Tan's Contact for Trapped Spin-Orbit-Coupled Fermi Gases at Finite Temperature

Abstract

Tan's contact, bridging microscopic features of many-body quantum systems to two-body contact interactions, is pivotal in the description of quantum gases. However, past studies of Tan's contact of the spin-orbit coupled Fermi gases have been exclusively discussed at zero temperature. The thermal effects on that, which cannot be neglected in most experimental conditions, has not been explored so far. Here, through a path integral Monte Carlo (PIMC) study, we show that thermal effects fundamentally alter the scaling relation between spin-orbit coupling (SOC) and Tan's contact. The linear scaling of Tan's contact with respect to the strength of SOC found at zero temperature is drastically undermined by the thermal effects. And a new mechanism based on the competition between the mean occupancy and the energy-reweighted occupancy of the opposite spin fermion pairs within the interaction range has been unveiled, which radically changes the above linear scaling to a non-monotonic dependence. Our work should be useful for future experiments to explore the preternatural thermal effects on Tan's contact of the strongly interacting spin-orbit-coupled Fermi gas.

The paper examines how finite temperature changes the dependence of Tan’s contact on spin–orbit coupling (SOC) in a harmonically trapped, strongly interacting Fermi gas. It studies isotropic three-dimensional SOC, proportional to $\mathbf p\cdot\boldsymbol\sigma$, using path-integral Monte Carlo for two and three fermions. Opposite-spin particles interact through an attractive square well tuned to divergent scattering length, with range $r_0=0.1a_{\mathrm{ho}}$. The temperatures are $k_BT/\hbar\omega=0.6,0.7,0.8$ for two particles and $0.8,0.9,1.0$ for three particles. The reported contact becomes nonmonotonic as SOC strength increases, while the approximately linear dependence associated with earlier zero-temperature results survives only in a weak-SOC region. To interpret this behavior, the calculation decomposes the scattering-length derivative of the thermal energy into two terms: one proportional to the mean imaginary-time-averaged number of opposite-spin pairs inside the interaction range, and another proportional to its covariance with an energy estimator. The first term decreases with SOC; the second initially decreases and subsequently increases, producing the reported turnover in their sum. The evidence consists of finite-temperature simulations of these few-particle systems; no thermodynamic-limit result or quantitative boundary for the linear regime is supplied. A distinction relevant to interpreting the observable is that the paper defines contact using $\partial E/\partial(-a_s^{-1})$ at fixed temperature, rather than the free-energy derivative appearing in the standard finite-temperature adiabatic relation. The covariance term follows from this energy derivative.