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EVOLUTION OF FERMIONIC SYSTEMS AS AN EXPECTATION OVER POISSON PROCESSES

Matteo Beccaria A, Gian Fabrizio De Angelis

Published Jan 1, 1999• 2 citations
OpenAlex

Abstract

We derive an exact probabilistic representation for the evolution of a Hubbard model with site- and spin-dependent hopping coefficients and site-dependent interactions in terms of an associated stochastic dynamics of a collection of Poisson processes. In a recent paper 1 we provided an exact probabilistic expression for the real time or the imaginary time evolution of a Fermi system, in particular a Hubbard model, in terms of an associated stochastic dynamics of a collection of Poisson processes. Here, we generalize the result 1 to the case of a Hubbard model with site- and spin-dependent hopping coefficients and site-dependent interactions. This situation is of interest if disorder is present 2. Let us consider the Hubbard Hamiltonian

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We derive an exact probabilistic representation for the evolution of a Hubbard model with site- and spin-dependent hopping coefficients and site-dependent interactions in terms of an associated stochastic dynamics of a collection of Poisson processes. In a recent paper 1 we provided an exact probabilistic expression for the real time or the imaginary time evolution of a Fermi system, in particular a Hubbard model, in terms of an associated stochastic dynamics of a collection of Poisson processes. Here, we generalize the result 1 to the case of a Hubbard model with site- and spin-dependent hopping coefficients and site-dependent interactions. This situation is of interest if disorder is present 2. Let us consider the Hubbard Hamiltonian
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.
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.
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.
A common approximate way to study the time evolution of a superfluid Fermi gas is to consider the mean-field time-dependent BCS equations. We show that, by inclusion of an appropriate fluctuating term in these mean-field equations, the exact quantum evolution of the Fermi gas is recovered, after an average over all possible realizations of the noise term: this leads to an exact quantum Monte Carlo technique for the real-time evolution of an interacting fermionic system. In practice, we derive the general conditions on the noise term, ensuring that the exact dynamics is reproduced, and we explore several explicit realizations of this noise term, trying to minimize the statistical spreading of the quantum Monte Carlo simulations. The convergence issue is discussed in detail, with upper bounds obtained on the statistical spreading, and the relation between the stochastic and the mean-field equations is analyzed. Finally, we apply the method to a simple two-site model, to exemplify that the simulations can display effects out of reach of the mean-field approximation.