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Physical review. B, Condensed matter

Criticality of the bond-diluted Ising ferromagnet in a semi-infinite simple-cubic lattice

L. R. da Silva, Constantino Tsallis, Enaldo F. Sarmento

Published May 1, 1988• 4 citations
Publisher

Abstract

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 ${p}_{c}^{2\mathrm{D}=(1/2}$, in fact down to ${p}_{c}$\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.

DOI
10.1103/physrevb.37.7832

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Physical review. B, Condensed matter•May 1, 1993•3 citations
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.
We use a Migdal-Kadanoff-like renormalization group approach to study the critical behaviour of a semi-infinite simple cubic Ising ferromagnet whose (1, 0, 0) free surface contains anisotropic (in spin space) Heisenberg ferromagnetic interactions. The phase diagram presents three phases (namely the paramagnetic, the bulk ferromagnetic and the surface ferromagnetic ones) which join on a multicritical point The location of this point is calculated as a function of the anisotropy. The various universality classes of the problem are exhibited.
Within a real-space renormalisation group framework, the authors discuss the criticality of a system constituted by two (not necessarily equal) semi-infinite ferromagnetic q-state Potts bulks separated by an interface. This interface is a bond-diluted Potts ferromagnet with a coupling constant which is in general different from those of both bulks. The phase diagram presents four physically different phases, namely the paramagnetic one, and the surface, single-bulk and double-bulk ferromagnetic ones. These various phases determine a multicritical surface which contains a higher-order multicritical line. Particular attention is devoted to the discussion of the critical concentration p c . Here, p c is the concentration of the interface bonds above which surface magnetic ordering is possible even if the bulks are disordered. An interesting feature comes out which is that p c varies continuously with J 1 /J s and J 2 /J s . The standard two-dimensional percolation concentration is recovered for J 1 =J 2 =0. From the analysis of the various fixed points obtained within the present formalism, a very rich set of critical universality classes emerges.
Within a real space renormalisation group framework which uses rather sophisticated clusters, the authors discuss the phase diagram and the universality classes of a semi-infinite cubic-lattice q-state Potts ferromagnet. In particular, they study the influence, on the surface magnetism, of q and Delta identical to J S /J B -1 (where J S and J B are respectively the surface and bulk coupling constants). The exact d=2 critical temperature T c 2D is recovered for all values of q. The q-dependence of the value Delta c above which surface magnetic order can exist even if the bulk is disordered, is calculated and, through a convenient extrapolation, reliable results are obtained (for the Ising particular case, i.e. q=2, they obtain the extrapolated value Delta c =0.569, which compares satisfactorily with the series result 0.6+or-0.1 and the Monte Carlo one, 0.50+or-0.03). At the surface-bulk (SB) multicritical point they calculate the q-dependences of the critical amplitude A and the crossover exponent phi (defined in the Delta to Delta c +0 limit through (T c S ( Delta )/T c 3D -1) approximately A( Delta / Delta c -1) 1 phi /, where T c S ( Delta ) and T c 3D identical to T c S ( Delta c ) respectively are the surface and the bulk critical temperatures), as well as the critical exponent y t1 SB . For the Ising particular case they obtain phi equivalent to 0.641 (which compares satisfactorily with the epsilon -expansion (0.68) and the Monte Carlo (0.56+or-0.04) results), A equivalent to 0.4 and 1/y t1 SB equivalent to 1.623. (As far as we know, no series or Monte Carlo results are available in the literature for A or 1/y t1 SB .).
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.
A semi-infinite Ising model on a simple-cubic lattice, bounded by two perpendicular planes which meet at an edge, is studied with the Migdal-Kadanoff renormalization-group approach. It is found that the edge spins can order at a temperature higher than the bulk transition temperature if one of the surfaces is also ordered, whereas no edge ordering is possible if none of the surfaces nor the bulk are ordered. A complete phase diagram is presented in the parameter space of the model considered. Correlation-length critical exponents \ensuremath{\nu} and the crossover exponent \ensuremath{\varphi} are calculated and various universality classes are established.