複素ランジュバン法におけるゲージ・クーリングの 一般化とその応用 西村 淳 （ KEK 理論センター、総研大） 「離散的手法による場と時空のダイナミクス」研究会 2015 ９月１５日（火）＠岡山 Ref.) J.N.-Shimasaki: arXiv: [hep-lat], Phys.Rev. D92 (2015) 1, Nagata-J.N-Shimasaki: arXiv: [hep-lat] Nagata-J.N-Shimasaki: in preparation

Complex action problem Path 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 over after 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 It works in many nontrivial cases, but converges to wrong results in the other cases.

theoretical developments One of the reasons for wrong convergence was clearly identified. gauge cooling as a remedy success in finite density QCD in the deconfined phase a possible problem in finite density QCD at low T (with light quarks) pointed out from studies of Random Matrix Theory (RMT) for QCD Aarts, James, Seiler, Stamatescu: Eur. Phys. J. C(’11) 71;1756 Seiler, Sexty, Stamatescu: PLB723 (’13) 213 Sexty: PLB 729 (2014) 108 = arXiv: [hep-lat] Mollgaard, Splittorff: PRD 88 (’13) Recent developments in the complex Langevin method

clarification of the problem anticipated to occur in finite density QCD at low T (with light quarks) J.N.-Shimasaki Phys. Rev. D 92, (R) (2015) (arXiv: [hep-lat]) Shimasaki’s parallel talk at LATTICE2015 generalization of gauge cooling to overcome this problem success in the studies of Random Matrix Theory for QCD Nagata-J.N.-Shimasaki, in preparation Nagata’s parallel talk at LATTICE2015 justification of the complex Langevin method including the gauge cooling procedure Nagata-J.N.-Shimasaki arXiv: [hep-lat] Our recent work

Plan of the talk 1. Justification of the complex Langevin method 2. Gauge cooling and its justification 3. Application to RMT for finite density QCD 4. Application to a simple IKKT-type model (Euclidean) 5. Summary and future prospects

1. Justification of the 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 system Parisi (’83), Klauder (’83) complex Langevin eq. assumed to be real here The solution becomes complex, so we denote it as: The crucial question is: ?

Proof of the key relation Aarts, James, Seiler, Stamatescu: Eur. Phys. J. C(’11) 71;1756 ? holds at t=0 if In fact, it also holds for t>0, under certain conditions, when evolves according to real positive Evolution of is derived from CLE as

integration by parts used here is justified if and only if : sufficiently sharp fall off of P(x,y;t) in y-direction. If L involves singularities, P(x,y;t) should be practically zero around the singularities. J.N.-Shimasaki: arXiv: [hep-lat], Phys.Rev. D92 (2015) 1, Aarts, James, Seiler, Stamatescu: Eur. Phys. J. C(’11) 71;1756

Failure of partial integration by singularities When the integrand involves singularities, integration by parts can fail.

A simple example Define the drift term: single-valued for any p J.N.-Shimasaki: arXiv: [hep-lat], Phys.Rev. D92 (2015) 1, real Gaussian noisedrift term complex Langevin eq. singularity at

The results of CLM

property of the distribution P(x,y)

2. Gauge cooling and its justification

An example of O(N) vector model Nagata-J.N-Shimasaki: arXiv: [hep-lat] Symmetry enhances from O(N) to O(N,C) !

(discretized) complex Langevin eq. complex Langevin eq. including “gauge cooling” Choose appropriately as a function of the config. before gauge cooling e.g.) In order to solve the problem of insufficient fall off of P(x,y;t) in y-directions, minimize: Nagata-J.N-Shimasaki: arXiv: [hep-lat]

complex Langevin eq. including “gauge cooling” (cont’d) the effects of gauge cooling Nagata-J.N-Shimasaki: arXiv: [hep-lat]

Evolution of P(x,y,;t) the effects of gauge cooling Justification of “gauge cooling” the effects of gauge cooling disappears as long as f(z) is O(N,C) invariant !!! Nagata-J.N-Shimasaki: arXiv: [hep-lat]

Important aspects of “gauge cooling” Gauge cooling changes the probability distribution P(x,y;t). However, its effect vanishes on the r.h.s. of the key relation : We have a chance to satisfy the properties of P(x,y;t) necessary for correct convergence of complex Langevin eq.. as long as one uses the complexified symmetry transformation, under which the action S(z) and the observable O(z) are invariant.

3. Application to random matrix theory for finite density QCD

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

Observables : chiral condensate baryon number density Basic properties of the Dirac operator

CLM for Random Matrix Theory Complexification of dynamical variables :

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

Using this complexified symmetry, we apply “gauge cooling” after each Langevin step so that problems are avoided. “gauge cooling” in random matrix theory Symmetry of the system : complexfication of variables Nagata-J.N.-Shimasaki, in prep. non-Hermiticity of the configurations zeroes of fermion determinant

Choices of the norm in the “gauge cooling” To avoid excursions into non-Hermite regime To avoid zero eigenvalues of D+m We take linear combinations like : tries to make D closer to anti-Hermitian

Results of CLM w/ and w/o “gauge-cooling” Nagata-J.N.-Shimasaki, in prep. Correct results for chiral condensate and baryon number density are reproduced by CLM with “gauge cooling” even in the small mass region !!!

Results of eigenvalue distribution of D “Gauge cooling” can be used to avoid the problem of the singular drift by choosing an appropriate norm. Nagata-J.N.-Shimasaki, in prep.

Relation to the chiral condensate eigenvalue distribution of D positive definite quantity (non-holomorphic) No relation to the eigenvalue distribution of D in the original path integral relation to the chiral condensate : eigenvalues of D Nagata-J.N.-Shimasaki, in prep.

Generalized Banks-Casher relation This relation is confirmed by direct measurements. Expected to be useful also in finite density QCD ! Nagata-J.N.-Shimasaki, in prep.

Testing the generalized Banks-Casher relation Nagata-J.N.-Shimasaki, in prep.

4. Application to a simple IKKT-type matrix model (Euclidean)

a simplified IKKT-type matrix model J.N. PRD 65, (2002), hep-th/

order parameters of the SSB of SO(4) symmetry SSB of SO(4) symmetry Note : for the phase-quenched model

Results of the Gaussian expansion method J.N., Okubo, Sugino: JHEP 0210 (2002) 043 hep-th/

Application of the complex Langevin method In order to investigate the SSB, we introduce an infinitesimal SO(4) breaking terms : and calculate : Ito-J.N., in preparation

Results of complex Langevin method The results of CLM reproduce the predictions from GEM !!! Problem of the singular drift occurs at small ε. (gauge cooling only for the Hermiticity norm) Ito-J.N., in preparation GEM results

large N extrapolation and the eigenvalue distribution Ito-J.N., in preparation

5. Summary and future prospects

Summary Complex Langevin method a very promising approach to the complex action problem the necessary and sufficient conditions for correct convergence Integration by parts should be justified Properties of P(x,y;t) :  sufficiently fast fall off in the y directions  sufficiently strong suppression around singularities Gauge cooling can be used to satisfy both of these conditions, thereby enlarging the range of applicability of CLM considerably.

Convergence of Fokker-Planck eq. in the case of complex action If the key relation is shown to hold, the above convergence is guaranteed if P(x,y;t) converges to some distribution uniquely. J.N.-Shimasaki: arXiv: [hep-lat], Phys.Rev. D92 (2015) 1,

Future prospects finite density QCD gauge cooling with the following norm may be useful at low T with light quarks. Lanczos method can be used for fast computation. Full QCD simulation at low temperature using KEK supercomputer H.Matsufuru-K.Nagata-J.N.-S.Shimasaki, work in progress  confirmation of Silver Blaze phenomenon : a mile stone in finite density QCD  new phases of QCD (color superconductivity, color-flavor locking,…)  determination of critical end point in the phase diagram

IKKT matrix model (Euclidean) Future prospects (cont’d) MC calculation based on the reweighting method (factorization method) in progress Anagnostoulos-Azuma-J.N., work in progress Results for 6d version : JHEP 1311 (2013) 009, arXiv: [hep-th] Complex Langevin method can be a more promising approach. Many other applications QCD with the theta term, Chern-Simons gauge theories, condensed matter physics (Hubbard model etc.) tested in a simplified model Ito-J.N., in preparation