# Complex Langevin and other approaches to the sign problem in quantum many-body physics

## TL;DR

This paper reviews complex Langevin and other methods addressing the sign problem in quantum many-body physics, discussing theoretical foundations, various approaches, and recent applications to quantum field theory.

## Contribution

It provides a comprehensive overview of complex stochastic quantization and compares it with alternative methods for solving the sign problem in quantum many-body systems.

## Key findings

- Complex Langevin can be applied to quantum field theories with sign problems.
- Various methods like Lefschetz thimbles and density-of-states improve convergence.
- The paper highlights open challenges and practical solutions in the field.

## Abstract

We review the theory and applications of complex stochastic quantization to the quantum many-body problem. Along the way, we present a brief overview of a number of ideas that either ameliorate or in some cases altogether solve the sign problem, including the classic reweighting method, alternative Hubbard-Stratonovich transformations, dual variables (for bosons and fermions), Majorana fermions, density-of-states methods, imaginary asymmetry approaches, and Lefschetz thimbles. We discuss some aspects of the mathematical underpinnings of conventional stochastic quantization, provide a few pedagogical examples, and summarize open challenges and practical solutions for the complex case. Finally, we review the recent applications of complex Langevin to quantum field theory in relativistic and nonrelativistic quantum matter, with an emphasis on the nonrelativistic case.