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arXiv:1306.4996 · Physical Review E

Monte Carlo algorithm for simulating the O (N) loop model on the square lattice

Antônio Márcio P. Silva, Adriaan M. J. Schakel, Giovani Lopes Vasconcelos

Published Aug 5, 2013• 1 citation
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Abstract

An efficient algorithm is presented to simulate the O(N) loop model on the square lattice for arbitrary values of N>0. The scheme combines the worm algorithm with a new data structure to resolve both the problem of loop crossings and the necessity of counting the number of loops at each Monte Carlo update. With the use of this scheme, the line of critical points (and other properties) of the O(N) model on the square lattice for 0<N≤2 have been determined.

DOI
10.1103/physreve.88.021301

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An efficient algorithm is presented to simulate the O(N) loop model on the square lattice for arbitrary values of N>0. The scheme combines the worm algorithm with a new data structure to resolve both the problem of loop crossings and the necessity of counting the number of loops at each Monte Carlo update. With the use of this scheme, the line of critical points (and other properties) of the O(N) model on the square lattice for 0<N≤2 have been determined.
The loop-gas approach to statistical physics provides an alternative, geometrical description of phase transitions in terms of line-like objects. The resulting statistical random-graph ensemble composed of loops and (open) chains can be e_ciently generated by Monte Carlo simulations using the so-called “worm” update algorithm. Concepts from percolation theory and the theory of self-avoiding random walks are used to derive estimators of physical observables that utilize the nature of the worm algorithm. The fractal structure of random loops and chains as well as their scaling properties encode the critical behavior of the statistical system. The general approach is illustrated for the hightemperature series expansion of the Ising model, or O(1) loop model, on a honeycomb lattice, with its known exact results as valuable benchmarks.
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.
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We present a new approach to path integral Monte Carlo (PIMC) simulations based on the worm algorithm, originally developed for lattice models and extended here to continuous-space many-body systems. The scheme allows for efficient computation of thermodynamic properties, including winding numbers and off-diagonal correlations, for systems of much greater size than that accessible to conventional PIMC simulations. As an illustrative application of the method, we simulate the superfluid transition of 4He in two dimensions.
I present a generalization of the Newman-Ziff algorithm, a standard technique in numerical percolation, to the random cluster model. The new method is straightfoward to implement, works for real q>0q>0q>0, and is completely free of the critical slowdown problem. By using the method to sample the critical polynomial, accurate estimates of Potts critical points can be found using relatively small lattice sizes. Furthermore, results for all values of qqq can be found at once within a single simulation. I demonstrate its utility here by computing critical points for non-integer values of qqq on the square lattice to compare with the exact solution, and on the non-planar square matching lattice.