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arXiv:0806.1410 · Physical Review E

Directed update for the stochastic Green function algorithm

V. G. Rousseau

Published Nov 24, 2008• 63 citations
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Abstract

In a recent paper we presented a stochastic Green function (SGF) algorithm, which has the properties of being general and easy to apply to any lattice Hamiltonian of the form $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{V}}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{T}}$, where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{V}}$ is diagonal in the chosen occupation number basis and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{T}}$ has only positive matrix elements. We propose here a modified version of the update scheme that keeps the simplicity and generality of the original SGF algorithm, and significantly enhances its efficiency.

DOI
10.1103/physreve.78.056707

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In a recent paper we presented a stochastic Green function (SGF) algorithm, which has the properties of being general and easy to apply to any lattice Hamiltonian of the form H\ifmmode^\elseˆ\fi=V\ifmmode^\elseˆ\fi\ensuremath−T\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{H}}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{V}}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{T}}H\ifmmode^\elseˆ\fi​=V\ifmmode^\elseˆ\fi​\ensuremath−T\ifmmode^\elseˆ\fi​, where V\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{V}}V\ifmmode^\elseˆ\fi​ is diagonal in the chosen occupation number basis and T\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{T}}T\ifmmode^\elseˆ\fi​ has only positive matrix elements. We propose here a modified version of the update scheme that keeps the simplicity and generality of the original SGF algorithm, and significantly enhances its efficiency.
A new method for the stabilization of the sign problem in the Green function Monte Carlo technique is proposed. The method is devised for real lattice Hamiltonians and is based on an iterative ``stochastic reconfiguration'' scheme which introduces some bias but allows a stable simulation with constant sign. The systematic reduction of this bias is possible in principle. The method is applied to the frustrated ${J}_{1}{\ensuremath{-}J}_{2}$ Heisenberg model, and tested against exact diagonalization data. Evidence of a finite spin gap for ${J}_{2}{/J}_{1}>\ensuremath{\sim}0.4$ is found in the thermodynamic limit.
The Stochastic Green Function (SGF) algorithm is able to simulate any Hamiltonian that does not suffer from the so-called "sign problem". We propose a new global space-time update scheme for the SGF algorithm which, in addition to being simpler than the previous formulation, reduces auto-correlation times. Using as a concrete example the extended Bose-Hubbard model and the complex Hamiltonian with six-site ring-exchange interactions which was recently studied in ArXiv:1206.2566v1, we present a comprehensive review of the SGF algorithm and the new updating scheme. Measurements of non-trivial physical quantities are presented in detail. While the SGF algorithm works in the canonical ensemble by nature, we give a simple extension that allows us to perform simulations in the grand-canonical ensemble too. We also discuss an optimized implementation which allows for access to large system sizes.
We study the tight-binding Hamiltonian H=∑j|j> ∊j<j| + ∑j,k |j>Vjk < k | defined on the Bethe lattice of an arbitrary coordination number; the hopping elements Vjk are nonzero (and constant) only for j,k being the nearest-neighbor sites and the energies ∊j are considered to be site-dependent. The Green's function (z-H)-1 problem is solved explicitly in the inverse form, with diagonal matrix elements {<j | (z-H)-1 | j >} as controlling (prescribed) variables. Namely: (i) the inverse profile relation, i.e. z-∊j versus diagonal matrix elements, is obtained in a local form; (ii) the off-diagonal matrix elements { <j | (z-H)-1 | k >} are shown to exhibit a simple factorization property in terms of the diagonal ones.