Two different definitions of cluster-size heterogeneity are investigated as well as correlation time of different quantities using the Metropolis algorithm and the Wolff algorithm. It is confirmed that the correlation time multiplied by the computation time is lower for the Wolff algorithm in an area around the critical temperature. It is also confirmed that one definition of the heterogeneity has a local maximum at the critical temperature where as the other has an abrupt change in derivative. The local maximum appears with L ≥ 64 and it is predicted but not verified that systems with L > 43 have such a maximum. The relationship between the number of distinct cluster sizes for clusters with spin-up and spin-down is investigated and it is observed that these transition from being significantly different at lower temperatures to being mostly similar at higher temperatures. The point of transition appears to be near the critical temperature.
Two different definitions of cluster-size heterogeneity are investigated as well as correlation time of different quantities using the Metropolis algorithm and the Wolff algorithm. It is confirmed that the correlation time multiplied by the computation time is lower for the Wolff algorithm in an area around the critical temperature. It is also confirmed that one definition of the heterogeneity has a local maximum at the critical temperature where as the other has an abrupt change in derivative. The local maximum appears with L ≥ 64 and it is predicted but not verified that systems with L > 43 have such a maximum. The relationship between the number of distinct cluster sizes for clusters with spin-up and spin-down is investigated and it is observed that these transition from being significantly different at lower temperatures to being mostly similar at higher temperatures. The point of transition appears to be near the critical temperature.
In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.
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).
A measure of cluster size heterogeneity (H), introduced by Lee et al. [Phys. Rev. E 84, 020101 (2011)] in the context of explosive percolation, was recently applied to random percolation and to domains of parallel spins in the Ising and Potts models. It is defined as the average number of different domain sizes in a given configuration and a new exponent was introduced to explain its scaling with the size of the system. In thermal spin models, however, physical clusters take into account the temperature-dependent correlation between neighboring spins and encode the critical properties of the phase transition. We here extend the measure of H to these clusters and, moreover, present new results for the geometric domains for both d=2 and 3. We show that the heterogeneity associated with geometric domains has a previously unnoticed double peak, thus being able to detect both the thermal and percolative transitions. An alternative interpretation for the scaling of H that does not introduce a new exponent is also proposed.
I simulated the two-dimensional Ising model at its exact critical temperature with two Markov chain Monte Carlo methods, the single-spin Metropolis algorithm and the Wolff cluster algorithm, on periodic L×L lattices from L = 8 to L = 64. For each method I measured the integrated autocorrelation time of the energy and of the absolute magnetization, using four independent runs per lattice size. Both methods reproduce the exact finite-lattice energy within statistical error. For Metropolis, the autocorrelation time of |M| grows as L^z with z = 2.11(3) for L ≥ 16 and 2.16(5) for L ≥ 24, in line with the accepted value z ≈ 2.17 for local dynamics. For Wolff, measured in sweep-equivalent units, I get z = 0.29(1) from the energy and 0.15(1) from |M| for L ≥ 24, within the range of published values. In CPU time per statistically independent sample of |M|, Wolff was 5 times faster at L = 8 and 275 times faster at L = 64, and the gap grows roughly as L^2. All code and raw results are available, and the full study runs in under an hour on two CPU cores.
We calculate the dynamic critical exponent for the Niedermayer algorithm applied to the two-dimensional Ising and XY models, for various values of the free parameter E 0 . For E 0 = − 1 we regain the Metropolis algorithm and for E 0 = 1 we regain the Wolff algorithm. For − 1 < E 0 < 1, we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, L , but eventually saturates at a given lattice size , which depends on E 0 . For , the Niedermayer algorithm is equivalent to the Metropolis one, i.e., they have the same dynamic exponent. For E 0 > 1, the autocorrelation time is always greater than for E 0 = 1 (Wolff) and, more important, it also grows faster than a power of L . Therefore, we show that the best choice of cluster algorithm is the Wolff one, when comparing against the Niedermayer generalization. We also obtain the dynamic behavior of the Wolff algorithm: although not conclusively, we propose a scaling law for the dependence of the autocorrelation time on L .
Phase transitions and critical phenomena in several spin models are studied. These models include a two and three dimensional XY model and Ising models in structured and random field. The methods of study are cluster Monte Carlo simulations and network flow optimization algorithms. The XY model and a related soft spin model belonging to the O(2) universality class are studied using a modified version of the Invaded Cluster algorithm. The static critical properties of the models and the dynamic characteristics of the algorithm are studied. The critical exponent η for the O(2) universality class in two and three dimensions is measured. The Invaded Cluster algorithm for the XY model does not show any critical slowing for global variables. Benzene adsorption in a nanoporous solid is modeled with an Ising model with both internal and external magnetic fields. The first order phase transition and the critical properties of the system are studied. The system is simulated using the two replica cluster algorithm. The critical temperature of the system is found to decrease with increasing difference of the values of the internal fields. It is found that the system undergoes a phase transition for a relatively wide, experimentally obtainable, region of values of its parameters. The random field Ising model in three dimensions is studied at zero temperature. The nature of randomness induced phase transition is studied using a numerical network flow optimization method. It is found that the latent heat at the phase transition vanishes in the limit of infinite system size which can be taken as a proof of a second order phase transition.