Fermion sign bounds theory in quantum Monte Carlo simulation
Xu Zhang, Gaopei Pan, Xiao Yan Xu, Zi Yang Meng
Abstract
Sign problems in fermion quantum Monte Carlo (QMC) simulation appears to be an extremely hard problem. Traditional lore passing around for years tells people that when there is a sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space. We, however, analytically show this is not always the case and manage to find physical bounds for the average sign. Our understanding is based on a direct connection between the sign bounds and a well-defined partition function of the reference system and could distinguish when the bounds have the usual exponential scaling and when they are bestowed on an algebraic scaling at the low-temperature limit. We analytically explain such algebraic sign problems found in flat-band moir\'e lattice models at the low-temperature limit. At finite temperature, a domain-size argument based on sign bounds also explains the connection between sign behavior and finite-temperature phase transition. Sign bounds, as a well-defined observable, may have the ability to ease or even make use of the sign problem.