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arXiv:2211.15625 · Journal of High Energy Physics

Towards learning optimized kernels for complex Langevin

Daniel Alvestad, Rasmus Larsen, Alexander Rothkopf

Published Apr 13, 2023• 18 citations

Abstract

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.

DOI: 10.1007/jhep04(2023)057

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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.

This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to construct a kernel. This allows us to simulate up to

1.5\beta

on the real-time Schwinger-Keldysh contour with the 0+1 dimensional anharmonic oscillator using

m=1

,

\lambda=24

, which was previously unattainable using the complex Langevin equation.

This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to construct a kernel. This allows us to simulate up to 2β on the real-time Schwinger-Keldysh contour with the 0 + 1 dimensional anharmonic oscillator using m = 1; λ = 24, which was previously unattainable using the complex Langevin equation.

This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to construct a kernel. This allows us to simulate up to

1.5β

on the real-time Schwinger-Keldysh contour with the 0+1 dimensional anharmonic oscillator using

m=1

,

λ=24

, which was previously unattainable using the complex Langevin equation.

This study explores the utility of a kernel in complex Langevin simulations of quantum real-time dynamics on the Schwinger-Keldysh contour. We give several examples where we use a systematic scheme to find kernels that restore correct convergence of complex Langevin. The schemes combine prior information we know about the system and the correctness of convergence of complex Langevin to construct a kernel. This allows us to simulate up to

1.5β

on the real-time Schwinger-Keldysh contour with the

0+1

dimensional anharmonic oscillator using

m=1,λ=24

, which was previously unattainable using the complex Langevin equation.

We investigate complex Langevin (CL) dynamics for real-time SU(

N

) gauge theories in 3+1 dimensions and introduce new kernel modifications tailored for lattice gauge theory (LGT). Using the anisotropic kernel as a baseline, we demonstrate that on a rectangular regularized Schwinger-Keldysh time contour, stable simulations are possible for pure SU(2) gauge theory up to a real-time extent of

2β

at a bare coupling

g=0.5

, where

β

denotes the inverse temperature. We show that a contour-dependent phase kernel extends this reach to

3β

, while a local, field-dependent kernel based on traced plaquettes achieves

3.5β

. These constructions represent the first application of such kernels to LGTs, especially in 3+1 dimensions. To validate the approach, we generalize the CL correctness criterion based on the drift magnitude to kernelled dynamics, providing a practical tool to assess correct convergence. Our results substantially advance the applicability of CL to real-time simulations in Minkowski space, establishing a framework for controlled studies of non-Abelian gauge theories at unprecedented time scales. A forthcoming paper applies these developments to the computation of unequal-time correlation functions.

A bstract This study explores the potential of modern implicit solvers for stochastic partial differential equations in the simulation of real-time complex Langevin dynamics. Not only do these methods offer asymptotic stability, rendering the issue of runaway solution moot, but they also allow us to simulate at comparatively large Langevin time steps, leading to lower computational cost. We compare different ways of regularizing the underlying path integral and estimate the errors introduced due to the finite Langevin time steps. Based on that insight, we implement benchmark (non-)thermal simulations of the quantum anharmonic oscillator on the canonical Schwinger-Keldysh contour of short real-time extent.