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: 1504.08359 [hep-lat] Nagata-J.N.-Shimasaki : in preparation

Complex action problemPath integral

Monte Carlo simulation

a power tool to study QFT in a fully nonperturbative manner

Generate ensemble of with the probability

Calculate VEV by taking an ensemble average of

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 !

VEV can be obtained by time averaging overafter thermalization

A real scalar field , then, becomes a complex scalar field.

complex Langevin equation

Stochastic quantization for real S Parisi-Wu (’81)

fictitious time evolution of

Langevin equation

Gaussian white noise

Use the same method for complex S Parisi (’83), Klauder(’83)

complex Langevin equation

drift term

Recent development in finite density QCD based on complex Langevin eq.

Finite density QCD in the deconfined phasewith relatively heavy quarks

Random Matrix Theory for finite density QCD (T=0) Mollgaard, Splittorff: 1412.2729 [hep-lat] 　　　

Sexty: PLB 729 (2014) 108 = arXiv:1307.7748 [hep-lat]

Using various techniques, CLE turned out to work successfully in:

with arbitrary quark mass (including the massless limit!)

See also reviewsSexty: 1410.8813 [hep-lat]Arts et al.: 1412.0847 [hep-lat] 　　　

related paper: Splittorff: 1412.0502 [hep-lat]

Problems with log singularies in the action (previous arguments) e.g., Mollgaard, Splittorff: 1309.4335 [hep-lat]

Using the drift term :

corresponds to considering as a multi-valued functionwith a branch cut

When the phase of rotates frequently,complex Langevin method gives wrong results.

“branch cut crossing problem” (Greensite: 1406.4558 [hep-lat])The issue of “non-holomorphicity of the action” (Sexty: 1410.8813 [hep-lat])

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

Plan of the talk1. The basic idea of complex Langevin method

2. Theoretical understanding of the problem with log singularities

3. Correct convergence even with severe sign problem

4. Non-logarithmic case

5. Two-variable case

6. “Gauge cooling” in random matrix theory

7. Summary and future prospects

stochastic quantizationcomplex 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: 1504.08359 [hep-lat]

Nagata-J.N.-Shimasaki :in preparation

1. THE BASIC IDEA OF COMPLEX LANGEVIN METHOD

Stochastic quantization Parisi-Wu (’81)For review, see Damgaard-Huffel (’87)

Langevin eq.

Gaussian white noise

View this as the stationary distribution of a stochastic process.

Proof

satisfies:

Fokker-Planck eq.

Define :

self-adjoint operator with non-negative spectrum

“Fokker-Planck Hamiltonian”

unique eigenfunction with zero eigenvalue

vanish at large t

Thus, we have shown:

Note:

VEV of observables can be obtained by taking an average over a long time of the Langevin process !

ergodicity

Extension to a complex-action systemParisi (’83), Klauder (’83)

Langevin eq.assumed to be real here

The solution becomes complex, so we denote it as:

The crucial question is:?

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.

?

should exist.

The most nontrivial assumption in the above argument is that

This may be a too strong requirement.

We may have to be satisfied with

for a specific class of operators.

A refined argumentAarts, James, Seiler, Stamatescu: Eur. Phys. J. C(’11) 71;1756

?

holds at t=0 if

Does it hold for t>0 when evolves according to

real positive

Evolution of is derived from CLE as

Integration by parts requires, in particular:

vanishes at large t

Hence, we needsufficiently sharp fall off of P(x,y;t) in y-direction

2. THEORETICAL UNDERSTANDING OF THE PROBLEM WITH LOG SINGULARITIES

A simple example

Define the drift term:

The action is multi-valued, but we DON’T need to use it !

single-valued for any p

Complex Langevin equationreal Gaussian noisedrift term

satisfies Fokker-Planck like equation :

The crucial relation to the complex weight

Fokker-Planck equation (FPE)

Solutions to the FPE

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.

The results of CLM

Eigenvalue spectrum of “FP Hamiltonian”

Fokker-Planck equation (FPE)

“Fokker-Planck Hamiltonian”

The eigenfunctions at .

An extra zero mode when p is a positive odd integer.

An implication of the negative modes

Fokker-Planck equation (FPE)

The most negative mode of the FP Hamiltonian

This behavior is incompatible with the relation :

This relation must be violated, at least, in the region where the negative modes appear !

Relation between and P

In order to derive this relation, one uses :

with the initial condition :

(#)

Naively, this vanishesthrough integratingby parts.

Proving eq.(#)

LHS of (#) RHS of (#) interpolated by the parameter

(#)

Define an interpolating function

When does the partial integration fail ?

Aarts-James-Seiler-Stamatescu,arXiv:1101.3270 [hep-lat]

slow fall off of the integrand at

c.f.) This lead to the idea of gauge-cooling.Seiler-Sexty-Stamatescu,arXiv:1211.3709 [hep-lat]

the presence of singularities in the integrandJ.N.-Shimasaki, arXiv:1504.08359 [hep-lat]

This is not a problem in the present case.

In the present example

The boundary terms :

Actual property of the distribution P(x,y)

3. CORRECT CONVERGENCE EVEN WITH SEVERE SIGN PROBLEM

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.

Results of CLM for large p

p

Radial distribution around the singularity

Severeness of the complex-action problem

phase quenched model

Reweighting method is extremely hard !

4. NON-LOGARITHMIC CASE

Non-logarithmic case

single-valuedness of the drift term

No issue of ambiguity associated with the branch cut !

Results of CLM in the non-logarithmic case

Results of CLM in the non-logarithmic case

These results support our argument.

5. TWO-VARIABLE CASE

Two-variable case

Negative modes appearfor p>1

Eigenvalue spectrum of the “FP Hamiltonian”

The relation between and P must be violated at least in this region !

Results of CLM for two-variable case

Results of CLM for two-variable case

6. GAUGE-COOLING IN RANDOM MATRIX THEORY

Application of CLM to finite density QCDFor a review, see: Aarts 1302.3028 [hep-lat]

complex action problem

This term may cause the problem ofthe singular drift we have been discussing !

Complex Langegin eq. (discretized ver.)

The action should be written in terms of holomorphic variables.

gauge symmetry:

“gauge cooling” Seiler, Sexty, Stamatescu: PLB723 (’13) 213

After each Langevin step, apply the gauge tr. in the direction of the steepest descent of the unitarity norm.

Unitarity norm :

d=0 only for SU(3) matrices.

Excursion in the imaginary directionsis avoided.

full QCD simulations with staggered fermion

Sexty: PLB729 (’14) 108

Full QCD simulations with staggered fermion

Lattice size:

Agreement with heavy-dense QCD(spatial hoppings are dropped)

Spectrum of

Random Matrix Theory for finite density QCD

The partition function is dominated by pions,which have zero quark charge.

CLM for Random Matrix Theory

Complexification of dynamical variables :

Previous results of CLM for random matrix theory

m=5 m=15Trajectory of det (D+m)

Convergence to wrong results !

Mollgaard, Splittorff: PRD 88 (’13) 116007

Using this complexified symmetry, we apply “gauge-cooling”after each Langevin step so that

“gauge-cooling” in random matrix theory

“anti-Hermiticity norm”

Symmetry of the system :

complexficationof variables

Nagata-J.N.-Shimasaki, in prep.

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 !

7. SUMMARY AND FUTURE PROSPECTS

Summary

Complex Langevin eq.

(2) convergence to

We have clarified (3), which was poorly understood before. The convergence to wrong results is due to violation of (1).

a promising approach to a system with complex S

Crucial questions are:(1)

(3) Violation of holomorphicity due to

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.

“Gauge cooling” + some coordinate transformation + nontrivial kernels for the random noise

Full QCD simulations in that region may be possiblein the near future with

Applications to other complex-action systemssuch as supersymmetric gauge theories and matrix models relevant to nonperturbative studies of superstring theory

We have generalized the idea of “gauge cooling”.