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

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.

Figure 1

Bulk viscosity for a weakly interacting spin-polarized 1D Fermi gas with odd-wave interactions as a function of density $n$. ${\lambda }_{\mathrm{th}}=\sqrt{2\pi /mT}$ is the thermal de Broglie wavelength with $\hslash$ and $k_B$ set to unity. The bulk viscosity at low temperatures is evaluated via the LL model, Eq. (37), while the high-temperature result follows the virial expansion, Eq. (44). The dashed-dotted line where $n{\lambda }_{\mathrm{th}}=1$ approximates the transition between the LL and virial expansion results. In both limits the bulk viscosity is proportional to ${\ell }^2$ as required by conformal symmetry. Our results in the high- and low-temperature limits smoothly connect with one another, consistent with the lack of a finite-temperature superfluid transition in one dimension,

Figure 2

Bulk viscosity of a spin-polarized Fermi gas near resonance. At high temperatures the bulk viscosity vanishes like $T^{-3/2}$, Eq. (45), while for low temperatures the bulk viscosity diverges as $T^{-4}$, Eq. (48). The dashed line acts as a guide for the eye, while the vertical dashed-dotted line separates the high- and low-temperature behaviors. From the duality relations, this plot also describes the bulk viscosity for a weakly interacting 1D Bose gas.