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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
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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 !
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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
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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]
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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])
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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
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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
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1. THE BASIC IDEA OF COMPLEX LANGEVIN METHOD
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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
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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
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Thus, we have shown: Note: ergodicity VEV of observables can be obtained by taking an average over a long time of the Langevin process !
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? 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: ?
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? 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.
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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.
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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
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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
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2. THEORETICAL UNDERSTANDING OF THE PROBLEM WITH LOG SINGULARITIES
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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 !
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Complex Langevin equation
drift term real Gaussian noise satisfies Fokker-Planck like equation :
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The crucial relation to the complex weight
Fokker-Planck equation (FPE)
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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.
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The results of CLM
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Eigenvalue spectrum of “FP Hamiltonian”
Fokker-Planck equation (FPE) “Fokker-Planck Hamiltonian”
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The eigenfunctions at An extra zero mode when p is a positive odd integer.
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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 !
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Relation between and P In order to derive this relation, one uses :
(#) with the initial condition :
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Proving eq.(#) Define an interpolating function
LHS of (#) RHS of (#) interpolated by the parameter Naively, this vanishes through integrating by parts.
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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]
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In the present example The boundary terms :
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Actual property of the distribution P(x,y)
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3. CORRECT CONVERGENCE EVEN WITH SEVERE SIGN PROBLEM
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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.
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Results of CLM for large p
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Radial distribution around the singularity
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Severeness of the complex-action problem
phase quenched model Reweighting method is extremely hard !
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4. NON-LOGARITHMIC CASE
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Non-logarithmic case single-valuedness of the drift term
No issue of ambiguity associated with the branch cut !
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Results of CLM in the non-logarithmic case
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Results of CLM in the non-logarithmic case
These results support our argument.
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5. TWO-VARIABLE CASE
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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 !
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Results of CLM for two-variable case
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Results of CLM for two-variable case
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6. GAUGE-COOLING IN RANDOM MATRIX THEORY
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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
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Complex Langegin eq. (discretized ver.)
The action should be written in terms of holomorphic variables. gauge symmetry:
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“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.
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Full QCD simulations with staggered fermion
Lattice size: Agreement with heavy-dense QCD (spatial hoppings are dropped)
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Spectrum of
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Random Matrix Theory for finite density QCD
The partition function is dominated by pions, which have zero quark charge.
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CLM for Random Matrix Theory
Complexification of dynamical variables :
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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
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“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”
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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 !
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7. SUMMARY AND FUTURE PROSPECTS
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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).
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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
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