Abstract
We point out that Monte Carlo simulations of theories with severe sign problems can be profitably performed over manifolds in complex space different from the one with fixed imaginary part of the action (“Lefschetz thimble”). We describe a family of such manifolds that interpolate between the tangent space at one critical point (where the sign problem is milder compared to the real plane but in some cases still severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal distribution function complicates the Monte Carlo sampling). We exemplify this approach using a simple 0+1 dimensional fermion model previously used on sign problem studies and show that it can solve the model for some parameter values where a solution using Lefschetz thimbles was elusive.
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References
AuroraScience collaboration, M. Cristoforetti, F. Di Renzo and L. Scorzato, 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].
M. Cristoforetti, F. Di Renzo, A. Mukherjee and L. Scorzato, Monte Carlo simulations on the Lefschetz thimble: taming the sign problem, Phys. Rev. D 88 (2013) 051501 [arXiv:1303.7204] [INSPIRE].
H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu and T. Sano, Hybrid Monte Carlo on Lefschetz thimbles — a study of the residual sign problem, JHEP 10 (2013) 147 [arXiv:1309.4371] [INSPIRE].
A. Mukherjee, M. Cristoforetti and L. Scorzato, Metropolis Monte Carlo integration on the Lefschetz thimble: application to a one-plaquette model, Phys. Rev. D 88 (2013) 051502 [arXiv:1308.0233] [INSPIRE].
M. Cristoforetti, F. Di Renzo, A. Mukherjee and L. Scorzato, Quantum field theories on the Lefschetz thimble, PoS(LATTICE 2013)197 [arXiv:1312.1052] [INSPIRE].
M. Cristoforetti et al., An efficient method to compute the residual phase on a Lefschetz thimble, Phys. Rev. D 89 (2014) 114505 [arXiv:1403.5637] [INSPIRE].
K. Fukushima and Y. Tanizaki, Hamilton dynamics for Lefschetz-thimble integration akin to the complex Langevin method, arXiv:1507.07351 [INSPIRE].
Y. Tanizaki, Lefschetz-thimble techniques for path integral of zero-dimensional O(n) σ-models, Phys. Rev. D 91 (2015) 036002 [arXiv:1412.1891] [INSPIRE].
Y. Tanizaki, Y. Hidaka and T. Hayata, Lefschetz-thimble analysis of the sign problem in one-site fermion model, New J. Phys. 18 (2016) 033002 [arXiv:1509.07146] [INSPIRE].
T. Kanazawa and Y. Tanizaki, Structure of Lefschetz thimbles in simple fermionic systems, JHEP 03 (2015) 044 [arXiv:1412.2802] [INSPIRE].
H. Fujii, S. Kamata and Y. Kikukawa, Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density, JHEP 11 (2015) 078 [Erratum ibid. 02 (2016) 036] [arXiv:1509.08176] [INSPIRE].
H. Fujii, S. Kamata and Y. Kikukawa, Monte Carlo study of Lefschetz thimble structure in one-dimensional Thirring model at finite density, JHEP 12 (2015) 125 [arXiv:1509.09141] [INSPIRE].
L. Scorzato, The Lefschetz thimble and the sign problem, in Proceedings, 33rd International Symposium on Lattice Field Theory (Lattice 2015), (2015) [arXiv:1512.08039] [INSPIRE].
A. Alexandru, G. Basar and P. Bedaque, Monte Carlo algorithm for simulating fermions on Lefschetz thimbles, Phys. Rev. D 93 (2016) 014504 [arXiv:1510.03258] [INSPIRE].
J.M. Pawlowski and C. Zielinski, Thirring model at finite density in 0 + 1 dimensions with stochastic quantization: crosscheck with an exact solution, Phys. Rev. D 87 (2013) 094503 [arXiv:1302.1622] [INSPIRE].
G. Aarts and I.-O. Stamatescu, Stochastic quantization at finite chemical potential, JHEP 09 (2008) 018 [arXiv:0807.1597] [INSPIRE].
M.V. Fedoryuk, The saddle-point method (in Russian), Izdat. Nauka, Moscow Russia (1977).
F. Pham, Vanishing homologies and the N variable saddlepoint method, Proc. Sympos. Pure Math. 40 (1983) 319.
E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
J.M. Pawlowski, I.-O. Stamatescu and C. Zielinski, Simple QED- and QCD-like models at finite density, Phys. Rev. D 92 (2015) 014508 [arXiv:1402.6042] [INSPIRE].
B. Shabat, Introduction to complex analysis: functions of several variables, American Mathematical Society, U.S.A. (1992).
R. Neal, Sampling from multimodal distributions using tempered transitions, Statist. Comput. 6 (1996) 353.
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ArXiv ePrint: 1512.08764v1
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Alexandru, A., Basar, G., Bedaque, P.F. et al. Sign problem and Monte Carlo calculations beyond Lefschetz thimbles. J. High Energ. Phys. 2016, 53 (2016). https://doi.org/10.1007/JHEP05(2016)053
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DOI: https://doi.org/10.1007/JHEP05(2016)053