1. Theoretical understanding of the problem with a singular drift term in the complex Langevin method
Jun Nishimura (KEK Theory Center, SOKENDAI)
May 13, 2015, QCD club at University of Tokyo, Hongo
Ref.) J.N.-Shimasaki: [hep-lat]
Nagata-J.N.-Shimasaki : in preparation

2. Complex action problem
Path integral
Monte Carlo simulation
Generate ensemble of with the probability
Calculate VEV by taking an ensemble average of
a power tool to study QFT in a fully nonperturbative manner
However, in many interesting examples, the action S becomes complex!
QCD at finite density or with theta term
supersymmetric gauge theories and matrix models
relevant to superstring theory
can no longer be regarded as the Boltzmann weight !

3. complex Langevin equation
Stochastic quantization for real S Parisi-Wu (’81)
fictitious time evolution of
Langevin equation
drift term
Gaussian white noise
VEV can be obtained by time averaging over
after thermalization
Use the same method for complex S Parisi (’83), Klauder(’83)
A real scalar field , then, becomes a complex scalar field.
complex Langevin equation

4. Recent development in finite density QCD based on complex Langevin eq.
Using various techniques, CLE turned out to work successfully in:
Finite density QCD in the deconfined phase with relatively heavy quarks
Random Matrix Theory for finite density QCD (T=0)
Sexty: PLB 729 (2014) = arXiv: [hep-lat]
See also reviews
Sexty: [hep-lat]
Arts et al.: [hep-lat]　　　
with arbitrary quark mass (including the massless limit!)
Mollgaard, Splittorff: [hep-lat]　　　
related paper: Splittorff: [hep-lat]

5. Problems with log singularies in the action (previous arguments) e. g
Problems with log singularies in the action (previous arguments) e.g., Mollgaard, Splittorff: [hep-lat]
Using the drift term :
corresponds to considering as a multi-valued function
with a branch cut
When the phase of rotates frequently,
complex Langevin method gives wrong results.
“branch cut crossing problem” (Greensite: [hep-lat])
The issue of “non-holomorphicity of the action” (Sexty: [hep-lat])

6. Our new theoretical understanding
Formulation of the method only requires:
The problem actually occurs due to
the singularity in the drift term !
Single-valuedness of the complex weight
Single-valuedness of the drift term after complexification
There is NO need to define the action through

7. Plan of the talk The basic idea of complex Langevin method
Theoretical understanding of the problem with log singularities
Correct convergence even with severe sign problem
Non-logarithmic case
Two-variable case
“Gauge cooling” in random matrix theory
Summary and future prospects
stochastic quantization
complex Langevin equation (CLE)
the criterion for giving correct results
Studies of the Fokker-Planck equation (FPE)
The breakdown of the correspondence between FPE and CLE
J.N.-Shimasaki: [hep-lat]
Nagata-J.N.-Shimasaki :
in preparation

8. 1. THE BASIC IDEA OF COMPLEX LANGEVIN METHOD

9. Stochastic quantization
Parisi-Wu (’81)
For review, see
Damgaard-Huffel (’87)
View this as the stationary distribution of a stochastic process.
Langevin eq.
Gaussian white noise

10. Proof satisfies: Fokker-Planck eq. Define :
“Fokker-Planck Hamiltonian”
self-adjoint operator
with non-negative spectrum
unique eigenfunction
with zero eigenvalue
vanish at large t

11. Thus, we have shown:
Note:
ergodicity
VEV of observables can be obtained by taking an average
over a long time of the Langevin process !

12. ? Extension to a complex-action system Parisi (’83), Klauder (’83)
Langevin eq.
assumed to be real here
The solution becomes complex, so we denote it as:
The crucial question is: ?

13. ? Formal argument Ambjorn-Yang (’85) Suppose exists.
Then, one can obtain:
Moreover, one can show
Fokker-Planck eq., but with S complex !
is a stationary solution.
If all the eigenvalues of this operator have negative real part,
the convergence is guaranteed.

14. The most nontrivial assumption in the above argument is that
should exist.
This may be a too strong requirement.
We may have to be satisfied with
for a specific class of operators.

15. A refined argument
Aarts, James, Seiler, Stamatescu: Eur. Phys. J. C(’11) 71;1756 ?
holds at t=0 if
real positive
Does it hold for t>0 when evolves according to
Evolution of is derived from CLE as

16. Integration by parts requires, in particular:
vanishes at large t
Hence, we need
sufficiently sharp fall off of P(x,y;t)
in y-direction

17. 2. THEORETICAL UNDERSTANDING OF THE PROBLEM WITH LOG SINGULARITIES

18. A simple example Define the drift term: single-valued for any p
The action is multi-valued, but we DON’T need to use it !

19. Complex Langevin equation
drift term
real Gaussian noise
satisfies Fokker-Planck like equation :

20. The crucial relation to the complex weight
Fokker-Planck equation
(FPE)

21. Solutions to the FPE Necessary and sufficient conditions
time-independent solution :
Necessary and sufficient conditions
for the correct convergence of CLM :
i) The relation holds.
ii) The solution of the FPE asymptotes to w(x) at large t.

22. The results of CLM

23. Eigenvalue spectrum of “FP Hamiltonian”
Fokker-Planck equation (FPE)
“Fokker-Planck Hamiltonian”

24. The eigenfunctions at
An extra zero mode when p is a positive odd integer.

25. An implication of the negative modes
The most negative mode of the FP Hamiltonian
Fokker-Planck equation (FPE)
This behavior is incompatible with the relation :
This relation must be violated, at least, in the region
where the negative modes appear !

26. Relation between and P In order to derive this relation, one uses :
(#)
with the initial condition :

27. Proving eq.(#) Define an interpolating function
LHS of (#)
RHS of (#)
interpolated by the parameter
Naively, this vanishes
through integrating
by parts.

28. When does the partial integration fail ?
slow fall off of the integrand at
Aarts-James-Seiler-Stamatescu,arXiv: [hep-lat]
c.f.) This lead to the idea of gauge-cooling.
Seiler-Sexty-Stamatescu,arXiv: [hep-lat]
This is not a problem in the present case.
the presence of singularities in the integrand
J.N.-Shimasaki, arXiv: [hep-lat]

29. In the present example
The boundary terms :

30. Actual property of the distribution P(x,y)

31. 3. CORRECT CONVERGENCE EVEN WITH SEVERE SIGN PROBLEM

32. Relation to the phase rotation ?
This causes the complex-action problem.
NO, not necessarily !
In order to demonstrate this point,
we study large p case.

33. Results of CLM for large p

34. Radial distribution around the singularity

35. Severeness of the complex-action problem
phase quenched model
Reweighting method is extremely hard !

36. 4. NON-LOGARITHMIC CASE

37. Non-logarithmic case single-valuedness of the drift term
No issue of ambiguity associated with the branch cut !

38. Results of CLM in the non-logarithmic case

39. Results of CLM in the non-logarithmic case
These results support our argument.

40. 5. TWO-VARIABLE CASE

41. Two-variable case Eigenvalue spectrum of the “FP Hamiltonian”
Negative modes appear
for p>1
The relation between and P
must be violated at least in this region !

42. Results of CLM for two-variable case

43. Results of CLM for two-variable case

44. 6. GAUGE-COOLING IN RANDOM MATRIX THEORY

45. Application of CLM to finite density QCD
For a review, see: Aarts [hep-lat]
This term may cause the problem of
the singular drift we have been discussing !
complex action problem

46. Complex Langegin eq. (discretized ver.)
The action should be written in terms of holomorphic variables.
gauge symmetry:

47. “gauge cooling” Seiler, Sexty, Stamatescu: PLB723 (’13) 213
Unitarity norm :
d=0 only for SU(3) matrices.
After each Langevin step, apply the gauge tr. in the direction of
the steepest descent of the unitarity norm.
full QCD simulations
with staggered fermion
Sexty: PLB729 (’14) 108
Excursion in the imaginary directions
is avoided.

48. Full QCD simulations with staggered fermion
Lattice size:
Agreement with heavy-dense QCD
(spatial hoppings are dropped)

49. Spectrum of

50. Random Matrix Theory for finite density QCD
The partition function is dominated by pions,
which have zero quark charge.

51. CLM for Random Matrix Theory
Complexification of dynamical variables :

52. Previous results of CLM for random matrix theory
Mollgaard, Splittorff: PRD 88 (’13)
Convergence to wrong results !
m=5
Trajectory of det (D+m)
m=15

53. “gauge-cooling” in random matrix theory
Nagata-J.N.-Shimasaki, in prep.
Symmetry of the system :
complexfication
of variables
Using this complexified symmetry, we apply “gauge-cooling”
after each Langevin step so that
“anti-Hermiticity norm”

54. Results of CLM w/ and w/o “gauge-cooling”
Nagata-J.N.-Shimasaki, in prep.
“Gauge cooling” can be used to avoid the problem of the singular drift !

55. 7. SUMMARY AND FUTURE PROSPECTS

56. Summary Complex Langevin eq.
a promising approach to a system with complex S
Crucial questions are:
(1)
(2) convergence to
(3) Violation of holomorphicity due to
We have clarified (3), which was poorly understood before.
The convergence to wrong results is due to violation of (1).

57. Future prospects Full QCD at finite density in the deconfined phase
with relatively heavy quarks : successful
“Gauge cooling” is the crucial technique.
Randam Matrix Theory serves as a testing ground for low T regime with light quarks.
We have generalized the idea of “gauge cooling”.
Full QCD simulations in that region may be possible in the near future with
“Gauge cooling” + some coordinate transformation
+ nontrivial kernels for the random noise
Applications to other complex-action systems
such as supersymmetric gauge theories and matrix models relevant to nonperturbative studies of superstring theory