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Exact Monte Carlo time dynamics in many-body lattice quantum systems

Massimo Ostilli, Carlo Presilla A

Published Jan 1, 2004• 5 citations
OpenAlex

Abstract

Abstract. For many-body lattice quantum systems the existence of a Feynman-Kac–type formula based on Poisson stochastic processes has recently allowed to derive an exact Monte Carlo algorithm for the evolution operator at real or imaginary times. We extend this algorithm to the study of exact time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and the Heisenberg model. PACS numbers: 02.50.-r, 02.70.SsExact Monte Carlo time dynamics in many-body lattice quantum systems 2 1.

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Abstract. For many-body lattice quantum systems the existence of a Feynman-Kac–type formula based on Poisson stochastic processes has recently allowed to derive an exact Monte Carlo algorithm for the evolution operator at real or imaginary times. We extend this algorithm to the study of exact time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and the Heisenberg model. PACS numbers: 02.50.-r, 02.70.SsExact Monte Carlo time dynamics in many-body lattice quantum systems 2 1.
On the basis of a Feynman–Kac-type formula involving Poisson stochastic processes, a Monte Carlo algorithm has recently been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.
An algorithm is developed for determining the exact ground state properties of quantum many-body systems which is equally applicable to bosons and fermions. The Schroedinger eigenvalue equation for the ground state energy is recast into the form of a many-dimensional integral through the use of the Hubbard-Stratonovitch representation of the imaginary time many- body evolution operator. The resulting functional integral is then evaluated stochastically. The algorithm is tested for an exactly soluble boson system and is then extended to include fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Improved computational efficiency is attained by performing the calculations in momentum space.
We develop a new implementation of the real-time full configuration interaction quantum Monte Carlo method, and apply it to the dynamics of the Heisenberg spin chain and the spectrum of the one-dimensional Fröhlich polaron. This is a general method for the numerical integration of the Schrödinger equation to evolve quantum systems in real time, introduced by Guther et al. as an extension of the imaginary-time version of the algorithm that finds ground-state properties. We introduce a new integrator for improved efficiency, and a scaling procedure for better control of simulations. We study the accuracy of the method, including the effect of discretising time and of the stochastic errors introduced by the Monte Carlo elements of the algorithm. Comparing the method to a hardware quantum simulator for the quench dynamics of a Heisenberg spin chain, we find that for this model the discretisation and stochastic errors grow quickly, limiting the effectiveness of the algorithm for finding accurate evolution over long times. We find that spectral information can be generated successfully using an energy-dependent damping, with either a tilted complex contour time evolution or a strong second-order damping. Specifically, we reproduce the low-energy spectrum of the Fr ̈ohlich polaron model and confirm the existence of a bound excited state.
We present a simple derivation of a Feynman-Kac type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal algorithm can be chosen which coincides with the Green Function Monte Carlo (GFMC) method in the limit when the latter becomes exact.