Lattice field theories with a sign problem

TL;DR

This paper reviews various approaches to address the sign problem in lattice QCD and related theories, including holomorphic methods, dual variables, tensor networks, and machine learning, assessing their potential for future solutions.

Contribution

It provides a comprehensive overview of current strategies to mitigate the sign problem, highlighting recent advances and future directions in lattice field theories.

Findings

Holomorphic methods like Lefschetz thimbles show promise in controlling the sign problem.

Dual variables and tensor networks offer alternative formulations to bypass the sign problem.

Machine learning approaches are emerging as potential tools for tackling the sign problem.

Abstract

The sign problem obstructs the determination of the QCD phase diagram in the temperature-baryon chemical potential plane using lattice QCD. We review the sign problem in QCD and related field theories, including applications to real-time dynamics. We focus on approaches where the sign problem can potentially be solved or controlled, irrespective of its severeness. These include holomorphic extensions -- Lefschetz thimbles, holomorphic flow, contour deformations, and complex Langevin dynamics --, and the introduction of new degrees of freedom -- dual variables and the tensor renormalisation group. We also highlight directions in which machine learning approaches have shown promise. Since many methods are first tested in simpler models, we provide an outlook on their feasibility for lattice systems.