Sign Problem Landscape of Dimer, Loop, and Ground-State Sectors of a U(1) Quantum Link Model

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-1/2 quantum links coupled to a single flavor of staggered fermions samples only two of the exponentially many Gauss law (GL) sectors at low temperatures, allowing the simulation of those two GL sectors at zero temperature in polynomial time. In this Letter, 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 explore the nature of phases in the GL sectors dominating at zero temperature. The ground state lives in a superselection sector free of the sign problem and maps to the quantum dimer model with mobile monomers. The usual zero-charge GL sector suffers from the fermion sign problem and maps to the fully packed loop model with mobile monomers. The role of the magnetic energy in causing transitions between GL sectors is outlined. We expect our results to be valid for truncated Kogut-Susskind gauge theories, beyond quantum link models.