Computational methods for the fermion determinant and the link between overlap and domain wall fermions

TL;DR

This paper reviews computational techniques for evaluating the fermion determinant in lattice QCD, focusing on matrix function methods, and explores the algebraic connection between overlap and domain wall fermions, including inversion algorithms.

Contribution

It provides a comprehensive review of matrix function evaluation methods and clarifies the formal relationship between overlap and domain wall fermions, including a multigrid inversion approach.

Findings

Krylov subspace methods improve matrix function evaluations.

Formal link established between overlap and domain wall fermions.

Multigrid algorithm effectively inverts the overlap operator.

Abstract

This paper reviews the most popular methods which are used in lattice QCD to compute the determinant of the lattice Dirac operator: Gaussian integral representation and noisy methods. Both of them lead naturally to matrix function problems. We review the most recent development in Krylov subspace evaluation of matrix functions. The second part of the paper reviews the formal relationship and algebraic structure of domain wall and overlap fermions. We review the multigrid algorithm to invert the overlap operator. It is described here as a preconditioned Jacobi iteration where the preconditioner is the Schur complement of a certain block of the truncated overlap matrix.