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Journal of Physics A Mathematical and General

The bond-dilutedZ(q) ferromagnetic model: criticality and break-collapse method

Mohamed Loulidi, N. Masaif

Published Feb 21, 1996• 0 citations
Publisher

Abstract

Within a real-space renormalization group (RG) which preserves two-site correlation functions, we study, on the square lattice, the criticality of the bond-diluted Z(q) ferromagnetic model. We generalize the `break-collapse method' which simplifies greatly the exact calculation of arbitrary Z(q) two-terminal clusters (commonly appearing in RG approaches) mainly for a large value of q. We reproduce, in the pure case, several known exact results. The structure of the phase diagrams, for all the values of q, is obtained with a good precision. The massless spin - wave-like phase, which evolves into the Kosterlitz - Touless phase for , occurs around (in agreement with the well known exact result). The structure of the phase diagrams in the diluted case is qualitatively similar to that obtained from the pure model. The massless spin - wave-like phase resists in an interval of concentration p which increases for the large values of q.

DOI
10.1088/0305-4470/29/4/008

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Within a real space renormalisation group (RG) scheme, the authors study the criticality of the ferromagnetic Z(4) model on an anisotropic square lattice. They use an RG cluster which has already proved to be very efficient for the Potts model on the same lattice. The establishment of the RG recurrence relations is greatly simplified through the break-collapse method. The phase diagram (exhibiting ferromagnetic, paramagnetic and nematic-like phases) recovers all the available exact results, and is believed to be of high precision everywhere. If the model is alternatively thought of as being associated with a particular hierarchical lattice rather than with the square lattice, then it is exact everywhere.
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We study the phase diagram and universality classes of the quenched bond-diluted spin-(1/2 Ising ferromagnet in a semi-infinite simple-cubic lattice with a (001) free surface. The approach (herein formulated for arbitrary values of the number of states q of the Potts model) is a real-space renormalization group which preserves two-spin correlation functions and uses clusters which have already proved to be very appropriate for the present structure. We observe that surface ferromagnetism persists below the two-dimensional (2D) percolation threshold pc2D=(1/2{p}_{c}^{2\mathrm{D}=(1/2}pc2D=(1/2​, in fact down to pc{p}_{c}pc​\ensuremath{\simeq}0.42. In addition to that we check that a conjectural extension of the Kasteleyn-Fortuin theorem is satisfied in the present particular case.
A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T , nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.
The phase diagrams of general qqq-state planar spin models on a square lattice are studied for $q=4\ensuremath{-}7$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of Z(q)Z(q)Z(q) models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for q=6and7q=6 \mathrm{and} 7q=6and7 but are inconclusive for q=5q=5q=5. More accurate methods are needed to determine the phase structure of the Z(5)Z(5)Z(5) model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).