Complex paths around the sign problem

The Monte Carlo evaluation of path integrals is one of a few general purpose methods to approach strongly coupled systems. It is used in all branches of physics, from QCD and nuclear physics to the correlated electron systems. However, many systems of great importance (dense matter inside neutron stars, the repulsive Hubbard model away from half filling, and dynamical and nonequilibrium observables) are not amenable to the Monte Carlo method as it currently stands due to the so-called sign problem. A new set of ideas recently developed to tackle the sign problem based on the complexification of field space and the Picard-Lefshetz theory accompanying it is reviewed. The mathematical ideas underpinning this approach, as well as the algorithms developed thus far, are described together with nontrivial examples where the method has already been proved successful. Directions of future work, including the burgeoning use of machine learning techniques, are delineated.

Automated summary

This article reviews a new approach to solve the sign problem based on the complexification of field space and the accompanying mathematical theory. The underlying mathematical ideas and developed algorithms are described, along with successful examples. The article also outlines future research directions, including the growing use of machine learning techniques.