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Journal of Physics C Solid State Physics

Critical behaviour of Ising magnets: infinitesimal Migdal-Kadanoff approximation

Ôjirô Nagai, M. Toyonaga

Published Jul 10, 1981• 25 citations
Publisher

Abstract

The critical behaviour of Ising magnets is theoretically studied by the use of the infinitesimal Migdal-Kadanoff approximation. The critical temperatures of ferromagnetic slabs, the phase boundary of semi-infinite Ising ferromagnets, the surface thermodynamic function of semi-infinite square lattice and the phase boundary of antiferromagnets in a magnetic field are investigated.

DOI
10.1088/0022-3719/14/19/005

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Journal of Physics Condensed Matter•Dec 18, 1989•13 citations
The critical behaviour of Ising magnets is theoretically studied by the use of the infinitesimal Migdal-Kadanoff approximation. The critical temperatures of ferromagnetic slabs, the phase boundary of semi-infinite Ising ferromagnets, the surface thermodynamic function of semi-infinite square lattice and the phase boundary of antiferromagnets in a magnetic field are investigated.
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.
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.
Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am -y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v -1 , a surface magnetic phase at the bulk critical temperature for y<v -1 , A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v -1 .
Using a recently introduced renormalisation group method, the authors study the behaviour of the spontaneous surface and bulk magnetisations as functions of the temperature for an Ising ferromagnet in a semi-infinite cubic lattice for various J S /J B ratios (J S and J B respectively are the surface and bulk coupling constants). In particular, they study the extraordinary transition where the surface maintains its magnetisation as the bulk disorders: the authors find a discontinuity in the first derivative of the surface magnetisation at the bulk transition temperature. The criticality of the system (universality classes, critical exponents and amplitudes) is discussed as well.