The virial expansion of attractively interacting Fermi gases in 1D, 2D, and 3D, up to fifth order

• Published in: arXiv.org 2020-05
• Main Authors: Hou, Y, Drut, J E
• Format: Article
• Language: English
• Subjects: Approximation; Fermi gases; Mathematical analysis; Virial coefficients
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1908.00174

The virial expansion characterizes the high-temperature approach to the quantum-classical crossover in any quantum many-body system. Here, we calculate the virial coefficients up to the fifth-order of Fermi gases in 1D, 2D, and 3D, with attractive contact interactions, as relevant for a variety of applications in atomic and nuclear physics. To that end, we discretize the imaginary-time direction and calculate the relevant canonical partition functions. In coarse discretizations, we obtain analytic results featuring relationships between the interaction-induced changes \(\Delta b_3\), \(\Delta b_4\), and \(\Delta b_5\) as functions of \(\Delta b_2\), the latter being exactly known in many cases by virtue of the Beth-Uhlenbeck formula. Using automated-algebra methods, we push our calculations to progressively finer discretizations and extrapolate to the continuous-time limit. We find excellent agreement for \(\Delta b_3\) with previous calculations in all dimensions and we formulate predictions for \(\Delta b_4\) and \(\Delta b_5\) in 1D and 2D. We also provide, for a range of couplings,the subspace contributions \(\Delta b_{31}\), \(\Delta b_{22}\), \(\Delta b_{41}\), and \(\Delta b_{32}\), which determine the equation of state and static response of polarized systems at high temperature. As a performance check, we compare the density equation of state and Tan contact with quantum Monte Carlo calculations, diagrammatic approaches, and experimental data where available. Finally, we apply Padé and Padé-Borel resummation methods to extend the usefulness of the virial coefficients to approach and in some cases go beyond the unit-fugacity point.