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arXiv:1605.08040 · Physical Review Letters

Monte Carlo Study of Real Time Dynamics on the Lattice

Andrei Alexandru, Gökçe Başar, Paulo F. Bedaque, Sohan Vartak, Neill C. Warrington

Published Aug 17, 2016• 106 citations
arXivHTMLPDF

Abstract

Monte Carlo studies involving real time dynamics are severely restricted by the sign problem that emerges from a highly oscillatory phase of the path integral. In this Letter, we present a new method to compute real time quantities on the lattice using the Schwinger-Keldysh formalism via Monte Carlo simulations. The key idea is to deform the path integration domain to a complex manifold where the phase oscillations are mild and the sign problem is manageable. We use the previously introduced "contraction algorithm" to create a Markov chain on this alternative manifold. We substantiate our approach by analyzing the quantum mechanical anharmonic oscillator. Our results are in agreement with the exact ones obtained by diagonalization of the Hamiltonian. The method we introduce is generic and, in principle, applicable to quantum field theory albeit very slow. We discuss some possible improvements that should speed up the algorithm.

DOI
10.1103/physrevlett.117.081602

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Monte Carlo studies involving real time dynamics are severely restricted by the sign problem that emerges from a highly oscillatory phase of the path integral. In this Letter, we present a new method to compute real time quantities on the lattice using the Schwinger-Keldysh formalism via Monte Carlo simulations. The key idea is to deform the path integration domain to a complex manifold where the phase oscillations are mild and the sign problem is manageable. We use the previously introduced "contraction algorithm" to create a Markov chain on this alternative manifold. We substantiate our approach by analyzing the quantum mechanical anharmonic oscillator. Our results are in agreement with the exact ones obtained by diagonalization of the Hamiltonian. The method we introduce is generic and, in principle, applicable to quantum field theory albeit very slow. We discuss some possible improvements that should speed up the algorithm.
Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.
We present a numerically exact procedure for the calculation of an important class of finite temperature quantum mechanical time correlation functions. The present approach is based around the stationary phase Monte Carlo (SPMC) method, a general mathematical tool for the calculation of high dimensional averages of oscillatory integrands. In the present context the method makes possible the direct numerical path integral calculation of real-time quantum dynamical quantities for times appreciably greater than the thermal time (βℏ). Illustrative applications involving finite temperature anharmonic motion are presented. Issues of importance with respect to future applications are identified and discussed.
Abstract The Markov chain Monte Carlo (MCMC) method is used to evaluate the imaginary-time path integral of a quantum oscillator with a potential that includes a quadratic term and a quartic term whose coupling is varied by several orders of magnitude. This path integral is discretized on a time lattice on which calculations for the energy and probability density of the ground state and energies of the first few excited states are carried out on lattices with decreasing spacing to estimate these quantities in the continuum limit. The variation of the quartic coupling constant produces corresponding variations in the optimum simulation parameters for the MCMC method and in the statistical uncertainty for a fixed number of paths used for measurement. The energies and probability densities are in excellent agreement with those obtained from numerical solutions of Schrödinger’s equation. The theoretical and computational framework presented here introduces undergraduates to the path integral formulations of quantum mechanics in real time and the partition function in statistical mechanics in imaginary time. The example of the anharmonic oscillator helps to build an intuition about the MCMC method of evaluating the partition function, which can then be used to solve other problems in physics and beyond.
Abstract We construct a lattice field theory method for computing the real-time dynamics of spin systems in a thermal bath. This is done by building on previous work of Takano with Schwinger-Keldysh and functional differentiation techniques. We derive a Schwinger-Keldysh path integral for generic spin Hamiltonians, then demonstrate the method on a simple system. Our path integral has a sign problem, which generally requires exponential run time in the system size, but requires only linear storage. The latter may place this method at an advantage over exact diagonalization, which is exponential in both. Our path integral is amenable to contour deformations, a technique for reducing sign problems.
Abstract Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy’s theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.
Quantum Monte Carlo is one of the most powerful numerical tools for studying nonpeturbative properties of quantum many-body systems. However, its application to real-time problems is limited since the complex and highly-oscillating path-integral weight of the real-time evolution harms the important sampling. % , which is the notorious sign problem. In this Letter, we show that some real-time problems in open fermion systems can be simulated using the quantum Monte Carlo. To this end, we prescribe a mapping between a real-time problem in open quantum systems and a statistical problem in non-Hermitian quantum systems; for some cases, the latter can be solved without suffering from the complex measure problem. To explain our idea and demonstrate how it works, we compute the real-time evolution of fidelities in open fermion systems under dissipation.