Abstract
We investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential. The lattice model is formulated with staggered fermions and a compact auxiliary vector boson (a link field), and the whole set of the critical points (the complex saddle points) are sorted out, where each critical point turns out to be in a one-to-one correspondence with a singular point of the effective action (or a zero point of the fermion determinant). For a subset of critical point solutions in the uniform-field subspace, we examine the upward and downward cycles and the Stokes phenomenon with varying the chemical potential, and we identify the intersection numbers to determine the thimbles contributing to the path-integration of the partition function. We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multiple thimbles must contribute to the path integration. Finally, reducing the model to a uniform field space, we study the relative importance of multi-thimble contributions and their behavior toward continuum and low-temperature limits quantitatively, and see how the rapid crossover behavior is recovered by adding the multi-thimble contributions at low temperatures. Those findings will be useful for performing Monte-Carlo simulations on the Lefschetz thimbles.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. de Forcrand, Simulating QCD at finite density, PoS(LAT2009)010 [arXiv:1005.0539] [INSPIRE].
G. Parisi, On complex probabilities, Phys. Lett. B 131 (1983) 393 [INSPIRE].
J.R. Klauder, A Langevin Approach to Fermion and Quantum Spin Correlation Functions, J. Phys. A 16 (1983) L317 [INSPIRE].
J.R. Klauder, Coherent State Langevin Equations for Canonical Quantum Systems With Applications to the Quantized Hall Effect, Phys. Rev. A 29 (1984) 2036 [INSPIRE].
E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv:1009.6032 [INSPIRE].
F. Pham, Vanishing homologies and the n-variable saddlepoint method, in Proceedings of Symposia in Pure Mathematics. Vol. 40, Part 2: Singularities, AMS Press, Providence U.S.A. (1983).
G. Aarts and I.-O. Stamatescu, Stochastic quantization at finite chemical potential, JHEP 09 (2008) 018 [arXiv:0807.1597] [INSPIRE].
G. Aarts, Can stochastic quantization evade the sign problem? The relativistic Bose gas at finite chemical potential, Phys. Rev. Lett. 102 (2009) 131601 [arXiv:0810.2089] [INSPIRE].
G. Aarts, Complex Langevin dynamics at finite chemical potential: Mean field analysis in the relativistic Bose gas, JHEP 05 (2009) 052 [arXiv:0902.4686] [INSPIRE].
G. Aarts, Can complex Langevin dynamics evade the sign problem?, PoS(LAT2009)024 [arXiv:0910.3772] [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, E. Seiler and I.-O. Stamatescu, The Complex Langevin method: When can it be trusted?, Phys. Rev. D 81 (2010) 054508 [arXiv:0912.3360] [INSPIRE].
G. Aarts and F.A. James, On the convergence of complex Langevin dynamics: The Three-dimensional XY model at finite chemical potential, JHEP 08 (2010) 020 [arXiv:1005.3468] [INSPIRE].
G. Aarts and K. Splittorff, Degenerate distributions in complex Langevin dynamics: one-dimensional QCD at finite chemical potential, JHEP 08 (2010) 017 [arXiv:1006.0332] [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].
G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Complex Langevin dynamics: criteria for correctness, PoS(LATTICE 2011)197 [arXiv:1110.5749] [INSPIRE].
G. Aarts and F.A. James, Complex Langevin dynamics in the SU(3) spin model at nonzero chemical potential revisited, JHEP 01 (2012) 118 [arXiv:1112.4655] [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].
G. Aarts, F.A. James, J.M. Pawlowski, E. Seiler, D. Sexty and I.-O. Stamatescu, Stability of complex Langevin dynamics in effective models, JHEP 03 (2013) 073 [arXiv:1212.5231] [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].
J.M. Pawlowski and C. Zielinski, Thirring model at finite density in 2+1 dimensions with stochastic quantization, Phys. Rev. D 87 (2013) 094509 [arXiv:1302.2249] [INSPIRE].
G. Aarts, Complex Langevin dynamics and other approaches at finite chemical potential, PoS(Lattice 2012)017 [arXiv:1302.3028] [INSPIRE].
G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty and I.-O. Stamatescu, Controlling complex Langevin dynamics at finite density, Eur. Phys. J. A 49 (2013) 89 [arXiv:1303.6425] [INSPIRE].
G. Aarts, P. Giudice and E. Seiler, Localised distributions and criteria for correctness in complex Langevin dynamics, Annals Phys. 337 (2013) 238 [arXiv:1306.3075] [INSPIRE].
D. Sexty, Simulating full QCD at nonzero density using the complex Langevin equation, Phys. Lett. B 729 (2014) 108 [arXiv:1307.7748] [INSPIRE].
G. Aarts, Lefschetz thimbles and stochastic quantization: Complex actions in the complex plane, Phys. Rev. D 88 (2013) 094501 [arXiv:1308.4811] [INSPIRE].
P. Giudice, G. Aarts and E. Seiler, Localised distributions in complex Langevin dynamics, PoS(LATTICE 2013)200 [arXiv:1309.3191] [INSPIRE].
A. Mollgaard and K. Splittorff, Complex Langevin Dynamics for chiral Random Matrix Theory, Phys. Rev. D 88 (2013) 116007 [arXiv:1309.4335] [INSPIRE].
D. Sexty, Extending complex Langevin simulations to full QCD at nonzero density, PoS(LATTICE 2013)199 [arXiv:1310.6186] [INSPIRE].
G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty and I.-O. Stamatescu, Complex Langevin simulation for QCD-like models, PoS(LATTICE 2013)451 [arXiv:1310.7412] [INSPIRE].
L. Bongiovanni, G. Aarts, E. Seiler, D. Sexty and I.-O. Stamatescu, Adaptive gauge cooling for complex Langevin dynamics, PoS(LATTICE 2013)449 [arXiv:1311.1056] [INSPIRE].
G. Aarts, L. Bongiovanni, E. Seiler and D. Sexty, Some remarks on Lefschetz thimbles and complex Langevin dynamics, JHEP 10 (2014) 159 [arXiv:1407.2090] [INSPIRE].
G. Aarts, E. Seiler, D. Sexty and I.-O. Stamatescu, Simulating QCD at nonzero baryon density to all orders in the hopping parameter expansion, Phys. Rev. D 90 (2014) 114505 [arXiv:1408.3770] [INSPIRE].
D. Sexty, Progress in complex Langevin simulations of full QCD at non-zero density, Nucl. Phys. A 931 (2014) 856 [arXiv:1408.6767] [INSPIRE].
D. Sexty, New algorithms for finite density QCD, PoS(LATTICE2014)016 [arXiv:1410.8813] [INSPIRE].
L. Bongiovanni, G. Aarts, E. Seiler and D. Sexty, Complex Langevin dynamics for SU(3) gauge theory in the presence of a theta term, PoS(LATTICE2014)199 [arXiv:1411.0949] [INSPIRE].
G. Aarts, F. Attanasio, B. Jäger, E. Seiler, D. Sexty and I.-O. Stamatescu, Exploring the phase diagram of QCD with complex Langevin simulations, PoS(LATTICE2014)200 [arXiv:1411.2632] [INSPIRE].
G. Aarts, B. Jäger, E. Seiler, D. Sexty and I.-O. Stamatescu, Systematic approximation for QCD at non-zero density, PoS(LATTICE2014)207 [arXiv:1412.5775] [INSPIRE].
G. Aarts, F. Attanasio, B. Jäger, E. Seiler, D. Sexty and I.-O. Stamatescu, QCD at nonzero chemical potential: recent progress on the lattice, arXiv:1412.0847 [INSPIRE].
A. Mollgaard and K. Splittorff, Full simulation of chiral random matrix theory at nonzero chemical potential by complex Langevin, Phys. Rev. D 91 (2015) 036007 [arXiv:1412.2729] [INSPIRE].
H. Makino, H. Suzuki and D. Takeda, Complex Langevin method applied to the 2D SU(2) Yang-Mills theory, Phys. Rev. D 92 (2015) 085020 [arXiv:1503.00417] [INSPIRE].
G. Aarts, E. Seiler, D. Sexty and I.O. Stamatescu, Hopping parameter expansion to all orders using the complex Langevin equation, arXiv:1503.08813 [INSPIRE].
J. Nishimura and S. Shimasaki, New insights into the problem with a singular drift term in the complex Langevin method, Phys. Rev. D 92 (2015) 011501 [arXiv:1504.08359] [INSPIRE].
G. Aarts, F. Attanasio, B. Jäger, E. Seiler, D. Sexty and I.-O. Stamatescu, The phase diagram of heavy dense QCD with complex Langevin simulations, Acta Phys. Polon. Supp. 8 (2015) 405 [arXiv:1506.02547] [INSPIRE].
K. Nagata, J. Nishimura and S. Shimasaki, Justification of the complex Langevin method with the gauge cooling procedure, arXiv:1508.02377 [INSPIRE].
Z. Fodor, S.D. Katz, D. Sexty and C. Török, Complex Langevin dynamics for dynamical QCD at nonzero chemical potential: a comparison with multi-parameter reweighting, arXiv:1508.05260 [INSPIRE].
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].
T.D. . Cohen, Functional integrals for QCD at nonzero chemical potential and zero density, Phys. Rev. Lett. 91 (2003) 222001 [hep-ph/0307089] [INSPIRE].
A. Mukherjee and M. Cristoforetti, Lefschetz thimble Monte Carlo for many-body theories: A Hubbard model study, Phys. Rev. B 90 (2014) 035134 [arXiv:1403.5680] [INSPIRE].
F. Di Renzo and G. Eruzzi, Thimble regularization at work: from toy models to chiral random matrix theories, Phys. Rev. D 92 (2015) 085030 [arXiv:1507.03858] [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].
T. Kanazawa and Y. Tanizaki, Structure of Lefschetz thimbles in simple fermionic systems, JHEP 03 (2015) 044 [arXiv:1412.2802] [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].
Y. Tanizaki and T. Koike, Real-time Feynman path integral with PicardLefschetz theory and its applications to quantum tunneling, Annals Phys. 351 (2014) 250 [arXiv:1406.2386] [INSPIRE].
Y. Tanizaki, H. Nishimura and K. Kashiwa, Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral, Phys. Rev. D 91 (2015) 101701 [arXiv:1504.02979] [INSPIRE].
A. Cherman, D. Dorigoni and M. Ünsal, Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles, JHEP 10 (2015) 056 [arXiv:1403.1277] [INSPIRE].
A. Behtash, T. Sulejmanpasic, T. Schäfer and M. Ünsal, Hidden topological angles and Lefschetz thimbles, Phys. Rev. Lett. 115 (2015) 041601 [arXiv:1502.06624] [INSPIRE].
K. Fukushima and Y. Tanizaki, Hamilton dynamics for the Lefschetz thimble integration akin to the complex Langevin method, arXiv:1507.07351 [INSPIRE].
S. Tsutsui and T.M. Doi, An improvement in complex Langevin dynamics from a view point of Lefschetz thimbles, arXiv:1508.04231 [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].
J.B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D 11 (1975) 395 [INSPIRE].
P. Hasenfratz and F. Karsch, Chemical Potential on the Lattice, Phys. Lett. B 125 (1983) 308 [INSPIRE].
H. Fujii, S. Kamata and Y. Kikukawa, Monte Carlo study of Lefschetz thimble structure in one-dimensional Thirring model at finite density, arXiv:1509.09141 [INSPIRE].
L.G. Molinari, Determinants of block tridiagonal matrices, Lin. Algebra Appl. 429 (2008) 2221.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1509.08176
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Fujii, H., Kamata, S. & Kikukawa, Y. Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density. J. High Energ. Phys. 2015, 78 (2015). https://doi.org/10.1007/JHEP11(2015)078
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)078