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arXiv:2206.08109 · Physical Review A

Viscous flow in a one-dimensional spin-polarized Fermi gas: The role of integrability on viscosity

Jeff Maki, Shizhong Zhang

Published Jan 24, 2023• 3 citations
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Abstract

The transport properties of one-dimensional Fermi gases at low temperatures are often described by the Luttinger liquid (LL) model. However, to study dissipative effects, one needs to examine interactions beyond the LL model. In this work, we provide a simple model that allows for a direct microscopic calculation of the bulk viscosity, namely, the one-dimensional spin polarized $p$-wave Fermi gas. To leading order in the finite interaction strength, we find that the bulk viscosity is finite and consistent with the requirement of scale symmetry. We further show that the bulk viscosity satisfies the Bose-Fermi duality relating the weakly interacting limit of the spin polarized Fermi gas to the strongly interacting limit of the Lieb-Liniger model and vice versa. This work establishes the bulk viscosity to leading order in scale-breaking interactions in both the strongly and weakly interacting limits for both high and low temperatures.

DOI
10.1103/physreva.107.013310

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The transport properties of one-dimensional Fermi gases at low temperatures are often described by the Luttinger liquid (LL) model. However, to study dissipative effects, one needs to examine interactions beyond the LL model. In this work, we provide a simple model that allows for a direct microscopic calculation of the bulk viscosity, namely, the one-dimensional spin polarized ppp-wave Fermi gas. To leading order in the finite interaction strength, we find that the bulk viscosity is finite and consistent with the requirement of scale symmetry. We further show that the bulk viscosity satisfies the Bose-Fermi duality relating the weakly interacting limit of the spin polarized Fermi gas to the strongly interacting limit of the Lieb-Liniger model and vice versa. This work establishes the bulk viscosity to leading order in scale-breaking interactions in both the strongly and weakly interacting limits for both high and low temperatures.
The scattering properties of spin-polarized Fermi gases are dominated by ppp-wave interactions. Besides their inherent angular dependence, these interactions differ from their sss-wave counterparts as they also require the presence of a finite effective range in order to understand the low-energy properties of the system. In this article we examine how the shear viscosity and thermal conductivity of a three-dimensional spin-polarized Fermi gas in the normal phase depend on the effective range and the scattering volume in both the weakly and the strongly interacting limits. We show that, although the shear viscosity and the thermal conductivity both explicitly depend on the effective range near resonance, the Prandtl number which parametrizes the ratio of momentum to thermal diffusivity does not have an explicit interaction dependence both at resonance and for weak interactions in the low-energy limit. In contrast to sss-wave systems, ppp-wave scattering exhibits an additional resonance at weak attraction from a quasi-bound state at positive energies, which leads to a pronounced dip in the shear viscosity at specific temperatures.
One-dimensional Bose and Fermi gases with contact interactions are known to exhibit the weak-strong duality, where the equilibrium thermodynamic properties of one system at weak coupling are identical to those of the other system at strong coupling. Here, we show that such duality extends beyond the thermodynamics to the frequency-dependent complex bulk viscosity, which is provided by the contact-contact response function. In particular, we confirm that the bulk viscosities of the Bose and Fermi gases agree in the high-temperature limit, where the systematic expansion in terms of fugacity is available at arbitrary coupling. We also compute their bulk viscosities perturbatively in the weak-coupling limit at arbitrary temperature, which via the duality serve as those of the Fermi and Bose gases in the strong-coupling limit.
We study the viscous properties of a system of weakly interacting spin-1/2 fermions in one dimension. Accounting for the effect of interactions on the quasiparticle energy spectrum, we obtain the bulk viscosity of this system at low temperatures. Our result is valid for frequencies that are small compared with the rate of fermion backscattering. For frequencies larger than this exponentially small rate, the excitations of the system become decoupled from the center of mass motion, and the fluid is described by two-fluid hydrodynamics. We calculate the three transport coefficients required to describe viscous dissipation in this regime.
A well-known feature of the classical monoatomic gas is that its bulk viscosity is strongly suppressed because the single-particle dispersion is quadratic. On the other hand, in condensed matter systems the effective single-particle dispersion is altered by lattice effects and interactions. In this work, we study the bulk viscosity of one-dimensional Fermi gases with generic energy-momentum dispersion relations. As an application, viscous dissipation arising from lattice effects is analyzed for the tight-binding model. In addition, we investigate how weak interactions affect the bulk viscosity. Finally, we discuss viscous dissipation in the regime in which the Fermi gas is not fully equilibrated, as can occur when the system is driven at frequencies that exceed the rate of fermion backscattering. In this case, the Fermi gas is described by three bulk viscosities, which we obtain for a generic single-particle dispersion.
In interaction-dominated two-dimensional electron gases at intermediate temperatures, electron transport is not diffusive as in the conventional Drude picture but instead hydrodynamic. The relevant transport coefficient in this regime is the shear viscosity. Here, we develop a numerically exact basis expansion to solve the Fermi liquid equation, and apply it to compute the shear viscosity of the electron gas with screened Coulomb interactions. Our calculations are valid at all temperatures and in particular describe the response beyond the asymptotic low-temperature limit, where perturbative approaches exist. We show that even in this low-temperature limit, there is a nonanalytic exchange contribution to the shear viscosity, highlighting the need for a full nonperturbative solution of the Fermi liquid equation. We hope that the techniques developed in this work will serve as a platform to determine the response of interacting Fermi liquids.
We investigate the role of the effective range on the bulk viscosity of s- and p-wave Fermi gases. At resonance, the presence of the effective range breaks the scale invariance of the system, and hence results in a nonzero bulk viscosity. However, we show that the effective range plays a very different role in the two cases. In the s-wave case, the role of the effective range is perturbative, and its contribution to the bulk viscosity vanishes in the limit of zero effective range. On the other hand, the effective range in p-wave Fermi gases leads to a nonzero bulk viscosity, even in the zero-range limit. We employ a general diagrammatic approach to compute the bulk viscosity spectral function that includes the effects of the effective range. We then compute the analytic expressions for the spectral function in the high temperature limit, at low and high frequencies. We also derive the sum rules for the bulk viscosity spectral function for both s- and p-wave gases.