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Cluster algorithms with empahsis on quantum spin systems

James E. Gubernatis, Naoki Kawashima

Published Oct 6, 1995• 0 citations
OpenAlex

Abstract

The purpose of this lecture is to discuss in detail the generalized approach of Kawashima and Gubernatis for the construction of cluster algorithms. We first present a brief refresher on the Monte Carlo method, describe the Swendsen-Wang algorithm, show how this algorithm follows from the Fortuin-Kastelyn transformation, and re=interpret this transformation in a form which is the basis of the generalized approach. We then derive the essential equations of the generalized approach. This derivation is remarkably simple if done from the viewpoint of probability theory, and the essential assumptions will be clearly stated. These assumptions are implicit in all useful cluster algorithms of which we are aware. They lead to a quite different perspective on cluster algorithms than found in the seminal works and in Ising model applications. Next, we illustrate how the generalized approach leads to a cluster algorithm for world-line quantum Monte Carlo simulations of Heisenberg models with S = 1/2. More succinctly, we also discuss the generalization of the Fortuin- Kasetelyn transformation to higher spin models and illustrate the essential steps for a S = 1 Heisenberg model. Finally, we summarize how to go beyond S = 1 to a general spin, XYZ model.

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The purpose of this lecture is to discuss in detail the generalized approach of Kawashima and Gubernatis for the construction of cluster algorithms. We first present a brief refresher on the Monte Carlo method, describe the Swendsen-Wang algorithm, show how this algorithm follows from the Fortuin-Kastelyn transformation, and re=interpret this transformation in a form which is the basis of the generalized approach. We then derive the essential equations of the generalized approach. This derivation is remarkably simple if done from the viewpoint of probability theory, and the essential assumptions will be clearly stated. These assumptions are implicit in all useful cluster algorithms of which we are aware. They lead to a quite different perspective on cluster algorithms than found in the seminal works and in Ising model applications. Next, we illustrate how the generalized approach leads to a cluster algorithm for world-line quantum Monte Carlo simulations of Heisenberg models with S = 1/2. More succinctly, we also discuss the generalization of the Fortuin- Kasetelyn transformation to higher spin models and illustrate the essential steps for a S = 1 Heisenberg model. Finally, we summarize how to go beyond S = 1 to a general spin, XYZ model.
We discuss the development of cluster algorithms from the viewpoint of probability theory and not from the usual viewpoint of a particular model. From the viewpoint of probability theory, we detail the nature of a cluster algorithm, make explicit the assumptions embodied in all clusters of which we are aware, and define the construction of free-cluster algorithms. We also illustrate these procedures by rederiving the Swendsen-Wang algorithm, presenting the details of the loop algorithm for a worldline simulation of a quantum S=1/2 model, and proposing a free-cluster version of the Swendsen-Wang replica method for the random Ising model. How the principle of maximum entropy might be used to aid the construction of cluster algorithms is also discussed.
This talk will review the history of both cluster and histogram methods for Monte Carlo simulations. Cluster methods are based on the famous exact mapping by Fortuin and Kasteleyn from general Potts models onto a percolation representation. I will discuss the Swendsen‐Wang algorithm, as well as its improvement and extension to more general spin models by Wolff. The Replica Monte Carlo method further extended cluster simulations to deal with frustrated systems. The history of histograms is quite extensive, and can only be summarized briefly in this talk. It goes back at least to work by Salsburg et al. in 1959. Since then, it has been forgotten and rediscovered several times. The modern use of the method has exploited its ability to efficiently determine the location and height of peaks in various quantities, which is of prime importance in the analysis of critical phenomena. The extensions of this approach to the multiple histogram method and multicanonical ensembles have allowed information to be obtained over a broad range of parameters. Histogram simulations and analyses have become standard techniques in Monte Carlo simulations.
These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S=1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
The recent generalization of the Fortuin-Kasteleyn percolation representation and the Swendsen-Wang (SW) cluster algorithm is reviewed. The new representation yields cluster algorithm for the XYZ model with arbitrary magnitude of spins, with arbitrary anisotropies, and in any dimensions. The new algorithm often reduces computational autocorrelation times by several orders of magnitude. By using the new algorithm together with methods of Bayesian statistical inference, we obtained the dynamical structure factor S (k, ω) for the S =1/2 XY model in two dimensions.
Featuring detailed explanations of the major algorithms used in quantum Monte Carlo simulations, this is the first textbook of its kind to provide a pedagogical overview of the field and its applications. The book provides a comprehensive introduction to the Monte Carlo method, its use, and its foundations, and examines algorithms for the simulation of quantum many-body lattice problems at finite and zero temperature. These algorithms include continuous-time loop and cluster algorithms for quantum spins, determinant methods for simulating fermions, power methods for computing ground and excited states, and the variational Monte Carlo method. Also discussed are continuous-time algorithms for quantum impurity models and their use within dynamical mean-field theory, along with algorithms for analytically continuing imaginary-time quantum Monte Carlo data. The parallelization of Monte Carlo simulations is also addressed. This is an essential resource for graduate students, teachers, and researchers interested in quantum Monte Carlo techniques.
A cluster algorithm is presented for the simulation of the q-state Potts models in which the number of spins is conserved in each state. The algorithm constructs Fortuin-Kasteleyn cluster configurations from spin configurations, in a way identical to the Swendsen-Wang algorithm; the spin assignment to these clusters is, however, different, and conserves the number of spins for each state. Compared to traditional nonlocal spin-exchange algorithms, the cluster algorithm presented here suffers less from critical slowing down, and consequently is more efficient near the critical temperature.