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Physical review. B, Condensed matter

Hybrid molecular-dynamics algorithm for the numerical simulation of many-electron systems

Richard T. Scalettar, Douglas J. Scalapino, Robert L. Sugar, D. Toussaint

Published Dec 1, 1987• 52 citations
Publisher

Abstract

We present an algorithm for the numerical simulation of many electron systems, which is based upon the hybrid stochastic approach presently being used to study lattice gauge theory. A method for preconditioning the fermion matrices is described which significantly reduces the computing time needed in the simulation. Results are presented for the three-dimensional Hubbard model on lattices ranging in size from ${4}^{3}$ to ${10}^{3}$.

DOI
10.1103/physrevb.36.8632

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We present an algorithm for the numerical simulation of many electron systems, which is based upon the hybrid stochastic approach presently being used to study lattice gauge theory. A method for preconditioning the fermion matrices is described which significantly reduces the computing time needed in the simulation. Results are presented for the three-dimensional Hubbard model on lattices ranging in size from 43{4}^{3}43 to 103{10}^{3}103.
The Hubbard model arises naturally when electron-electron interactions are added to the tight-binding descriptions of many condensed matter systems. For instance, the two-dimensional Hubbard model on the honeycomb lattice is central to the ab initio description of the electronic structure of carbon nanomaterials, such as graphene. Such low-dimensional Hubbard models are advantageously studied with Markov chain Monte Carlo methods, such as hybrid Monte Carlo (HMC). HMC is the standard algorithm of the lattice gauge theory community, as it is well suited to theories of dynamical fermions. As HMC performs continuous, global updates of the lattice degrees of freedom, it provides superior scaling with system size relative to local updating methods. A potential drawback of HMC is its susceptibility to ergodicity problems due to so-called exceptional configurations, for which the fermion operator cannot be inverted. Recently, ergodicity problems were found in some formulations of HMC simulations of the Hubbard model. Here, we address this issue directly and clarify under what conditions ergodicity is maintained or violated in HMC simulations of the Hubbard model. We study different lattice formulations of the fermion operator and provide explicit, representative calculations for small systems, often comparing to exact results. We show that a fermion operator can be found which is both computationally convenient and free of ergodicity problems.
To reduce the rapidly growing computational cost of the dual-fermion lattice calculation with increasing system size, we introduce two embedding schemes. One is the real fermion embedding, and the other is the dual-fermion embedding. Our numerical tests show that the real fermion and dual-fermion embedding approaches converge to essentially the same result. The application on the Anderson disorder and Hubbard models shows that these embedding algorithms converge more quickly with system size as compared to the conventional dual-fermion method, for the calculation of both single- and two-particle quantities.
I discuss algorithms for simulating many-fermion systems via global updatings of auxiliary fields followed by an accept-reject stage which eliminates finite-step-size errors. When the system size is larger than the correlation length, these procedures should require computer time growing only slightly faster than linearly with the system volume V. A corrected Langevin scheme should asymptotically display a V4/3{V}^{4/3}V4/3 behavior, while the hybrid Monte Carlo scheme can behave as V5/4{V}^{5/4}V5/4. I present some tests of the latter algorithm on a simple model of interacting electrons on a two-dimensional lattice.