Overcoming the sign problem in 1-dimensional QCD by new integration rules with polynomial exactness

TL;DR

This paper introduces a new polynomially exact integration method for $U(N)$ and $SU(N)$ groups, effectively overcoming the sign problem in 1D QCD with a chemical potential, especially where traditional methods fail.

Contribution

The paper presents a novel integration technique with polynomial exactness that successfully addresses the sign problem in 1D QCD, outperforming Markov Chain Monte Carlo methods in challenging parameter regions.

Findings

Method is polynomially exact for N ≤ 3

Successfully overcomes the sign problem in 1D QCD

Achieves significantly reduced errors across parameter space

Abstract

In this paper we describe a new integration method for the groups $U(N)$ and $SU(N)$, for which we verified numerically that it is polynomially exact for $N\le 3$. The method is applied to the example of 1-dimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov Chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that our new method leads to orders of magnitude reduced errors also in other regions of parameter space.