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AIP conference proceedings

Improved directed loop method for Quantum Monte Carlo simulations

Fabien Alet

Published Jan 1, 2003• 0 citations
Publisher

Abstract

Efficient schemes called directed loops for cluster updates in Quantum Monte Carlo simulations have recently been proposed, which improve both the efficiency and precision for simulations of quantum models. In this work we show that local detailed balance is not necessary in order to fulfill global detailed balance during the construction of these loops. We therefore propose to insert additional degrees of freedom into the directed loop equations, resulting in even more efficient algorithms. Our approach works directly in the natural representation of an extended space where the matrix elements of the operators defining the worm (broken worldline segments) are taken into account.

DOI
10.1063/1.1632150

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Efficient schemes called directed loops for cluster updates in Quantum Monte Carlo simulations have recently been proposed, which improve both the efficiency and precision for simulations of quantum models. In this work we show that local detailed balance is not necessary in order to fulfill global detailed balance during the construction of these loops. We therefore propose to insert additional degrees of freedom into the directed loop equations, resulting in even more efficient algorithms. Our approach works directly in the natural representation of an extended space where the matrix elements of the operators defining the worm (broken worldline segments) are taken into account.
Efficient quantum Monte Carlo update schemes called directed loops have recently been proposed, which improve the efficiency of simulations of quantum lattice models. We propose to generalize the detailed balance equations at the local level during the loop construction by accounting for the matrix elements of the operators associated with open world-line segments. Using linear programming techniques to solve the generalized equations, we look for optimal construction schemes for directed loops. This also allows for an extension of the directed loop scheme to general lattice models, such as high-spin or bosonic models. The resulting algorithms are bounce free in larger regions of parameter space than the original directed loop algorithm. The generalized directed loop method is applied to the magnetization process of spin chains in order to compare its efficiency to that of previous directed loop schemes. In contrast to general expectations, we find that minimizing bounces alone does not always lead to more efficient algorithms in terms of autocorrelations of physical observables, because of the nonuniqueness of the bounce-free solutions. We therefore propose different general strategies to further minimize autocorrelations, which can be used as supplementary requirements in any directed loop scheme. We show by calculating autocorrelation times for different observables that such strategies indeed lead to improved efficiency; however, we find that the optimal strategy depends not only on the model parameters but also on the observable of interest.
We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.
The directed‐loop scheme is a framework for generalized loop‐type updates in quantum Monte Carlo, applicable both to world‐line and stochastic series expansion methods. Here, the directed‐loop equations, the solution of which gives the probabilities of the various loop‐building steps, are discussed in the context of the anisotropic S = 1/2 Heisenberg model in a uniform magnetic field. This example shows how the directed‐loop concept emerges as a natural generalization of the conventional loop algorithm, where the loops are selfavoiding, to cases where selfintersection must be allowed in order to satisfy detailed balance.
We review the development of update schemes for quantum lattice models simulated using world line quantum Monte Carlo algorithms. Starting from the Suzuki‐Trotter mapping we discuss limitations of local update algorithms and highlight the main developments beyond Metropolis‐style local updates: the development of cluster algorithms, their generalization to continuous time, the worm and directed‐loop algorithms and finally a generalization of the flat histogram method of Wang and Landau to quantum systems.