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arXiv:2105.02735 · Journal of High Energy Physics

Stable solvers for real-time Complex Langevin

Daniel Alvestad, Rasmus Larsen, Alexander Rothkopf

Published Aug 1, 2021• 18 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep08(2021)138

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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.
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 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 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.
Quantum dynamical systems typically deal with huge num-bers of degrees of freedom, particularly functional integrals, which makes them very difficult to simulate using deter-ministic methods. Modeling such systems with stochas-tic differential equations (SDEs) is an attractive alterna-tive to Markov chain Monte-Carlo, but there remain stabil-ity problems. An example of a multi-dimensional complex SDE arises from an anharmonic oscillator with a one-mode Bose-Einstein condensate (BEC) Hamiltonian given in [1], for which the numerical solution becomes unreliable after approximatively t = 0.3 for all tested numerical methods. Splitting of drift methods as introduced in [3] have been tried on this problem and seemed to improve the stability of the results, but not to the desired extent (see [1]). To improve the results, new numerical splitting methods have recently been developed. However, it seems that the choice of the numerical scheme alone might not guarantee the reliability of the results. The use of carefully chosen stochastic gauges as in [1] improves the accuracy of the simulation greatly by taking into account some aspects of the dynamics of the system. Non-vanishing boundary terms from partial integration can be problematic and have to be taken care of, for example considering them as constraints and using projected SDEs techniques [2]. 1. Simulation of Bose-Einstein condensates Consider the case of an anharmonic oscillator with one-mode BEC Hamiltonian treated in [1]. From the master equation ∂ρ̂
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.