New Numerical Method for Fermion Field Theory

TL;DR

This paper introduces a new deterministic numerical method for solving fermion field theories that avoids many issues of traditional stochastic techniques, enabling faster and more flexible calculations of fermionic propagators.

Contribution

The paper presents a novel deterministic approach based on lattice functional equations and Grassmann polynomial expansions, offering an alternative to Monte Carlo methods for fermion systems.

Findings

Calculates fermion propagators efficiently without Monte Carlo.

Avoids difficulties related to fermion determinants and dynamical fermions.

Demonstrates the method with simple illustrative examples.

Abstract

A new deterministic, numerical method to solve fermion field theories is presented. This approach is based on finding solutions $Z[J]$ to the lattice functional equations for field theories in the presence of an external source $J$. Using Grassmann polynomial expansions for the generating functional $Z$, we calculate propagators for systems of interacting fermions. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. Because it is not based on a statistical technique, it does not have many of the difficulties often encountered when simulating fermions. Since no determinant is ever calculated, solutions to problems with dynamical fermions are handled more easily. This approach is very flexible, and can be taylored to specific problems based on convenience and computational constraints. We present simple examples to illustrate the method; more general schemes are desirable for more complicated systems.