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1 The Loop-Cluster Algorithm for the Case of the 6 Vertex Model ∗

Hans Gerd Evertz, Mihai Marcu B

Published Jan 1, 1992• 2 citations
OpenAlex

Abstract

We present the loop algorithm, a new type of cluster algorithm that we recently introduced for the F model. Using the framework of Kandel and Domany, we show how to generalize the algorithm to the arrow flip symmetric 6 vertex model. We propose the principle of least possible freezing as the guide to choosing the values of free parameters in the algorithm. Finally, we briefly discuss the application of our algorithm to simulations of quantum spin systems. In particular, all necessary information is provided for the simulation of spin 1 2 models. 1.

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We present an algorithmic framework for a variant of the quantum Monte Carlo operator-loop algorithm, where nonlocal cluster updates are constructed in a way that makes each individual loop smaller. The algorithm is designed to increase simulation efficiency in cases where conventional loops become very large, do not close altogether, or otherwise behave poorly. We demonstrate and characterize some aspects of the short loop on a square lattice spin-1/2 XXZ model where, remarkably, a significant increase in simulation efficiency is observed in some parameter regimes. The simplicity of the model provides a prototype for the use of short loops on more complicated quantum systems.

arXiv:0709.1718•Physical Review E•Mar 25, 2008•2 citations
We present the loop algorithm, a new type of cluster algorithm that we recently introduced for the F model. Using the framework of Kandel and Domany, we show how to generalize the algorithm to the arrow flip symmetric 6 vertex model. We propose the principle of least possible freezing as the guide to choosing the values of free parameters in the algorithm. Finally, we briefly discuss the application of our algorithm to simulations of quantum spin systems. In particular, all necessary information is provided for the simulation of spin 1 2 models. 1.
We present a new type of cluster algorithm that strongly reduces critical slowing down in simulations of vertex models. Since the clusters are closed paths of bonds, we call it the loop algorithm. The basic steps in constructing a cluster are the breakup and the freezing of vertices. We concentrate on the case of the F model, which is a subset of the six-vertex model exhibiting a Kosterlitz-Thouless transition. The loop algorithm is also applicable to simulations of other vertex models and of one- and two-dimensional quantum spin systems.
Recently, Syljuåsen and Sandvik [Phys. Rev. E. (to be published)] proposed a new framework for constructing algorithms of quantum Monte Carlo simulation. While it includes new classes of powerful algorithms, it is not straightforward to find an efficient algorithm for a given model. Based on their framework, we propose an algorithm that is a natural extension of the conventional loop algorithm with the split-spin representation. A complete table of the vertex density and the worm-scattering probability is presented for the general XXZ model of an arbitrary S with a uniform magnetic field.
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.
We show how the directed-loop Monte Carlo algorithm can be applied to study vertex models. The algorithm is employed to calculate the arrow polarization in the six-vertex model with the domain wall boundary conditions. The model exhibits spatially separated ordered and "disordered" regions. We show how the boundary between these regions depends on parameters of the model. We give some predictions on the behavior of the polarization in the thermodynamic limit and discuss the relation to the Arctic Circle theorem.
We present an algorithmic framework for a variant of the quantum Monte Carlo operator-loop algorithm, where nonlocal cluster updates are constructed in a way that makes each individual loop smaller. The algorithm is designed to increase simulation efficiency in cases where conventional loops become very large, do not close altogether, or otherwise behave poorly. We demonstrate and characterize some aspects of the short loop on a square lattice spin-1/2 XXZ model where, remarkably, a significant increase in simulation efficiency is observed in some parameter regimes. The simplicity of the model provides a prototype for the use of short loops on more complicated quantum systems.