Lattice theory for nonrelativistic fermions in one spatial dimension

TL;DR

This paper develops a sign-problem-free loop representation for a one-dimensional four-component Fermi gas, enabling numerical and mean-field studies of its universal many-body phenomena, unlike the problematic higher-dimensional cases.

Contribution

It introduces a novel loop representation for 1D four-component Fermi gases that avoids the sign problem, facilitating new theoretical and numerical analyses.

Findings

Sign-problem-free representation for 1D Fermi gases

Existence of a phase with a well-defined continuum limit

Potential for numerical simulation of universal phenomena

Abstract

I derive a loop representation for the canonical and grand-canonical partition functions for an interacting four-component Fermi gas in one spatial dimension and an arbitrary external potential. The representation is free of the "sign problem" irrespective of population imbalance, mass imbalance, and to a degree, sign of the interaction strength. This property is in sharp contrast with the analogous three-dimensional two-component interacting Fermi gas, which exhibits a sign problem in the case of unequal masses, chemical potentials, and repulsive interactions. The one-dimensional system is believed to exhibit many phenomena in common with its three-dimensional counterpart, including an analog of the BCS-BEC crossover, and nonperturbative universal few- and many-body physics at scattering lengths much larger than the range of interaction, making the theory an interesting candidate for numerical study. Positivity of the probability measure for the partition function allows for a mean-field treatment of the model; here, I present such an analysis for the interacting Fermi gas in the SU(4) (unpolarized, mass-symmetric) limit, and demonstrate that there exists a phase in which a continuum limit may be defined.