Abstract
We present a novel strategy aimed at restoring correct convergence in complex Langevin simulations. The central idea is to incorporate system-specific prior knowledge into the simulations, in order to circumvent the NP-hard sign problem. In order to do so, we modify complex Langevin using kernels and propose the use of modern auto-differentiation methods to learn optimal kernel values. The optimization process is guided by functionals encoding relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. For the strongly coupled quantum anharmonic oscillator we achieve correct convergence up to three-times the real-time extent of the previous benchmark study. An appendix sheds light on the fact that for correct convergence not only the absence of boundary terms, but in addition the correct Fokker-Plank spectrum is crucial.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. Busza, K. Rajagopal and W. van der Schee, Heavy Ion Collisions: The Big Picture, and the Big Questions, Ann. Rev. Nucl. Part. Sci. 68 (2018) 339 [arXiv:1802.04801] [INSPIRE].
P. Foka and M.A. Janik, An overview of experimental results from ultra-relativistic heavy-ion collisions at the CERN LHC: Bulk properties and dynamical evolution, Rev. Phys. 1 (2016) 154 [arXiv:1702.07233] [INSPIRE].
C.-C. Chien, S. Peotta and M.D. Ventra, Quantum transport in ultracold atoms, Nature Phys. 11 (2015) 998.
M. Qin et al., The Hubbard model: A computational perspective, Ann. Rev. Condens. Mat. Phys. 13 (2022) 275 [arXiv:2104.00064] [INSPIRE].
C. Gattringer and K. Langfeld, Approaches to the sign problem in lattice field theory, Int. J. Mod. Phys. A 31 (2016) 1643007 [arXiv:1603.09517] [INSPIRE].
G. Pan and Z.Y. Meng, Sign Problem in Quantum Monte Carlo Simulation, arXiv:2204.08777 [INSPIRE].
A. Rothkopf, Bayesian inference of real-time dynamics from lattice QCD, Front. Phys. 10 (2022) 1. [arXiv:2208.13590] [INSPIRE].
M. Troyer and U.-J. Wiese, Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations, Phys. Rev. Lett. 94 (2005) 170201 [cond-mat/0408370] [INSPIRE].
C.E. Berger et al., Complex Langevin and other approaches to the sign problem in quantum many-body physics, Phys. Rept. 892 (2021) 1 [arXiv:1907.10183] [INSPIRE].
S. Chandrasekharan and U.-J. Wiese, Meron cluster solution of a fermion sign problem, Phys. Rev. Lett. 83 (1999) 3116 [cond-mat/9902128] [INSPIRE].
Y. Delgado Mercado, H.G. Evertz and C. Gattringer, The QCD phase diagram according to the center group, Phys. Rev. Lett. 106 (2011) 222001 [arXiv:1102.3096] [INSPIRE].
T. Kloiber and C. Gattringer, Dual Methods for Lattice Field Theories at Finite Density, PoS LATTICE2013 (2014) 206 [arXiv:1310.8535] [INSPIRE].
P. de Forcrand and O. Philipsen, Constraining the QCD phase diagram by tricritical lines at imaginary chemical potential, Phys. Rev. Lett. 105 (2010) 152001 [arXiv:1004.3144] [INSPIRE].
J. Braun et al., Imaginary polarization as a way to surmount the sign problem in Ab Initio calculations of spin-imbalanced Fermi gases, Phys. Rev. Lett. 110 (2013) 130404 [arXiv:1209.3319] [INSPIRE].
J. Braun, J.E. Drut and D. Roscher, Zero-temperature equation of state of mass-imbalanced resonant Fermi gases, Phys. Rev. Lett. 114 (2015) 050404 [arXiv:1407.2924] [INSPIRE].
J.N. Guenther et al., The QCD equation of state at finite density from analytical continuation, Nucl. Phys. A 967 (2017) 720 [arXiv:1607.02493] [INSPIRE].
F. Wang and D.P. Landau, Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States, Phys. Rev. Lett. 86 (2001) 2050 [cond-mat/0011174] [INSPIRE].
K. Langfeld, B. Lucini and A. Rago, The density of states in gauge theories, Phys. Rev. Lett. 109 (2012) 111601 [arXiv:1204.3243] [INSPIRE].
C. Gattringer and P. Törek, Density of states method for the ℤ3 spin model, Phys. Lett. B 747 (2015) 545.
R. Orús, Tensor networks for complex quantum systems, APS Physics 1 (2019) 538 [arXiv:1812.04011] [INSPIRE].
J. Hauschild and F. Pollmann, Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes 5 (2018) 1 [arXiv:1805.00055].
N. Rom, D.M. Charutz and D. Neuhauser, Shifted-contour auxiliary-field Monte Carlo: circumventing the sign difficulty for electronic-structure calculations, Chem. Phys. Lett. 270 (1997) 382.
AuroraScience collaboration, New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble, Phys. Rev. D 86 (2012) 074506 [arXiv:1205.3996] [INSPIRE].
A. Alexandru, G. Basar, P.F. Bedaque and N.C. Warrington, Complex paths around the sign problem, Rev. Mod. Phys. 94 (2022) 015006 [arXiv:2007.05436] [INSPIRE].
P.H. Damgaard and H. Hüffel, Stochastic Quantization, Phys. Rept. 152 (1987) 227 [INSPIRE].
M. Namiki et al., Stochastic quantization, Lect. Notes Phys. Monogr. 9 (1992) 1 [INSPIRE].
E. Seiler, Status of Complex Langevin, Eur. Phys. J. Web Conf. 175 (2018) 01019 [arXiv:1708.08254] [INSPIRE].
D. Alvestad, R. Larsen and A. Rothkopf, Stable solvers for real-time Complex Langevin, JHEP 08 (2021) 138 [arXiv:2105.02735] [INSPIRE].
G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Adaptive stepsize and instabilities in complex Langevin dynamics, Phys. Lett. B 687 (2010) 154 [arXiv:0912.0617] [INSPIRE].
G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Complex Langevin: Etiology and Diagnostics of its Main Problem, Eur. Phys. J. C 71 (2011) 1756 [arXiv:1101.3270] [INSPIRE].
K. Nagata, J. Nishimura and S. Shimasaki, Argument for justification of the complex Langevin method and the condition for correct convergence, Phys. Rev. D 94 (2016) 114515 [arXiv:1606.07627] [INSPIRE].
D. Sexty, E. Seiler, I.-O. Stamatescu and M.W. Hansen, Complex Langevin boundary terms in lattice models, PoS LATTICE2021 (2022) 194 [arXiv:2112.02924] [INSPIRE].
E. Seiler, D. Sexty and I.-O. Stamatescu, Gauge cooling in complex Langevin for QCD with heavy quarks, Phys. Lett. B 723 (2013) 213 [arXiv:1211.3709] [INSPIRE].
J. Berges and D. Sexty, Real-time gauge theory simulations from stochastic quantization with optimized updating, Nucl. Phys. B 799 (2008) 306 [arXiv:0708.0779] [INSPIRE].
F. Attanasio and B. Jäger, Dynamical stabilisation of complex Langevin simulations of QCD, Eur. Phys. J. C 79 (2019) 16 [arXiv:1808.04400] [INSPIRE].
G. Aarts, F. Attanasio, B. Jäger and D. Sexty, Complex Langevin in Lattice QCD: dynamic stabilisation and the phase diagram, Acta Phys. Polon. Supp. 9 (2016) 621 [arXiv:1607.05642] [INSPIRE].
G. Aarts et al., Open charm mesons at nonzero temperature: results in the hadronic phase from lattice QCD, arXiv:2209.14681 [INSPIRE].
P. Hotzy, K. Boguslavski and D.I. Müller, A stabilizing kernel for complex Langevin simulations of real-time gauge theories, PoS LATTICE2022 (2023) 279 [arXiv:2210.08020] [INSPIRE].
A. Alexandru et al., Monte Carlo Study of Real Time Dynamics on the Lattice, Phys. Rev. Lett. 117 (2016) 081602 [arXiv:1605.08040] [INSPIRE].
A. Alexandru, G. Basar, P.F. Bedaque and G.W. Ridgway, Schwinger-Keldysh formalism on the lattice: A faster algorithm and its application to field theory, Phys. Rev. D 95 (2017) 114501 [arXiv:1704.06404] [INSPIRE].
J. Berges, S. Borsányi, D. Sexty and I.-O. Stamatescu, Lattice simulations of real-time quantum fields, Phys. Rev. D 75 (2007) 045007 [hep-lat/0609058] [INSPIRE].
G. Aarts et al., Stability of complex Langevin dynamics in effective models, JHEP 03 (2013) 073 [arXiv:1212.5231] [INSPIRE].
H. Okamoto, K. Okano, L. Schulke and S. Tanaka, The Role of a Kernel in Complex Langevin Systems, Nucl. Phys. B 324 (1989) 684 [INSPIRE].
A. Gunes Baydin, B.A. Pearlmutter, A. Andreyevich Radul and J.M. Siskind, Automatic differentiation in machine learning: a survey, Journal of Marchine Learning Research 18 (2018) 1 [arXiv:1502.05767] [https://doi.org/10.48550/arXiv.1502.05767].
D. Alvestad, alvestad10/KernelCL: Towards learning optimized kernels for complex Langevin, (2022) [https://doi.org/10.5281/zenodo.7373498].
J.R. Klauder and W.P. Petersen, Numerical Integration of Multiplicative Noise Stochastic Differential Equations, SIAM J. Num. Anal. 22 (1985) 1153 [INSPIRE].
P. Giudice, G. Aarts and E. Seiler, Localised distributions in complex Langevin dynamics, PoS LATTICE2013 (2014) 200 [arXiv:1309.3191] [INSPIRE].
Y. Abe and K. Fukushima, Analytic studies of the complex Langevin equation with a Gaussian ansatz and multiple solutions in the unstable region, Phys. Rev. D 94 (2016) 094506 [arXiv:1607.05436] [INSPIRE].
E. Seiler and J. Wosiek, Positive Representations of a Class of Complex Measures, J. Phys. A 50 (2017) 495403 [arXiv:1702.06012] [INSPIRE].
L.L. Salcedo, Positive representations of complex distributions on groups, J. Phys. A 51 (2018) 505401 [arXiv:1805.01698] [INSPIRE].
S. Woodward, P.M. Saffin, Z.-G. Mou and A. Tranberg, Optimisation of Thimble simulations and quantum dynamics of multiple fields in real time, JHEP 10 (2022) 082 [arXiv:2204.10101] [INSPIRE].
H. Nakazato and Y. Yamanaka, Minkowski Stochastic Quantization, in 23rd International Conference on High-Energy Physics, Berkeley U.S.A., July 16–23 1986 [Phys. Rev. D 34 (1986) 492].
H. Hüffel and P.V. Landshoff, Stochastic Diagrams and Feynman Diagrams, Nucl. Phys. B 260 (1985) 545 [INSPIRE].
N. Matsumoto, Comment on the subtlety of defining a real-time path integral in lattice gauge theories, PTEP 2022 (2022) 093B03 [arXiv:2206.00865] [INSPIRE].
M. Scherzer, E. Seiler, D. Sexty and I.-O. Stamatescu, Complex Langevin and boundary terms, Phys. Rev. D 99 (2019) 014512 [arXiv:1808.05187] [INSPIRE].
D. Harrison, A Brief Introduction to Automatic Differentiation for Machine Learning, arXiv:2110.06209 [https://doi.org/10.48550/arXiv.2110.06209].
M. Innes, Don’t Unroll Adjoint: Differentiating SSA-Form Programs, CoRR abs/1810.07951 (2018) [arXiv:1810.07951].
Y. Cao, S. Li, L. Petzold and R. Serban, Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution, SIAM J. Sci. Comput. 24 (2003) 1076.
F. Schäfer, M. Kloc, C. Bruder and N. Lörch, A differentiable programming method for quantum control, Mach. Learn. Sci. Tech. 1 (2020) 035009 [arXiv:2002.08376].
C. Rackauckas et al., Universal Differential Equations for Scientific Machine Learning, arXiv:2001.04385.
Q. Wang, R. Hu and P. Blonigan, Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations, J. Comput. Phys. 267 (2014) 210 [arXiv:1204.0159].
A. Ni and Q. Wang, Sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Shadowing (NILSS), J. Comput. Phys. 347 (2017) 56.
K. Okano, L. Schulke and B. Zheng, Kernel controlled complex Langevin simulation: Field dependent kernel, Phys. Lett. B 258 (1991) 421 [INSPIRE].
G. Aarts, Lefschetz thimbles and stochastic quantization: Complex actions in the complex plane, Phys. Rev. D 88 (2013) 094501 [arXiv:1308.4811] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2211.15625
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Alvestad, D., Larsen, R. & Rothkopf, A. Towards learning optimized kernels for complex Langevin. J. High Energ. Phys. 2023, 57 (2023). https://doi.org/10.1007/JHEP04(2023)057
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2023)057