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Physics Reports Complex Langevin and Other Approaches to the Sign Physics Reports 892 (2021) 1–54 Contents lists available at ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Complex Langevin and other approaches to the sign problem in quantum many-body physics ∗ C.E. Berger a,1, L. Rammelmüller b,c,1, A.C. Loheac a, F. Ehmann b, J. Braun b,e,d, , J.E. Drut a a Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, USA b Institut für Kernphysik (Theoriezentrum), Technische Universität Darmstadt, D-64289 Darmstadt, Germany c GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, D-64291 Darmstadt, Germany d FAIR, Facility for Antiproton and Ion Research in Europe GmbH, Planckstraße 1, D-64291 Darmstadt, Germany e ExtreMe Matter Institute EMMI, GSI, Planckstraße 1, D-64291 Darmstadt, Germany article info a b s t r a c t Article history: We review the theory and applications of complex stochastic quantization to the quan- Received 3 September 2019 tum many-body problem. Along the way, we present a brief overview of a number of Received in revised form 25 August 2020 ideas that either ameliorate or in some cases altogether solve the sign problem, including Accepted 8 September 2020 the classic reweighting method, alternative Hubbard–Stratonovich transformations, dual Available online 7 October 2020 variables (for bosons and fermions), Majorana fermions, density-of-states methods, Editor: A. Schwenk imaginary asymmetry approaches, and Lefschetz thimbles. We discuss some aspects Keywords: of the mathematical underpinnings of conventional stochastic quantization, provide Sign problem a few pedagogical examples, and summarize open challenges and practical solutions Stochastic quantization for the complex case. Finally, we review the recent applications of complex Langevin Complex Langevin to quantum field theory in relativistic and nonrelativistic quantum matter, with an emphasis on the nonrelativistic case. ' 2020 Elsevier B.V. All rights reserved. Contents 1. Introduction...............................................................................................................................................................................................2 1.1. The challenge of many-body quantum mechanics: memory and statistics.........................................................................2 1.2. Path integrals and the sign problem.........................................................................................................................................3 2. Approaches to the sign problem: from reweighting to complex Langevin ......................................................................................6 2.1. Reweighting..................................................................................................................................................................................6 2.2. Alternative Hubbard–Stratonovich transformations................................................................................................................7 2.3. Density-of-states methods..........................................................................................................................................................9 2.4. Dual variables for bosons ........................................................................................................................................................... 10 2.5. Dual variables for fermions and fermion bags ....................................................................................................................... 12 2.6. Majorana fermions....................................................................................................................................................................... 14 2.7. Imaginary asymmetry ................................................................................................................................................................. 16 2.8. Lefschetz thimbles ....................................................................................................................................................................... 18 ∗ Corresponding author at: Institut für Kernphysik (Theoriezentrum), Technische Universität Darmstadt, D-64289 Darmstadt, Germany. E-mail addresses: [email protected] (C.E. Berger), [email protected] (L. Rammelmüller), [email protected] (A.C. Loheac), [email protected] (F. Ehmann), [email protected] (J. Braun), [email protected] (J.E. Drut). 1 Contributed equally to this work. https://doi.org/10.1016/j.physrep.2020.09.002 0370-1573/' 2020 Elsevier B.V. All rights reserved. C.E. Berger, L. Rammelmüller, A.C. Loheac et al. Physics Reports 892 (2021) 1–54 2.9. Path optimization method.......................................................................................................................................................... 20 3. The Langevin method for real and complex variables ........................................................................................................................ 21 3.1. Complex Langevin: origins and modern re-emergence .......................................................................................................... 21 3.2. Stochastic quantization: path integrals and the Langevin process........................................................................................ 22 3.3. A practical guide to real Langevin............................................................................................................................................. 24 3.3.1. Toy problem I: a pedagogical example of real Langevin ........................................................................................ 25 3.4. A practical guide to complex Langevin..................................................................................................................................... 27 3.4.1. Toy problem II: a pedagogical example of complex Langevin ............................................................................... 28 3.5. Formal aspects and justification ................................................................................................................................................ 30 3.5.1. Practical monitoring of the boundary terms ............................................................................................................ 31 3.6. Challenges and some solutions.................................................................................................................................................. 32 3.6.1. Mathematical shortcomings........................................................................................................................................ 32 3.6.2. Instabilities.................................................................................................................................................................... 32 3.6.3. Convergence to incorrect values ................................................................................................................................ 32 3.6.4. Practical solutions ........................................................................................................................................................ 33 3.6.5. Schwinger–Dyson representation............................................................................................................................... 34 4. Applications in relativistic physics: from toy models to finite density QCD.................................................................................... 35 4.1. QCD-inspired toy models............................................................................................................................................................ 35 4.1.1. XY model....................................................................................................................................................................... 35 4.1.2. Polyakov and SU(3) spin models................................................................................................................................ 35 4.1.3. Random matrix theory (RMT)..................................................................................................................................... 36 4.1.4. Thirring model.............................................................................................................................................................. 36 4.1.5. Heavy-dense QCD (HDQCD) and hopping parameter expansion ........................................................................... 36 4.1.6. Low-dimensional QCD at non-zero chemical potential........................................................................................... 37 4.2. 3 C 1 Dimensional QCD at finite chemical potential .............................................................................................................. 37 4.3. Other applications in relativistic field theories........................................................................................................................ 38 4.3.1. Real-time dynamics ..................................................................................................................................................... 38 4.3.2. Theories with topological terms................................................................................................................................
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