Abstract
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.
Highlights
Are variants of reweighing, extrapolation from complex parameters and the reformulation of the system of interest in new degrees of freedom unaffected by a sign problem
This study explores the potential of modern implicit solvers for stochastic partial differential equations in the simulation of real-time complex Langevin dynamics
We focus in this study solely on the question of stability, returning to the remaining two in future work. (I.e. in order to remain in the parameter range where the complex Langevin method itself is known to converge to the correct results, we limit ourselves to a short real-time extent in this study.)
Summary
The task at hand is to compute quantum statistical expectation values of an observable O. Note that the delta function in the correlator of the noise encompasses all dimensions of x. This prescription places an independent stochastic process at each space-time point. Due to the complex drift term, the field degrees complexify and we can rewrite the evolution equations instead in terms of the real and imaginary part of the field as φ(x, τL) = φR(x, τL) + iφI (x, τL),. We have here used the standard construction in which the noise term η(x, τL) is real These are the stochastic partial differential evolution equations we will solve in the subsequent sections. The first central task is to find appropriate numerical solvers to accommodate these equations
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