# Third-order perturbative lattice and complex Langevin analyses of the finite-temperature equation of state of non-relativistic fermions in one dimension

## TL;DR

This paper develops and applies third-order lattice perturbation theory and complex Langevin methods to study the finite-temperature equation of state of one-dimensional non-relativistic fermions, achieving agreement with non-perturbative results and extending to strong couplings.

## Contribution

It introduces an unconventional lattice perturbation approach using Hubbard-Stratonovich transformation and automates Wick's theorem, along with a dimension-independent technique for Matsubara sums, and applies these to fermions at finite temperature.

## Key findings

- Excellent agreement between perturbative and non-perturbative results at weak couplings.
- Predictions for strong couplings using complex Langevin methods.
- Perturbative calculations of up to fifth-order virial coefficients.

## Abstract

We analyze the pressure and density equations of state of unpolarized non-relativistic fermions at finite temperature in one spatial dimension. For attractively interacting regimes, we perform a third-order lattice perturbation theory calculation, assess its convergence properties by comparing with hybrid Monte Carlo results (there is no sign problem in this regime), and demonstrate agreement with real Langevin calculations. For repulsive interactions, we present lattice perturbation theory results as well as complex Langevin calculations, with a modified action to prevent uncontrolled excursions in the complex plane. Although perturbation theory is a common tool, our implementation of it is unconventional; we use a Hubbard-Stratonovich transformation to decouple the system and automate the application of Wick's theorem, thus generating the diagrammatic expansion, including symmetry factors, at any desired order. We also present an efficient technique to tackle nested Matsubara frequency sums without relying on contour integration, which is independent of dimension and applies to both relativistic and non-relativistic systems, as well as all energy-independent interactions. We find exceptional agreement between perturbative and non-perturbative results at weak couplings, and furnish predictions based on complex Langevin at strong couplings. We additionally present perturbative calculations of up to the fifth-order virial coefficient for repulsive and attractive couplings. Both the lattice perturbation theory and complex Langevin formalisms can easily be extended to a variety of situations including polarized systems, bosons, and higher dimension.