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arXiv:hep-lat/9310025 · Physics Letters B

Ising model universality for two-dimensional lattices

Wolfhard Janke E, Mohammad W. Katoot, Ramon Villanova

Published Oct 1, 1993• 38 citations
arXivHTMLPDF
DOI
10.1016/0370-2693(93)91633-x

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In this work, we evaluate the probability distribution function of the order parameter for the two-dimensional site-diluted Ising model. Extensive Monte Carlo simulations have been performed for different spin concentrations p (0.70<or=p<or=1) on square lattices with linear sizes L (20<or=L<or=100). The results indicate that the universality class of the diluted Ising model seems to be independent of the amount of dilution. Logarithmic corrections of the finite-size critical temperature behavior of the model can also be inferred even for such small lattices.
The loop model is an important model of statistical mechanics and has been extensively studied in two-dimensional lattices. However, it is still difficult to simulate the loop model directly in three-dimensional lattices, especially in lattices with coordination numbers larger than 3. In this paper, a cluster weight Ising model is proposed by introducing an additional cluster weight n in the partition function of the traditional Ising model. This model is equivalent to the loop model on the two-dimensional lattice, but on the three-dimensional lattice, it is still not very clear whether or not these models have the same universality. By using a Monte Carlo method with cluster updates and color assignment, we obtain the global phase diagram containing the paramagnetic and ferromagnetic phases. The phase transition between the two phases is second order at 1≤n<n_{cri} and first order at n≥n_{cri}, where n_{cri}≈2. The thermal exponent y_{t} is equal to the system dimension d when the first-order transition occurs. For the second-order transitions, the numerical estimation of y_{t} and the magnetic exponent y_{m}, shows that the universalities of the two models on the three-dimensional lattice are different. Our results are helpful in the understanding of some traditional statistical mechanics models.
We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.