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arXiv:0711.3839 · Physical Review E

Stochastic Green function algorithm

V. G. Rousseau

Published May 20, 2008• 67 citations
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Abstract

We present a stochastic Green function algorithm designed for bosons on lattices. This quantum Monte Carlo algorithm is independent of the dimension of the system, works in continuous imaginary time, and is exact (no error beyond statistical errors). Hamiltonians with several species of bosons (and one-dimensional Bose-Fermi Hamiltonians) can be easily simulated. Some important features of the algorithm are that it works in the canonical ensemble and gives access to n -body Green functions.

DOI
10.1103/physreve.77.056705

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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.
A novel scheme to solve the quantum eigenvalue problem through the imaginary-time Green function Monte Carlo method is presented. This method is applicable not only to the ground states, but also to the excited states of generic system. We demonstrate the actual utility of the method with the numerical examples on three simple systems. Typeset using REVTEX 1 The stochastic quantization has been successfully applied in recent years to the various problems of physics ranging from quantum chromo dynamics to the plasmon oscillation [1-3]. While it can handle very complex systems which is usually beyond the approach either by the direct diagonalization or by the perturbation, it has been only applicable to the calculation of the ground state properties of the system. In this paper, we propose a new method based on the imaginary-time Green function Monte Carlo approach, which is extendable to the excited states of Hamiltonian system. We have primarily in mind the non-relativistic quantum few body problem such as encountered in nuclear physics [4-6].
An efficient quantum Monte Carlo algorithm for the simulation of bosonic systems on a lattice in a grand canonical ensemble is proposed. It is based on the mapping of bosonic models to the spin models in the limit of the infinite total spin quantum number. It is demonstrated how this limit may be taken explicitly in the algorithm, eliminating the systematic errors. The efficiency of the algorithm is examined for the noninteracting lattice boson model and compared with the stochastic series expansion method with the heat-bath-type scattering probability of the random walker.
We present a numerically stable Quantum Monte Carlo algorithm to calculate zero-temperature imaginary-time Green functions $ G(\vec{r}, τ) $ for Hubbard type models. We illustrate the efficiency of the algorithm by calculating the on-site Green function $ G(\vec{r}=0, τ) $ on 4×44 \times 44×4 to 12×1212 \times 1212×12 lattices for the two-dimensional half-filled repulsive Hubbard model at U/t=4U/t = 4U/t=4. By fitting the tail of $ G(\vec{r}=0, τ) $ at long imaginary time to the form e−τΔce^{-τΔ_c}e−τΔc​, we obtain a precise estimate of the charge gap: Δc=0.67±0.02Δ_c = 0.67 \pm 0.02Δc​=0.67±0.02 in units of the hopping matrix element. We argue that the algorithm provides a powerful tool to study the metal-insulator transition from the insulator side.