Wilson Fermion Determinant in Lattice QCD

TL;DR

This paper introduces a formula to reduce the computational complexity of Wilson fermion determinants in lattice QCD, analyzes eigenvalues and coefficients relevant for finite density calculations, and provides numerical methods to handle large variations in these coefficients.

Contribution

A novel reduction formula for Wilson fermion determinants that simplifies eigenvalue analysis and aids finite density lattice QCD computations.

Findings

Eigenvalues of the reduced matrix are analyzed.

Coefficients $C_n$ exhibit exponential growth with lattice volume.

A numerical method is proposed to manage large variations in $C_n$.

Abstract

We present a formula for reducing the rank of Wilson fermions from $4 N_c N_x N_y N_z N_t$ to $4 N_c N_x N_y N_z$ keeping the value of its determinant. We analyse eigenvalues of a reduced matrix and coefficients $C_n$ in the fugacity expansion of the fermion determinant $\sum_n C_n (\exp(\mu/T))^n$, which play an important role in the canonical formulation, using lattice QCD configurations on a $4^4$ lattice. Numerically, $\log |C_n|$ varies as $N_x N_y N_z$, and goes easily over the standard numerical range; We give a simple cure for that. The phase of $C_n$ correlates with the distribution of the Polyakov loop in the complex plain. These results lay the groundwork for future finite density calculations in lattice QCD.