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World-Line Monte Carlo Methods

Naoki Kawashima, Kenji Harada

Published Dec 29, 2003• 0 citations
OpenAlex

Abstract

World-line quantum Monte Carlo methods are reviewed with an emphasis on breakthroughs made in recent years. In particular, three algorithms -- the loop algorithm, the worm algorithm, and the directed-loop algorithm -- for updating world-line configurations are presented in a unified perspective. Detailed descriptions of the algorithms in specific cases are also given.

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arXiv:cond-mat/9903375•Journal of the Physical Society of Japan•Aug 15, 1999•9 citations
World-line quantum Monte Carlo methods are reviewed with an emphasis on breakthroughs made in recent years. In particular, three algorithms -- the loop algorithm, the worm algorithm, and the directed-loop algorithm -- for updating world-line configurations are presented in a unified perspective. Detailed descriptions of the algorithms in specific cases are also given.
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.
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.
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.
A quantum Monte Carlo method combining update of the loop algorithm with the global flip of the world line is proposed as an efficient method to study the magnetization process in an external field, which has been difficult because of inefficiency of the update of the total magnetization. The method is demonstrated in the one dimensional antiferromagnetic Heisenberg model and the trimer model. We attempted various other Monte Carlo algorithms to study systems in the external field and compared their efficiency.