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arXiv:2211.10728

Kernel controlled real-time Complex Langevin simulation

Daniel Alvestad, Larsen, Rasmus, Alexander Rothkopf

Published Nov 19, 2022• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2211.10728

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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 1.5β1.5β1.5β on the real-time Schwinger-Keldysh contour with the 0+10+10+1 dimensional anharmonic oscillator using m=1,λ=24m=1,λ=24m=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 1.5β1.5β1.5β on the real-time Schwinger-Keldysh contour with the 0+1 dimensional anharmonic oscillator using m=1m=1m=1, λ=24λ=24λ=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 1.5β1.5\beta1.5β on the real-time Schwinger-Keldysh contour with the 0+1 dimensional anharmonic oscillator using m=1m=1m=1, λ=24\lambda=24λ=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.
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.
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.
The real time evolution of a scalar field in 0+1 dimensions is investigated on a complex time contour. The path integral formulation of the system has a sign problem, which is circumvented using the Complex Langevin equation. Measurement of the boundary terms allow for the detection of correct results (for contours with small real time extents) or incorrect results (at large real time extents), as confirmed by comparison to exact results calculated using diagonalization of the Hamiltonian. We introduce a constant matrix kernel in the Complex Langevin equation, which is optimized with the requirement that distributions of the fields on the complexified manifold remain close to the real manifold. We observe that reachable real times are roughly twice as large with the optimal kernel. We also investigate field dependent kernels represented by a neural network for a toy model as well as for the scalar field, providing promising first results.