Complex Langevin simulations of quantum systems with the sign problem

Florian Ehmann, C.E. Berger, Lukas Rammelmüller +3 more

Casey Berger, Lukas Rammelmüller, Andrew C. Loheac +3 more

Mitsuaki Hirasawa, ミツアキ ヒラサワ

The non-perturbative dynamics plays important roles in quantum field theories and superstring theory.Monte Carlo calculation of the systems is one of the useful methods to study non-perturbative dynamics.There are many successful works.However, the usual Monte Carlo methods are not applicable, for example, in the Lorentzian metric case, theories with a θ term, theories at finite density, and so on.In such cases, since the Boltzmann weight becomes complex, it is not possible to interpret the weight as the probability.This problem is called the sign problem.Some methods to overcome the sign problem have been proposed.The complex Langevin method (CLM) is one of such methods.An advantage of the method compared with others is that the numerical cost for increasing the system size is reasonable.This method was applied to many toy models, in which it worked well.However, the applications to physically interesting models are not so many.In our work, we apply the CLM to the gauge theories with a θ term and the Lorentzian type IIB matrix model.As a validity test of the CLM for the gauge theories with a θ term, we first apply the CLM to the 2d U(1) gauge theory on the torus with a θ term which is solvable with finite lattice spacing and finite volume on an arbitrary manifold.We find that a naive implementation of the method fails because of the topological nature of the θ term.In order to circumvent this problem, we simulate the same theory on a punctured torus, which is equivalent to the original model in the infinite volume limit for |θ| < π.Rather surprisingly, we find that the CLM works and reproduces the exact results for a punctured torus even at large θ, where the link variables near the puncture become very far from being unitary.Since the application of the CLM to the 2d U(1) case is successful, we next apply the CLM to the 4d SU(2) case.There are interesting predictions for the phase structure around θ = π by using 't Hooft anomaly matching condition.Therefore our aim is to investigate the phase structure by the first principle calculation.We apply the CLM to the theory on a torus and find that, in the coarse lattice case, there is no freezing of topological charge.Although the CLM works well even at large θ, we cannot see the 2π periodicity due to the finite lattice spacing effects.Therefore, we need to decrease the lattice spacing.However, there is the freezing problem in the fine lattice case.We try to solve the problem by imposing the periodic boundary conditions We find that imposing the open boundary condition in all spatial directions alleviates the topology freezing sufficiently, and the CLM works well at large θ.However, the finite volume effect occurs as a drawback of the open boundary conditions.We also discuss the possible way to investigate the phase structure.The Lorentzian type IIB matrix model was proposed as a non-perturbative formulation of superstring theory in 1996.The emergence of (3+1)-dimensional expanding space-time in the model is an intriguing phenomenon which was observed in Monte Carlo studies of

Arata Yamamoto, Tomoya Hayata

The complex Langevin method is one hopeful candidate to tackle the sign problem. This method is applicable not only to QCD but also to nonrelativistic field theory, such as condensed matter physics. We present the simulation results of a rotating Bose gas and an imbalanced Fermi-Hubbard model.

Daniel Alvestad, Rasmus Larsen, Alexander Rothkopf

A bstract We present a novel strategy aimed at restoring correct convergence in complex Langevin simulations. The central idea is to incorporate system-specific prior knowledge into the simulations, in order to circumvent the NP-hard sign problem. In order to do so, we modify complex Langevin using kernels and propose the use of modern auto-differentiation methods to learn optimal kernel values. The optimization process is guided by functionals encoding relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. For the strongly coupled quantum anharmonic oscillator we achieve correct convergence up to three-times the real-time extent of the previous benchmark study. An appendix sheds light on the fact that for correct convergence not only the absence of boundary terms, but in addition the correct Fokker-Plank spectrum is crucial.

G. W. Ford, Mark Kac