A subset solution to the sign problem in random matrix simulations

TL;DR

This paper introduces a subset method that effectively solves the sign problem in random matrix simulations at nonzero chemical potential, enabling accurate computations of physical observables with manageable statistical errors.

Contribution

The authors develop a subset approach that ensures positivity of determinants, providing a new solution to the sign problem in matrix models and related QCD-like theories.

Findings

The subset method maintains a mild error dependence on matrix size and chemical potential.

It outperforms standard reweighting methods by avoiding exponential error growth.

The method successfully resolves the Silver Blaze puzzle in the microscopic limit.

Abstract

We present a solution to the sign problem in dynamical random matrix simulations of a two-matrix model at nonzero chemical potential. The sign problem, caused by the complex fermion determinants, is solved by gathering the matrices into subsets, whose sums of determinants are real and positive even though their cardinality only grows linearly with the matrix size. A detailed proof of this positivity theorem is given for an arbitrary number of fermion flavors. We performed importance sampling Monte Carlo simulations to compute the chiral condensate and the quark number density for varying chemical potential and volume. The statistical errors on the results only show a mild dependence on the matrix size and chemical potential, which confirms the absence of sign problem in the subset method. This strongly contrasts with the exponential growth of the statistical error in standard reweighting methods, which was also analyzed quantitatively using the subset method. Finally, we show how the method elegantly resolves the Silver Blaze puzzle in the microscopic limit of the matrix model, where it is equivalent to QCD.