# Reaching the continuum limit in finite-temperature ab initio field-theory computations in many-fermion systems

## TL;DR

This paper introduces a low-rank factorization method that significantly reduces computational costs in finite-temperature field-theory calculations for many-fermion systems, enabling more effective continuum limit extrapolations.

## Contribution

The authors develop a systematic low-rank factorization approach that lowers the computational scaling from cubic to linear in lattice size, improving finite-temperature many-body simulations.

## Key findings

- Achieved exact calculations of 2D Fermi gas properties across temperature ranges.
- Demonstrated the method's effectiveness in normal and superfluid states.
- Enabled more feasible continuum limit extrapolations in correlated fermion systems.

## Abstract

Finite-temperature, grand-canonical computations based on field theory are widely applied in areas including condensed matter physics, ultracold atomic gas systems, and lattice gauge theory. However, these calculations have computational costs scaling as $N_s^3$ with the size of the lattice or basis set, $N_s$. We report a new approach based on systematically controllable low-rank factorization which reduces the scaling of such computations to $N_s N_e^2$, where $N_e$ is the average number of fermions in the system. In any realistic calculations aiming to describe the continuum limit, $N_s/N_e$ is large and needs to be extrapolated effectively to infinity for convergence. The method thus fundamentally changes the prospect for finite-temperature many-body computations in correlated fermion systems. Its application, in combination with frameworks to control the sign or phase problem as needed, will provide a powerful tool in {\it ab initio} quantum chemistry and correlated electron materials. We demonstrate the method by computing exact properties of the two-dimensional Fermi gas with zero-range attractive interaction, as a function of temperature in both the normal and superfluid states.