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

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.

68 pages, 19 figures, review article, version submitted to Physics Reports

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• Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations
• Mitigating the Hubbard Sign Problem with Complex-Valued Neural Networks
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• Stable solvers for real-time Complex Langevin
• Bayesian inference of real-time dynamics from lattice QCD
• Complex Langevin Simulations of Zero-dimensional Supersymmetric Quantum Field Theories
• A modular implementation of an effective interaction approach for harmonically trapped fermions in 1D
• Simulating Yang-Mills theories with a complex coupling
• Time-space duality in 2D quantum gravity
• Pairing patterns in polarized unitary Fermi gases above the superfluid transition
• Complex Langevin approach to interacting Bose gases
• Density-Induced Hadron-Quark Crossover via the Formation of Cooper Triples
• Contribution to understanding the phase structure of strong interaction matter: Lee-Yang edge singularities from lattice QCD
• Truly Lorentzian quantum cosmology
• Complex Langevin Dynamics and Supersymmetric Quantum Mechanics
• Partition Function Estimation: Quantum and Quantum-Inspired Algorithms
• Regularization of Complex Langevin Method
• Towards sampling complex actions
• A lattice pairing-field approach to ultracold Fermi gases
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