muL$, we obtain the equations

[TABLE]

(In the first equation here one may recognize the continuity equation in the special case $\partial_{x}n=0$.)

When the system size changes, collisions between the quasiparticles cause a change of the occupation numbers $f_{l}$. Because collisions preserve the number of quasiparticles and the energy, the time-dependent distribution function obeys the conservation laws LP_X

[TABLE]

If the size changes slowly, $f_{l}$ remains close to equilibrium at all times. For $f_{l}=f_{0l}$ Eqs. (7) and (9) imply that $\dot{S}[f_{0}]=\sum_{l}\dot{f}_{0l}\ln(1/\mkern-1.0muf_{0l}-1)=0$, which corresponds to an adiabatic process. Using Eqs. (7)-(9) and the expression for $\mu$ given above, we find

[TABLE]

in the leading order in $T/m_{\ast}v^{2}\ll 1$. Differentiation of Eq. (7) with the help of these relations yields

[TABLE]

with

[TABLE]

where we took into account Eq. (3).

The dependence on time of the full distribution function $f_{l}=f_{0l}+\delta\mkern-2.0muf_{l}$ is governed by the equation

[TABLE]

where the functional $I[f]$ is the collision integral LP_X . The detailed balance LP_X ensures that $I[f_{0}]=0$ for any Fermi-Dirac distribution, including that given by Eq. (7). Therefore, whereas in the left-hand side of Eq. (13) the nonadiabatic correction $\delta\mkern-2.0muf_{l}$ can be neglected, it must be retained in the right-hand side.

Solutions of Eq. (13) satisfy the conservation laws (9). Since $f_{0l}$ obeys these laws, so does $\delta\mkern-2.0muf_{l}=f_{l}-f_{0l}$, which yields the relation $\sum_{l}(\varepsilon_{l}-\mu)\mkern 1.0mu\delta\mkern-1.0mu\dot{f}_{l}=0$. With this relation taken into account, the heating rate $W=T\dot{S}$ in the lowest nonvanishing order in $\delta\mkern-1.0muf_{l}$ assumes the form

[TABLE]

In order to estimate the viscosity, we use the relaxation time approximation for the collision integral, $I[f]=-\mkern 2.0mu\delta\mkern-1.0muf_{l}/\tau$, which leads to $\delta\mkern-1.0muf_{l}=-\mkern 2.0mu\tau\mkern-1.0mu\dot{f}_{0l}$. Equations (11) and (14) then yield Eq. (1) with

[TABLE]

where $c$ is a numerical coefficient of order unity. (The accuracy of the relaxation time approximation is insufficient to allow its evaluation.) Equation (15) is the main result of our paper. It relates the bulk viscosity $\zeta$ to the lifetime $\tau$ of fermionic quasiparticles. In the remainder of the paper we employ Eq. (15) to estimate the viscosity in several typical situations.

The decay rate $1/\tau$ is determined by higher-order terms in the gradient expansion (2), which in turn depend on the interaction potential in the underlying microscopic model. For short-range potentials that fall off with the distance faster than any power, the dominant contribution to the rate comes from scattering processes involving three quasiparticles that do not move in the same direction ISG ; LFG-07 ; KPKG-07 ; PWA-09 ; MF-13 ; RM-14 ; ABG-14 ; PGM-14 . At quasiparticle energies of order $T$, this contribution has the form

[TABLE]

whereas collisions between three quasiparticles moving in the same direction yield $1/\tau\propto T^{14}$ PGM-14 . The dimensionless coefficient $\mathcal{B}$ in Eq. (16) depends on the interaction potential and can be related MF-13 to the cubic term in the expansion of the excitation spectrum $\varepsilon_{l}$. Equations (15) and (16) yield

[TABLE]

which diverges at $T\to 0\mkern 1.0mu$.

It is instructive to apply Eq. (17) to weakly interacting fermions and bosons. We start with spinless fermions. In the absence of interactions $m_{\ast}=m$ and $v$ coincides with the Fermi velocity $v_{F}=\pi\hbar\mkern 1.0mun/m$, hence $\nu=1$ [see Eq. (12)], and Eq. (15) gives $\mathcal{A}=0$. This property can be traced back to the excitation spectrum, which in this case is given by $\varepsilon_{l}=v_{F}p_{l}\mkern 1.0mu+\mkern 1.0mup_{l}^{2}/2m$, so that $\varepsilon_{l}$ scales as $1/L^{2}$ with the system size. Therefore, the occupation numbers given by the Fermi-Dirac distribution (7) with both $T$ and $\mu$ rescaled as $1/L^{2}$ are independent of time for any $L(t)$, in agreement with Eq. (11) for $\nu=1$. Importantly, the relation $\mathcal{A}=0$ does not imply vanishing viscosity. Indeed, for free fermions there is no quasiparticle decay; hence, $\mathcal{B}$ in Eq. (16) also vanishes, rendering Eq. (17) inapplicable.

The uncertainty is resolved by considering the limit of vanishing (as opposed to neglected from the outset) interaction between fermions. Correction to $\nu$ appears in the first order of perturbation theory in the interaction strength $V(x)$, which gives $\mathcal{A}\propto V^{2}$ [see Eq. (15)]. The coefficient $\mathcal{B}$ in Eq. (16) scales as the probability of three-particle scattering. In perturbation theory, the nonvanishing amplitude of such scattering arises in the second order KPKG-07 , hence $\mathcal{B}\propto V^{4}$. Therefore, the ratio $\mathcal{A}/\mathcal{B}$ in Eq. (17) scales as the inverse square of the interaction strength and diverges when the interaction is taken to zero, indicating a breakdown of the hydrodynamic description. Consider, for example, spinless fermions with interaction potential $V(x)$ that falls off rapidly with $|x|$ at $|x|\gtrsim a_{0}$ and is smooth and featureless at $|x|\lesssim a_{0}$ (cf. Refs. ISG ; LFG-07 ; KPKG-07 ; PGM-14 ). For $a_{0}n\gg 1$ we find $\mathcal{A}\sim(V_{0}/\hbar v_{F})^{2}$ and KPKG-07 ; PGM-14 $\mathcal{B}\sim(V_{0}/\hbar v_{F})^{4}(a_{0}n)^{4}$ with $V_{0}=\int\!dx\mkern 2.0muV(x)\ll\hbar v_{F}$. Accordingly, $\mathcal{A}\mkern 1.0mu/\mathcal{B}$ in Eq. (17) indeed diverges in the free-fermion limit $V_{0}/\hbar v_{F}\to 0$.

As an example of a system of bosons with short-range repulsion, we consider bosonic atoms confined in a one-dimensional trap BDZ ; PS . Relevant microscopic parameters in this case are the $s$-wave scattering length in three dimensions $a$ and the amplitude of zero-point motion in transverse direction $b$ PS ; Olshanii ; bosons . In the first order in interaction, the ground state energy can be estimated as PS $E\sim\hbar^{2}N^{2}a/mV_{\mkern 1.0mu\text{3D}}$, where $V_{\mkern 1.0mu\text{3D}}$ is the three-dimensional volume occupied by the bosons. For $V_{\mkern 1.0mu\text{3D}}\sim b^{2}\mkern-1.0muL$ this gives $E_{0}=E/L\sim(\hbar^{2}\mkern-2.0mu/m)(an^{2}\mkern-2.0mu/b^{2})$ for the ground state energy per length. Equation (3) then yields $K\sim\sqrt{b^{2}n/a\mkern 1.0mu}\,$ for the Luttinger-liquid parameter. (For weak repulsion considered here $K\gg 1$.) Substituting this expression into Eqs. (12) and (15), we find $\mathcal{A}\sim 1$. Because the dimensionless amplitude of three-particle scattering is of order $(a/b)^{2}$ bosons , the decay rate of fermionic quasiparticles evaluated in Refs. RM-14 ; ABG-14 assumes the form of Eq. (16) with $\mathcal{B}\sim(bn)^{4}$. The viscosity is then given by Eq. (17) with $\mathcal{A}/\mathcal{B}\sim(bn)^{-4}$.

Interestingly, the viscosity diverges in the limit $bn\to 0$ taken with both $K$ and $\hbar nv/\mkern 1.0muT$ kept constant. This limit corresponds to the Lieb-Liniger model Lieb ; Sutherland_book describing bosons with contact repulsion, which is well known to be integrable. In fact, divergent viscosity is a general property of integrable models. Indeed, in all such models the fermionic quasiparticles introduced in Eqs. (4) and (5) represent exact eigenstates; hence, their decay rate vanishes identically ISG ; LFG-07 ; KPKG-07 ; MF-13 ; RM-14 ; ABG-14 ; PM_soliton . Because $\mathcal{A}$ is finite for any nonvanishing interaction regardless of integrability, this observation and Eq. (17) imply that $\zeta$ is infinite even at finite temperature.

Equations (16) and (17) are applicable only when interactions decay rapidly with the distance between particles. An important exception is the system of spinless fermions with interaction potential that falls off with the distance as $1/|x|^{3}$. Such interaction describes electrons in quantum wires RM-13 ; wire and atoms in ultracold dipolar gases dipolar gas . Electrons, for instance, interact via the Coulomb potential screened by a metal gate at distance $d$ from the wire. For simplicity, we assume that $d\sim 1/n$. Correction to the velocity then reads $\delta v\sim e^{2}\mkern-2.0mu/\hbar\mkern 2.0mu$. Substituting $K=1-\delta v/v_{F}$ into Eqs. (12) and (15), we find $\mathcal{A}\sim(na_{B})^{-2}$, where $a_{B}=\hbar^{2}\mkern-2.0mu/me^{2}$ is the Bohr radius. Because of the long-range nature of the potential $V(x)\propto|x|^{-3}$, the quasiparticle decay rate due to scattering of three electrons moving in the same direction scales as $1/\tau\propto T^{6}$ RM-13 as opposed to $1/\tau\propto T^{14}$ PGM-14 for rapidly decaying potentials. At low temperatures, this contribution dominates the decay rate. Using the result of Ref. RM-13 and Eq. (15), we obtain

[TABLE]

for the viscosity. For the same reason as for fermions with short-range interaction, $\zeta$ given by Eq. (18) diverges in the free-fermion limit $na_{B}\to\infty$. However, the temperature dependence is different from that in Eq. (17).

We now discuss potential applications of our theory. Our main result, Eq. (15), relates the viscosity $\zeta$ to the quasiparticle lifetime $\tau$. In principle, the lifetime can be measured in momentum-resolved tunneling experiments Yacoby-10 . However, this method is effective only for high-energy excitations with decay rate $\hbar/\tau\gtrsim T$. On the other hand, $\tau$ in Eq. (15) is the lifetime of quasiparticles with energy of order $T$ for which $\hbar/\tau\ll T$. Thus, our theory relates the decay rate of such thermal excitations evaluated in Refs. ISG ; RM-13 ; ABG-14 ; PGM-14 to an experimentally accessible quantity, the viscosity.

Bulk viscosity $\zeta$ can be measured in transport experiments with quantum wires. Indeed, when electron liquid flows past a very smooth obstacle its density changes. The resulting viscous dissipation yields a contribution to the electrical resistance of the wire proportional to $\zeta$ Vignale ; AKS ; DM-15 . Alternatively, viscous dissipation can be measured in cold atomic gases in one-dimensional traps BDZ . In this case, slow changes of longitudinal confinement will heat the gas, see Eq. (1). This heating can be measured in time-of-flight experiments BDZ .

Although our consideration relied on Galilean invariance, some aspects of our theory are applicable to chiral one-dimensional systems as well. Indeed, the gradient expansion (2) coincides PGM-14 with the well-known Wen effective low-energy description of gapless edge excitations in quantum Hall systems. Consider, for example, a quantum Hall antidot, i.e., a gate-defined depleted region in a two-dimensional electron gas in the quantum Hall regime antidot . Because bulk degrees of freedom are gapped, a low-frequency signal applied to the gate that controls the size of the antidot will pump energy to propagating states at its edge. The resulting temperature rise may be detected by transport spectroscopy antidot . Chiral edge excitations carry charge and interact via Coulomb potential screened by the gate. Our theory then suggests that the energy transfer rate depends on temperature according to $W\propto(T\ln T)^{-2}$, see Eqs. (1) and (18).