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Physical review. B, Condensed matter

Percolation renormalization-group approach to the hard-square model

Chin‐Kun Hu, Chi-Ning Chen

Published Mar 1, 1991• 11 citations
Publisher

Abstract

Based on the definition of clusters for the lattice hard-core particles considered by Hu and Mak, we use a percolation renormalization-group method with various cell sizes to calculate the correlation-length exponent \ensuremath{\nu} for the hard-square model. The results suggest that the hard-square model is in the universality class of the simple Ising model.

DOI
10.1103/physrevb.43.6184

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Physical review. B, Condensed matter•Jun 1, 1984•6 citations
Based on the definition of clusters for the lattice hard-core particles considered by Hu and Mak, we use a percolation renormalization-group method with various cell sizes to calculate the correlation-length exponent \ensuremath{\nu} for the hard-square model. The results suggest that the hard-square model is in the universality class of the simple Ising model.
We present a renormalization-group approach for treating the percolation properties of the nearest-neighbor triangular Ising model. We obtain exponents for the line of percolation transitions ${T}_{c}<~T<~\ensuremath{\infty}$. In particular, we find a possibly exact result for the connectedness-length exponent ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}$ in the "pure" percolation ($T=\ensuremath{\infty}$) limit, which holds for ${T}_{c}<T<~\ensuremath{\infty}$. At T=TcT={T}_{c}T=Tc​ we find the connectedness-length exponents of the percolation problem to be identical to the correlation-length exponents for the thermal problem.
A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.
The correlation length of the Ising model on the simple cubic lattice is calculated directly from the magnetization profiles which are determined in the pair, the square, and the cube approximations in the simplified cluster variation method. The critical exponent ν is obtained by applying the coherent anomaly method to the correlation length. The wave-number dependent magnetic susceptibility or the spin-pair correlation function is not required in obtaining the expression for the correlation length.
We study the percolation properties of geometrical clusters defined in the overlap space of two statistically independent replicas of a square-lattice Ising model that are simulated at the same temperature. In particular, we consider two distinct types of clusters in the overlap, which we dub soft- and hard-constraint clusters, and which are subsets of the regions of constant spin overlap. By means of Monte Carlo simulations and a finite-size scaling analysis we estimate the transition temperature as well as the set of critical exponents characterizing the percolation transitions undergone by these two cluster types. The results suggest that both soft- and hard-constraint clusters percolate at the critical temperature of the Ising model and their critical behavior is governed by the correlation-length exponent ν=1 found by Onsager. At the same time, they exhibit nonstandard and distinct sets of exponents for the average cluster size and percolation strength.
The large-cell Monte Carlo renormalization group introduced by Reynolds, Stanley, and Klein [Phys. Rev. B 20, 1223 (1980)] is applied with a new weight function to random percolation in two and three dimensions. Our renormalization-group results show different behavior with lattice size than that observed previously. Specifically, we find that ${Y}_{p}=0.73\ifmmode\pm\else\textpm\fi{}0.01$ (the correlation length exponent $\ensuremath{\nu}=1.37$) in two dimensions and ${Y}_{p}=1.124\ifmmode\pm\else\textpm\fi{}0.004$ ($\ensuremath{\nu}=0.89$) in three dimensions. This weight function allows for easy analysis of percolation in higher dimensions.