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Physical review. B, Condensed matter

Ergodicity at large couplings with the determinant Monte Carlo algorithm

Richard T. Scalettar, R. M. Noack, Rajiv R. P. Singh

Published Nov 15, 1991• 65 citations
Publisher

Abstract

It is known that ergodicity problems are encountered in simulating the Hubbard model at large ratios of the Coulomb interaction strength U to the hopping matrix element t using the ``determinant Monte Carlo'' algorithm with single-spin-flip moves. We discuss the potential barriers that cause this difficulty in sampling, and introduce global moves of a line of Hubbard-Stratonovich variables on a single site and all time slices to equilibrate the system. We show that at half filling these moves allow the system to explore fully the singly occupied subspace without having to move out of it, in contrast to the single-spin-flip moves of the Hubbard-Stratonovich variables where the system must pass through configurations with appreciable weight for doubly occupied or empty sites in order to go from one state in the singly occupied sector to another. We use these global moves to obtain the antiferromagnetic structure factor at large U/t.

DOI
10.1103/physrevb.44.10502

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It is known that ergodicity problems are encountered in simulating the Hubbard model at large ratios of the Coulomb interaction strength U to the hopping matrix element t using the ``determinant Monte Carlo'' algorithm with single-spin-flip moves. We discuss the potential barriers that cause this difficulty in sampling, and introduce global moves of a line of Hubbard-Stratonovich variables on a single site and all time slices to equilibrate the system. We show that at half filling these moves allow the system to explore fully the singly occupied subspace without having to move out of it, in contrast to the single-spin-flip moves of the Hubbard-Stratonovich variables where the system must pass through configurations with appreciable weight for doubly occupied or empty sites in order to go from one state in the singly occupied sector to another. We use these global moves to obtain the antiferromagnetic structure factor at large U/t.
A sensible application of the hybrid Monte Carlo (HMC) method is often hindered by the presence of large—or even infinite—potential barriers. These potential barriers separate the configuration space into distinct sectors and can lead to ergodicity violations that bias measurements. In this work, we address this problem by augmenting HMC with a multiplicative Metropolis-Hastings update in a “radial direction” of the fields which enables crossing the potential barriers and ensures ergodicity of the sampling algorithm at comparably low computational cost. We demonstrate the algorithm on a simple toy model and show how it can be applied to the fermionic Hubbard model describing physics ranging from an exactly solvable two-site system to the C 20 H 12 perylene molecule. Our numerical results show that the radial updates successfully remove ergodicity violations, while simultaneously reducing autocorrelation times.
Two cluster algorithms, based on constructing and flipping loops, are presented for world-line quantum Monte Carlo simulations of fermions and are tested on the one-dimensional repulsive Hubbard model. We call these algorithms the loop-flip and loop-exchange algorithms. For these two algorithms and the standard world-line algorithm, we calculated the autocorrelation times for various physical quantities and found that the ordinary world-line algorithm, which uses only local moves, suffers from very long correlation times that makes not only the estimate of the error difficult but also the estimate of the average values themselves difficult. These difficulties are especially severe in the low-temperature, large-U regime. In contrast, we find that new algorithms, when used alone or in combinations with themselves and the standard algorithm, can have significantly smaller autocorrelation times, in some cases being smaller by three orders of magnitude. The new algorithms, which use nonlocal moves, are discussed from the point of view of a general prescription for developing cluster algorithms. The loop-flip algorithm is also shown to be ergodic and to belong to the grand canonical ensemble. Extensions to other models and higher dimensions are briefly discussed.
Monte Carlo methods for the Hubbard model rely on a Hubbard-Stratonovich (HS) decomposition (auxiliary field method) to perform importance sampling on classical variables. Freedom in the choice of the local HS fields can be formally seen as a gauge choice. While the choice of gauge does not influence observable quantities, it may influence intermediate quantities in the calculation, such as the famous “fermion sign”, and it may also influence the efficiency with which the algorithm explores phase space. The effect of arbitrary gauge choices on both aspects of the algorithm are investigated. It is found that in the single spin-flip determinantal approach, certain gauges lead to a better exploration of phase space. This improvement is demonstrated, in the intermediate coupling regime, by histograms which for the first time show the behavior expected from grand canonical simulations. It is also found that the improved phase space exploration can in practice offset the apparent disadvantage of a smaller fermion sign.
We have performed numerical studies of the Hubbard-Holstein model in two dimensions using determinant quantum Monte Carlo (DQMC). Here, we present details of the method, emphasizing the treatment of the lattice degrees of freedom, and then study the filling and behavior of the fermion sign as a function of model parameters. We find a region of parameter space with large Holstein coupling where the fermion sign recovers despite large values of the Hubbard interaction. This indicates that studies of correlated polarons at finite carrier concentrations are likely accessible to DQMC simulations. We then restrict ourselves to the half-filled model and examine the evolution of the antiferromagnetic structure factor, other metrics for antiferromagnetic and charge-density-wave order, and energetics of the electronic and lattice degrees of freedom as a function of electron-phonon coupling. From this we find further evidence for a competition between charge-density-wave and antiferromagnetic orders at half-filling.
We have studied the effects of random-hopping matrix elements and random potentials on the properties of the one-dimensional Hubbard model. Using a quantum Monte Carlo technique, disorder-averaged static spin- and charge-density susceptiblities have been evaluated for various strengths of the disorder. Results for the spin susceptibility at wave number q=2kF{\mathit{k}}_{\mathit{F}}kF​ indicate that this quantity, which is the fastest diverging susceptibility of the pure system, diverges as T\ensuremath{\rightarrow}0 also when there is randomness in the hopping matrix elements, but not in the presence of random potentials. Both types of disorder cause a divergence of the uniform magnetic susceptibility. However, for random potentials a finite critical strength of the disorder appears to be required. At half-filling the transition from Mott (gapped) to Anderson (gapless) insulating behavior has been studied. A critical disorder strength is needed to destroy the gap, in agreement with Ma's renormalization group calculations.