Isothermal compressibility in Fermi gases in an optical lattice

Abstract

Isothermal compressibility ${\ensuremath{\kappa}}_{T}$, which measures the fluidity of the liquid or the stiffness of the solids, is one of the most important physical quantities in thermodynamic physics. In ultracold atoms, this quantity has been widely explored in the measurement of the transition from normal gas to Bose-Einstein condensate, as well as the transition from superfluids to solid phases, such as supersolids and Mott insulators. This physical quantity was also examined in the spin-orbit-coupled degenerate gases in free space, showing that the combined contribution of spin-orbit coupling (SOC) and the Zeeman field may pronouncedly enhance the isothermal compressibility in the gapless Weyl phases. Motivated by the differences between the free space and the optical lattice, in this work we examine this compressibility in the spin-orbit-coupled degenerate Fermi gases in a square lattice model, in which the filling factor and particle-hole symmetry about half filling become an important influencing factor. In this case, the particle density in compressibility should be replaced by the density of a particle or hole according to their filling to ensure symmetry about half filling. With increasing particle density $n$, the compressibility decreases to a small magnitude, while ${n}^{2}{\ensuremath{\kappa}}_{T}$ may increase monotonically with increasing $n$ from zero to half filling. In the strong-coupling regime, when the chemical potential and pairing strength are much larger than the tunneling strength and Zeeman field, this compressibility will approach ${n}^{2}{\ensuremath{\kappa}}_{T}=2/U$, where $U$ is the on-site interacting strength. This limit can be explained by a one-site Hubbard model. We show that SOC and the Zeeman field can play opposite roles in the behavior of compressibilities. The combination of these two terms can give rise to pronouncedly enhanced isothermal compressibility in some proper parameter regime. The anomalous compressibilities also give rise to a anomaly in the exponent of pressure. In the lattice model, the enhanced peak can be found in both the fully gapped phase and gapless Weyl phases, which is different from that in free space, which can be realized only in the gapless phases. We derive the formalisms from the Gibbs-Duhem formalism. These observations unveil important differences between the continuous model and discrete models. Our theory may also be useful to study the physics in other superfluids, including topological superfluids with finite-momentum pairings, in which the finite-momentum pairing may also have an implicit contribution to this compressibility. In this way, we may realize superfluids with some extraordinary physical properties.