Argument for justification of the complex Langevin method and the condition for correct convergence - INSPIRE

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Argument for justification of the complex Langevin method and the condition for correct convergence

Keitaro Nagata
(
- KEK, Tsukuba
)
,
Jun Nishimura
(
- KEK, Tsukuba and
- Sokendai, Tsukuba
)
,
Shinji Shimasaki
(
- Keio U. and
- KEK, Tsukuba
)

Jun 24, 2016

19 pages

Published in:

Phys.Rev.D 94 (2016) 11, 114515

Published: Dec 27, 2016

e-Print:

1606.07627 [hep-lat]

DOI:

10.1103/PhysRevD.94.114515

Report number:

KEK-TH-1911

View in:

ADS Abstract Service

reference search148 citations

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Abstract: (APS)

The complex Langevin method is a promising approach to the complex-action problem based on a fictitious time evolution of complexified dynamical variables under the influence of a Gaussian noise. Although it is known to have a restricted range of applicability, the use of gauge cooling made it applicable to various interesting cases including finite density QCD in certain parameter regions. In this paper we revisit the argument for justification of the method. In particular, we point out a subtlety in the use of time-evolved observables, which play a crucial role in the previous argument. This requires that the probability of the drift term should fall off exponentially or faster at large magnitude. We argue that this is actually a necessary and sufficient condition for the method to be justified. Using two simple examples, we show that our condition tells us clearly whether the results obtained by the method are trustable or not. We also discuss a new possibility for the gauge cooling, which can reduce the magnitude of the drift term directly.

Note:

34 pages, 8 figures. The version accepted for publication in PRD

quantum chromodynamics: density
density: finite
action: complex
Langevin equation: complex
continuum limit
suppression
toy model
dimension: 0
gauge field theory: SU(3)
lattice field theory

References(35)

Figures(16)

[1]

ON COMPLEX PROBABILITIES

G. Parisi
(
- Rome U.
)

- Phys.Lett.B 131 (1983) 393-395
•
DOI:
- 10.1016/0370-2693(83)90525-7

[2]

Coherent State Langevin Equations for Canonical Quantum Systems With Applications to the Quantized Hall Effect

John R. Klauder
(
- Bell Labs
)

- Phys.Rev.A 29 (1984) 2036-2047
•
DOI:
- 10.1103/PhysRevA.29.2036

[3]

Simulating QCD at nonzero baryon density to all orders in the hopping parameter expansion

Gert Aarts
(
- Swansea U.
)
,
Erhard Seiler
(
- Munich, Max Planck Inst.
)
,
Dénes Sexty
(
- Heidelberg U.
)
,
Ion-Olimpiu Stamatescu
(
- Heidelberg U.
)

- Phys.Rev.D 90 (2014) 11, 114505
•
e-Print:
- 1408.3770
•
DOI:
- 10.1103/PhysRevD.90.114505

[4]

Simulating full QCD at nonzero density using the complex Langevin equation

Dénes Sexty
(
- U. Heidelberg, ITP
)

- Phys.Lett.B 729 (2014) 108-111
•
e-Print:
- 1307.7748
•
DOI:
- 10.1016/j.physletb.2014.01.019

[5]

Gauge cooling in complex Langevin for QCD with heavy quarks

Erhard Seiler
(
- Munich, Max Planck Inst.
)
,
Denes Sexty
(
- Heidelberg U.
)
,
Ion-Olimpiu Stamatescu
(
- Heidelberg U.
)

- Phys.Lett.B 723 (2013) 213-216
•
e-Print:
- 1211.3709
•
DOI:
- 10.1016/j.physletb.2013.04.062

[6]

Complex Langevin Dynamics for chiral Random Matrix Theory

A. Mollgaard
(
- Bohr Inst.
)
,
K. Splittorff
(
- Bohr Inst.
)

- Phys.Rev.D 88 (2013) 11, 116007
•
e-Print:
- 1309.4335
•
DOI:
- 10.1103/PhysRevD.88.116007

[7]

Full simulation of chiral random matrix theory at nonzero chemical potential by complex Langevin

A. Mollgaard
(
- Bohr Inst.
)
,
K. Splittorff
(
- Bohr Inst.
)

- Phys.Rev.D 91 (2015) 3, 036007
•
e-Print:
- 1412.2729
•
DOI:
- 10.1103/PhysRevD.91.036007

[8]

Complex Langevin dynamics for dynamical QCD at nonzero chemical potential: A comparison with multiparameter reweighting

Z. Fodor
(
- Wuppertal U. and
- IAS, Julich and
- Eotvos U.
)
,
S.D. Katz
(
- Eotvos U. and
- Res. Group Stat. Biol., Budapest
)
,
D. Sexty
(
- Wuppertal U.
)
,
C. Török
(
- Eotvos U. and
- Res. Group Stat. Biol., Budapest
)

- Phys.Rev.D 92 (2015) 9, 094516
•
e-Print:
- 1508.05260
•
DOI:
- 10.1103/PhysRevD.92.094516

[9]

Exploring Complex-Langevin Methods for Finite-Density QCD

Donald K. Sinclair
(
- Argonne, PHY
)
,
J.B. Kogut
(
- Dept. of Energy, Wash., D.C. and
- Maryland U.
)

- PoS LATTICE2015 (2016) 153
•
e-Print:
- 1510.06367
•
DOI:
- 10.22323/1.251.0153

[10]

Gauge cooling for the singular-drift problem in the complex Langevin method - a test in Random Matrix Theory for finite density QCD

Keitaro Nagata
(
- KEK, Tsukuba
)
,
Jun Nishimura
(
- KEK, Tsukuba and
- Tsukuba, Graduate U. Adv. Studies
)
,
Shinji Shimasaki
(
- KEK, Tsukuba and
- Keio U.
)

- JHEP 07 (2016) 073
•
e-Print:
- 1604.07717
•
DOI:
- 10.1007/JHEP07(2016)073

[11]

Perturbation Theory Without Gauge Fixing

G. Parisi
(
- Beijing, Inst. Theor. Phys. and
- Frascati
)
,
Yong-shi Wu
(
- Beijing, Inst. Theor. Phys.
)

- Sci.Sin. 24 (1981) 483,
- Sci. Sin. 24 ( 1981) 483-496

[12]

Stochastic Quantization

Poul H. Damgaard
(
- CERN
)
,
Helmuth Huffel
(
- CERN
)

- Phys.Rept. 152 (1987) 227
•
DOI:
- 10.1016/0370-1573(87)90144-X

[13]

The Complex Langevin method: When can it be trusted?

Gert Aarts
(
- Swansea U.
)
,
Erhard Seiler
(
- Munich, Max Planck Inst.
)
,
Ion-Olimpiu Stamatescu
(
- Heidelberg U.
)

- Phys.Rev.D 81 (2010) 054508
•
e-Print:
- 0912.3360
•
DOI:
- 10.1103/PhysRevD.81.054508

[14]

Complex Langevin: Etiology and Diagnostics of its Main Problem

Gert Aarts
(
- Swansea U.
)
,
Frank A. James
(
- Swansea U.
)
,
Erhard Seiler
(
- Munich, Max Planck Inst.
)
,
Ion-Olimpiu Stamatescu
(
- Heidelberg U. and
- Heidelberg, FEST
)

- Eur.Phys.J.C 71 (2011) 1756
•
e-Print:
- 1101.3270
•
DOI:
- 10.1140/epjc/s10052-011-1756-5

[15]

New Insights into the Problem with a Singular Drift Term in the Complex Langevin Method

Jun Nishimura
(
- KEK, Tsukuba and
- Tsukuba, Graduate U. Adv. Studies
)
,
Shinji Shimasaki
(
- KEK, Tsukuba
)

- Phys.Rev.D 92 (2015) 1, 011501
•
e-Print:
- 1504.08359
•
DOI:
- 10.1103/PhysRevD.92.011501

[16]

Controlling complex Langevin dynamics at finite density

Gert Aarts
(
- Swansea U.
)
,
Lorenzo Bongiovanni
(
- Swansea U.
)
,
Erhard Seiler
(
- Munich, Max Planck Inst.
)
,
Denes Sexty
(
- Heidelberg U.
)
,
Ion-Olimpiu Stamatescu
(
- Heidelberg U. and
- Heidelberg, FEST
)

- Eur.Phys.J.A 49 (2013) 89
•
e-Print:
- 1303.6425
•
DOI:
- 10.1140/epja/i2013-13089-4

[17]

The QCD phase diagram in the limit of heavy quarks using complex Langevin dynamics

Gert Aarts
(
- Swansea U.
)
,
Felipe Attanasio
(
- Swansea U.
)
,
Benjamin Jäger
(
- Swansea U.
)
,
Dénes Sexty
(
- Eotvos Lorand U., Budapest, Inst. Theor. Phys. and
- Bergische U., Wuppertal (main)
)

- JHEP 09 (2016) 087
•
e-Print:
- 1606.05561
•
DOI:
- 10.1007/JHEP09(2016)087

[18]

Justification of the complex Langevin method with the gauge cooling procedure

Keitaro Nagata
(
- KEK, Tsukuba
)
,
Jun Nishimura
(
- KEK, Tsukuba and
- Sokendai, Tsukuba
)
,
Shinji Shimasaki
(
- KEK, Tsukuba
)

- PTEP 2016 (2016) 1, 013B01
•
e-Print:
- 1508.02377
•
DOI:
- 10.1093/ptep/ptv173

[19]

Localised distributions and criteria for correctness in complex Langevin dynamics

Gert Aarts
(
- Swansea U.
)
,
Pietro Giudice
(
- Swansea U.
)
,
Erhard Seiler
(
- Munich, Max Planck Inst.
)

- Annals Phys. 337 (2013) 238-260
•
e-Print:
- 1306.3075
•
DOI:
- 10.1016/j.aop.2013.06.019

[20]

Comparison of complex Langevin and mean field methods applied to effective Polyakov line models

Jeff Greensite
(
- San Francisco State U.
)

- Phys.Rev.D 90 (2014) 11, 114507
•
e-Print:
- 1406.4558
•
DOI:
- 10.1103/PhysRevD.90.114507

[21]

Analytic Continuation Of Chern-Simons Theory

Edward Witten
(
- Princeton, Inst. Advanced Study
)

- AMS/IP Stud.Adv.Math. 50 (2011) 347-446
•
e-Print:
- 1001.2933

[22]

New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble

AuroraScience

Collaboration

•

Marco Cristoforetti
(
- ECT, Trento
)

et al.

- Phys.Rev.D 86 (2012) 074506
•
e-Print:
- 1205.3996
•
DOI:
- 10.1103/PhysRevD.86.074506

[23]

Monte Carlo simulations on the Lefschetz thimble: Taming the sign problem

Marco Cristoforetti
(
- Trento U. and
- ECT, Trento
)
,
Francesco Di Renzo
(
- INFN, Parma and
- Parma U.
)
,
Abhishek Mukherjee
(
- Trento U. and
- ECT, Trento
)
,
Luigi Scorzato
(
- Trento U. and
- ECT, Trento
)

- Phys.Rev.D 88 (2013) 5, 051501
•
e-Print:
- 1303.7204
•
DOI:
- 10.1103/PhysRevD.88.051501

[24]

Hybrid Monte Carlo on Lefschetz thimbles - A study of the residual sign problem

et al.

- JHEP 10 (2013) 147
•
e-Print:
- 1309.4371
•
DOI:
- 10.1007/JHEP10(2013)147

[25]

Lefschetz thimble Monte Carlo for many-body theories: A Hubbard model study

Abhishek Mukherjee
(
- ECT, Trento and
- Fond. Bruno Kessler, Povo
)
,
Marco Cristoforetti
(
- ECT, Trento and
- Fond. Bruno Kessler, Povo
)

- Phys.Rev.B 90 (2014) 3, 035134
•
e-Print:
- 1403.5680
•
DOI:
- 10.1103/PhysRevB.90.035134

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