Equation of state of non-relativistic matter from automated perturbation theory and complex Langevin

TL;DR

This paper develops an automated perturbation theory approach combined with complex Langevin methods to calculate the equation of state for non-relativistic fermionic systems, including polarized and strongly interacting regimes, in one and three dimensions.

Contribution

It introduces an automated diagrammatic perturbation theory framework using Hubbard-Stratonovich transformation and Wick's theorem, validated against stochastic methods for non-relativistic fermions.

Findings

Excellent agreement between perturbation theory and Monte Carlo in weak coupling.

Predictions for strong coupling and polarized systems.

First estimate of the 3D polarized unitary Fermi gas equation of state.

Abstract

We calculate the pressure and density of polarized non-relativistic systems of two-component fermions coupled via a contact interaction at finite temperature. For the unpolarized one-dimensional system with an attractive interaction, we perform a third-order lattice perturbation theory calculation and assess its convergence by comparing with hybrid Monte Carlo. In that regime, we also demonstrate agreement with real Langevin. For the repulsive unpolarized one-dimensional system, where there is a so-called complex phase problem, we present lattice perturbation theory as well as complex Langevin calculations. For our studies, we employ a Hubbard-Stratonovich transformation to decouple the interaction and automate the application of Wick's theorem for perturbative calculations, which generates the diagrammatic expansion at any order. We find excellent agreement between the results from our perturbative calculations and stochastic studies in the weakly interacting regime. In addition, we show predictions for the strong coupling regime as well as for the polarized one-dimensional system. Finally, we show a first estimate for the equation of state in three dimensions where we focus on the polarized unitary Fermi gas.