The fermion sign problem in Gauss law sectors of quantum link models with dynamical matter

TL;DR

This paper identifies specific Gauss law sectors in quantum link models that can be simulated without the fermion sign problem, and explores their phase structure using numerical methods.

Contribution

It analytically characterizes sign-problem-free Gauss law sectors in quantum link models and examines their phases with large-scale simulations.

Findings

Ground states are in sectors satisfying a staggered Gauss law.

Sign problem is absent in certain Gauss law sectors but not in the zero-charge sector.

Transitions between sectors are influenced by magnetic energy.

Abstract

The fermion sign problem poses a formidable challenge to the use of Monte Carlo methods for lattice gauge theories with dynamical fermionic matter fields. A meron cluster algorithm recently formulated for gauge fields represented as spin-$\frac{1}{2}$ quantum links coupled to a single flavour of staggered fermions samples only two of the exponentially many Gauss law (GL) sectors at low temperatures, making it possible to simulate either of those two GL sectors at zero temperature in polynomial time. In this article, we analytically identify GL sectors which can be simulated without encountering the fermion sign problem in arbitrary spatial dimensions. Using large-scale exact diagonalization and cluster Monte Carlo methods, we further explore the nature of phases in the GL sectors dominating at zero temperature. The vacuum states lie in sectors which satisfy a staggered Gauss law, in contrast to the zero GL sector familiar in particle physics. Moreover, we prove that while the ground state GL sectors do not suffer from the fermion sign problem, the usual zero-charge GL sector (often considered the physical sector) does. We outline the role of the magnetic energy in causing transitions between GL sectors. We expect our results to be valid for truncated Kogut-Susskind gauge theories, beyond quantum link models.