A general approach to the sign problem - the factorization method with multiple observables

TL;DR

This paper presents a generalized factorization method with multiple observables to address the sign problem in Monte Carlo simulations, demonstrating its effectiveness and scalability through explicit examples and comparison with analytic results.

Contribution

It introduces a flexible approach controlling multiple observables to overcome the sign problem, extending the applicability of the factorization method to more complex systems.

Findings

The method successfully reproduces known analytic results.

Controlling multiple observables improves sampling efficiency.

Scalability to large systems is demonstrated.

Abstract

The sign problem is a notorious problem, which occurs in Monte Carlo simulations of a system with the partition function whose integrand is not real positive. The basic idea of the factorization method applied on such a system is to control some observables in order to determine and sample efficiently the region of configuration space which gives important contribution to the partition function. We argue that it is crucial to choose appropriately the set of the observables to be controlled in order for the method to work successfully in a general system. This is demonstrated by an explicit example, in which it turns out to be necessary to control more than one observables. Extrapolation to large system size is possible due to the nice scaling properties of the factorized functions, and known results obtained by an analytic method are shown to be consistently reproduced.