Fermion sign bounds theory in quantum Monte Carlo simulation

TL;DR

This paper challenges the traditional view that the fermion sign problem in quantum Monte Carlo always decays exponentially with system size, providing analytical bounds that reveal algebraic decay in certain models and conditions.

Contribution

The authors analytically derive bounds for the fermion sign in QMC, distinguishing exponential and algebraic decay regimes, and connect these bounds to physical properties and phase transitions.

Findings

Sign bounds can be algebraic or exponential depending on the system and temperature.

Analytical explanation for algebraic sign problems in flat band moiré models.

Sign bounds relate to phase transitions at finite temperature.

Abstract

Sign problem 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 reference system and could distinguish when the bounds have the usual exponential scaling, and when they are bestowed on an algebraic scaling at low temperature limit. We analytically explain such algebraic sign problems found in flat band moir\'e lattice models at 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 ability to ease or even make use of the sign problem.