# Leading- and next-to-leading order semiclassical approximation to the first seven virial coefficients of spin-1/2 fermions across spatial dimensions

## TL;DR

This paper develops a semiclassical approximation method to calculate virial coefficients of spin-1/2 fermions with contact interactions across different dimensions, providing analytic formulas and analyzing the virial expansion's convergence.

## Contribution

It introduces leading- and next-to-leading order semiclassical formulas for virial coefficients up to seventh order, relating them to the second-order coefficient across dimensions.

## Key findings

- Derived analytic expressions for virial coefficients up to seventh order.
- Analyzed the virial expansion's convergence and its dependence on coupling strength.
- Applied the results to calculate pressure and Tan's contact in 2D Fermi gases.

## Abstract

Following up on recent calculations, we investigate the leading- and next-to-leading order semiclassical approximation to the virial coefficients of a two-species fermion system with a contact interaction. Using the analytic result for the second-order virial coefficient as a renormalization condition, we derive expressions for up to the seventh-order virial coefficient $\Delta b_7$. Our results at leading order, though approximate, furnish simple analytic formulas that relate $\Delta b_n$ to $\Delta b_2$ for arbitrary dimension, providing a glimpse into the behavior of the virial expansion across dimensions and coupling strengths. As an application, we calculate the pressure and Tan's contact of the 2D attractive Fermi gas and examine the radius of convergence of the virial expansion as a function of the coupling strength.