The QCD sign problem and dynamical simulations of random matrices

TL;DR

This paper investigates the sign problem in lattice QCD at nonzero chemical potential using random matrix theory, deriving a new, simpler formula for the average phase of the fermion determinant applicable to finite matrices and comparing it with simulations.

Contribution

The paper provides an alternative derivation of the average phase formula, simplifying calculations and extending applicability beyond the microscopic limit, with validation through simulations.

Findings

New formula for average phase at finite matrix size

Demonstrated convergence to microscopic limit

Comparison of analytical results with simulation data

Abstract

At nonzero quark chemical potential dynamical lattice simulations of QCD are hindered by the sign problem caused by the complex fermion determinant. The severity of the sign problem can be assessed by the average phase of the fermion determinant. In an earlier paper we derived a formula for the microscopic limit of the average phase for general topology using chiral random matrix theory. In the current paper we present an alternative derivation of the same quantity, leading to a simpler expression which is also calculable for finite-sized matrices, away from the microscopic limit. We explicitly prove the equivalence of the old and new results in the microscopic limit. The results for finite-sized matrices illustrate the convergence towards the microscopic limit. We compare the analytical results with dynamical random matrix simulations, where various reweighting methods are used to circumvent the sign problem. We discuss the pros and cons of these reweighting methods.