A note on the Loop Formula for the fermionic determinant

TL;DR

This paper derives a formula that expresses the fermionic determinant as an infinite product of fixed-size determinants indexed by loops, providing a potentially useful computational approach.

Contribution

It introduces a novel loop formula for the fermionic determinant, decomposing it into a product of smaller, fixed-size determinants independent of lattice size.

Findings

The fermionic determinant can be expressed as an infinite product of loop-indexed determinants.

The smaller determinants are of fixed size, regardless of lattice size.

The formula offers new insights into fermionic determinant computations.

Abstract

A formula expressing the fermionic determinant as an infinite product of smaller determinants is derived and discussed. These smaller determinants are of a fixed size, independent of the size of the lattice and are indexed by loops of increasing length.