Complex Langevin: Etiology and Diagnostics of its Main Problem

The complex Langevin method is a leading candidate for solving the so-called sign problem occurring in various physical situations. Its most vexing problem is that in some cases it produces `convergence to the wrong limit'. In the first part of the paper we go through the formal justification of the method, identify points at which it may fail and identify a necessary and sufficient criterion for correctness. This criterion would, however, require checking infinitely many identities, and therefore is somewhat academic. We propose instead a truncation to the check of a few identities; this still gives a necessary criterion, but a priori it is not clear whether it remains sufficient. In the second part we carry out a detailed study of two toy models: first we identify the reasons why in some cases the method fails, second we test the efficiency of the truncated criterion and find that it works perfectly at least in the toy models studied.

Note:

39 pages, 15 figures; typos corrected and reference added

Langevin equation: complex
toy model
numerical calculations
measure: complex
gauge field theory: U(1)
noise
Fokker-Planck equation
Dyson-Schwinger equation

References(24)

Figures(0)

Simulating QCD at finite density

Philippe de Forcrand
(
- Zurich, ETH and
- CERN
)

- PoS LAT2009 (2009) 010
•
e-Print:
- 1005.0539
•
DOI:
- 10.22323/1.091.0010

A Langevin Approach to Fermion and Quantum Spin Correlation Functions

John R. Klauder
(
- Bell Labs
)

- J.Phys.A 16 (1983) L317
•
DOI:
- 10.1088/0305-4470/16/10/001

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

ON COMPLEX PROBABILITIES

G. Parisi
(
- Rome U.
)

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

Complex Langevin Simulation of the SU(3) Spin Model With Nonzero Chemical Potential

F. Karsch
(
- Illinois U., Urbana
)
,
H.W. Wyld
(
- Illinois U., Urbana
)

- Phys.Rev.Lett. 55 (1985) 2242,
- IN *DAMGAARD, P.H. (ED.), HUEFFEL, H. (ED.): STOCHASTIC QUANTIZATION*, 413-416. (PHYS. REV. LETT. 55 (1985) 2242-2245) AND ILLINOIS UNIV. URBANA - ILL-(TH)-85-60 (85,REC.SEP.) 18 P. (SEE BOOK INDEX)
•
DOI:
- 10.1103/PhysRevLett.55.2242

Stochastic Quantization

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

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

Simulating nonequilibrium quantum fields with stochastic quantization techniques

J. Berges
(
- Heidelberg U.
)
,
I.-O. Stamatescu
(
- Heidelberg U. and
- Heidelberg, FEST
)

- Phys.Rev.Lett. 95 (2005) 202003
•
e-Print:
- hep-lat/0508030
•
DOI:
- 10.1103/PhysRevLett.95.202003

Lattice simulations of real-time quantum fields

- Phys.Rev.D 75 (2007) 045007
•
e-Print:
- hep-lat/0609058
•
DOI:
- 10.1103/PhysRevD.75.045007

Real-time gauge theory simulations from stochastic quantization with optimized updating

- Nucl.Phys.B 799 (2008) 306-329
•
e-Print:
- 0708.0779
•
DOI:
- 10.1016/j.nuclphysb.2008.01.018

Stochastic quantization at finite chemical potential

Gert Aarts
(
- Swansea U.
)
,
Ion-Olimpiu Stamatescu
(
- Heidelberg U. and
- Heidelberg, FEST
)

- JHEP 09 (2008) 018
•
e-Print:
- 0807.1597
•
DOI:
- 10.1088/1126-6708/2008/09/018

Can stochastic quantization evade the sign problem? The relativistic Bose gas at finite chemical potential

Gert Aarts
(
- Swansea U.
)

- Phys.Rev.Lett. 102 (2009) 131601
•
e-Print:
- 0810.2089
•
DOI:
- 10.1103/PhysRevLett.102.131601

Adaptive stepsize and instabilities in complex Langevin dynamics

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

- Phys.Lett.B 687 (2010) 154-159
•
e-Print:
- 0912.0617
•
DOI:
- 10.1016/j.physletb.2010.03.012

Complex Langevin dynamics at finite chemical potential: Mean field analysis in the relativistic Bose gas

Gert Aarts
(
- Swansea U.
)

- JHEP 05 (2009) 052
•
e-Print:
- 0902.4686
•
DOI:
- 10.1088/1126-6708/2009/05/052

Degenerate distributions in complex Langevin dynamics: one-dimensional QCD at finite chemical potential

Gert Aarts
(
- Swansea U.
)
,
K. Splittorff
(
- Bohr Inst.
)

- JHEP 08 (2010) 017
•
e-Print:
- 1006.0332
•
DOI:
- 10.1007/JHEP08(2010)017

On the convergence of complex Langevin dynamics: The Three-dimensional XY model at finite chemical potential

Gert Aarts
(
- Swansea U.
)
,
Frank A. James
(
- Swansea U.
)

- JHEP 08 (2010) 020
•
e-Print:
- 1005.3468
•
DOI:
- 10.1007/JHEP08(2010)020

Numerical Problems in Applying the Langevin Equation to Complex Effective Actions

Jan Ambjorn
(
- Bohr Inst.
)
,
S.K. Yang
(
- Bohr Inst.
)

- Phys.Lett.B 165 (1985) 140,
- IN *DAMGAARD, P.H. (ED.), HUEFFEL, H. (ED.): STOCHASTIC QUANTIZATION*, 406-412. (PHYS. LETT. B165 (1985) 140-146) AND COPENHAGEN UNIV. - NBI-HE-85-27 (85,REC.OCT.) 15p
•
DOI:
- 10.1016/0370-2693(85)90708-7

- J.Statist.Phys. 39 53

- Phys.Rev.B 34 1964

The Complex Langevin Equation and Monte Carlo Simulations of Actions With Static Charges

- Nucl.Phys.B 275 (1986) 375-397
•
DOI:
- 10.1016/0550-3213(86)90605-X

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

Effective Potential for Complex Langevin Equations

Gerald Guralnik
(
- Brown U.
)
,
Cengiz Pehlevan
(
- Brown U.
)

- Nucl.Phys.B 822 (2009) 349-366
•
e-Print:
- 0902.1503
•
DOI:
- 10.1016/j.nuclphysb.2009.06.016

Complex Langevin Equations and Schwinger-Dyson Equations

Cengiz Pehlevan
(
- Brown U.
)
,
Gerald Guralnik
(
- Brown U.
)

- Nucl.Phys.B 811 (2009) 519-536
•
e-Print:
- 0710.3756
•
DOI:
- 10.1016/j.nuclphysb.2008.11.034