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arXiv:cond-mat/9612091 · Journal of Experimental and Theoretical Physics Letters

Exact quantum Monte Carlo process for the statistics of discrete systems

Nikolay V. Prokof'ev, B. V. Svistunov, I. S. Tupitsyn

Published Dec 1, 1996• 104 citations
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Abstract

We propose an exact Monte Carlo approach for the statistics of discrete quantum systems that does not employ the standard partition of the imaginary time into a mesh and does not contain small parameters. The method operates with discrete objects — kinks, describing virtual transitions at different moments in time. The global statistics of the kinks is reproduced by exact local procedures, the main one being based on the known solution for an asymmetric two-level system.

DOI
10.1134/1.567243

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We propose an exact Monte Carlo approach for the statistics of discrete quantum systems that does not employ the standard partition of the imaginary time into a mesh and does not contain small parameters. The method operates with discrete objects — kinks, describing virtual transitions at different moments in time. The global statistics of the kinks is reproduced by exact local procedures, the main one being based on the known solution for an asymmetric two-level system.
We show how the worldline quantum Monte Carlo procedure, which usually relies on an artificial time discretization, can be formulated directly in continuous time, rendering the scheme exact. For an arbitrary system with discrete Hilbert space, none of the configuration update procedures contain small parameters. We find that the most effective update strategy involves the motion of worldline discontinuities (both in space and time), i.e., the evaluation of the Green’s function. Being based on local updates only, our method nevertheless allows one to work with the grand canonical ensemble and nonzero winding numbers, and to calculate any dynamical correlation function as easily as expectation values of, e.g., total energy. The principles found for the update in continuous time generalize to any continuous variables in the space of discrete virtual transitions, and in principle also make it possible to simulate continuous systems exactly.
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.
It is shown that Monte Carlo methods can be fruitfully applied to the computation of quantum processes in the real-time (Minkowski) regime. The method is first explained in the context of a simple quantum-mechanical model, then applied to the problem of the quantized electromagnetic field interacting with a large number of discrete two-state systems (the problem of superradiance). Numerical results previously obtained by operator methods are reproduced in this framework and the calculations extended to much larger systems.
We present a general formulation of the Green's function Monte Carlo method in imaginary-time quantum Monte Carlo which employs exact propagators. This algorithm has no time-step errors and is obtained by minimal modifications of the time-independent Green's function Monte Carlo method. We describe how the method can be applied to the many-body Schrödinger equation, lattice Hamiltonians, and simple field theories. Our modification of the Green's function Monte Carlo algorithm is applied to the ground state of liquid 4He. We calculate the zero-temperature imaginary-time diffusion constant and relate that to the effective mass of a mass-four "impurity" atom in liquid 4He.