Stable and Efficient Algorithms for the Fermion Determinant

TL;DR

This paper summarizes algorithms for the exact numerical treatment of fermion determinants in quantum Monte Carlo, emphasizing methods suitable for different volume regimes and temperature conditions.

Contribution

It introduces stable dense matrix methods for low-temperature regimes and scalable sparse matrix approaches for large volumes, tailored to fermion determinant calculations.

Findings

Dense matrix method is numerically stable for low temperatures.

Sparse matrix approach is efficient and scalable for large volumes.

Different algorithms are recommended based on volume and temperature regimes.

Abstract

Some algorithms for the numerically exact treatment of fermion determinants are summarised. This is not supposed to be a review, rather a concise handbook. The audience is expected to have a basic understanding of how to put fermions on a computer. We primarily discuss different ways to work with the fermion matrix in the "sausage" (Green's function) formulation for quantum Monte Carlo (QMC). We emphasise the need for varied approaches in different space-time volume regimes. In particular, for small spatial volumes we describe a numerically stable method based on dense matrix operations. It is designed specifically to deal with very low temperature regimes. On the other hand, for (relatively) large volumes we describe a highly efficient and scalable sparse matrix approach.