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Physical review. B, Condensed matter

Cluster Monte Carlo dynamics for the fully frustrated Ising model

Daniel Kandel, Radel Ben-Av, Eytan Domany

Published Mar 1, 1992• 33 citations
Publisher

Abstract

We have developed a cluster algorithm for Monte Carlo simulations of the fully frustrated Ising model on the square lattice. The method does not suffer from problems of metastability, and is extremely efficient even at T=0, as we demonstrated previously. Here we describe results of more extensive simulations of the model at T=0 which allow us to extract the effective dynamical critical exponent of the algorithm, z\ensuremath{\approxeq}0.55, as well as the static exponent \ensuremath{\eta}=0.5. We also provide an argument that explains why our method works.

DOI
10.1103/physrevb.45.4700

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We have developed a cluster algorithm for Monte Carlo simulations of the fully frustrated Ising model on the square lattice. The method does not suffer from problems of metastability, and is extremely efficient even at T=0, as we demonstrated previously. Here we describe results of more extensive simulations of the model at T=0 which allow us to extract the effective dynamical critical exponent of the algorithm, z\ensuremath{\approxeq}0.55, as well as the static exponent \ensuremath{\eta}=0.5. We also provide an argument that explains why our method works.
The critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice has been studied using the short-time dynamics method. Particles with the periodic boundary conditions containing N = 262144 spins have been studied. Calculations have been performed by the standard Metropolis Monte Carlo algorithm. The static critical exponents of the magnetization and correlation radius have been obtained. The dynamic critical exponent of the model under study has been calculated.
Critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice is studied by the short-time dynamics method. Cubic systems with periodic boundary conditions and linear sizes of L = 32, 64, 96, and 128 in each crystallographic direction are studied. Calculations were carried out by the Monte Carlo method using the standard Metropolis algorithm. The static critical exponents for the magnetization and correlation radius and the dynamic critical exponents are calculated.
We study the critical behavior of the fully frustrated XY\mathrm{XY}XY model on a square lattice by means of extensive histogram Monte Carlo simulations. We find a single transition in contrast to the scenario of two distinct transitions. From a finite-size scaling we determine the critical exponents associated with the Ising-like order parameter and find ${T}_{c}(\ensuremath{\infty})=0.4552(2)$, $\ensuremath{\nu}=0.852(2),$ and $\ensuremath{\gamma}=1.531(3)$. These exponents, different from the pure two-dimensional Ising critical exponents, are in agreement with the ones obtained for a mixed XY\mathrm{XY}XY Ising system.
Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).