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 simple real space renormalization group scheme, we study the phase diagram of the d = 3 semi-infinite anisotropic Heisenberg ferromagnet with surface (bulk) coupling constant J s ( J B ) and anisotropy η s (η B ). We exhibit, in particular, the interesting effects of η s and η B on the (multicritical) ratio J s / J B above which surface order becomes observable even if bulk order is absent. Enhancement occurs for an Ising-like free surface (η s ≃ 1) on top of an isotropic-Heisenberg-like bulk (η B ≃ 0).
Within a real-space renormalisation group framework, the authors treat the q-state Potts ferromagnet in a simple cubic lattice consisting of two different semi-infinite bulks (respectively characterised by the coupling constants J1and J2) separated by a (1,0,0) interface (characterised by Js). The use of a rather sophisticated two-terminal cluster allows a quite reliable discussion of the phase diagram and its universality classes. Four physically different phases are obtained, namely the paramagnetic and the double-bulk, single-bulk and surface ferromagnetic ones. The critical surface contains a multicritical line (associated with J1not=J2) which in turn contains a special point associated with J1=J2.
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.
We study the anisotropic Heisenberg spin-glass model in a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction (
Jij), Gaussian and uniform, are used, with zero mean and width
Jˉ. The
(kT/Jˉ)×Δ0 phase diagram is obtained, where
T is the temperature,
Δ0 is the first moment of the probability distribution for the uniaxial anisotropy, and
k is the Boltzmann constant. For the Ising model (
Δ0=1), there is a spin-glass phase at low temperatures (high
Jˉ) and a paramagnetic phase at high temperatures (low
Jˉ). For the isotropic Heisenberg model (
Δ0=0), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with
Δ0, as expected, but goes to zero at a finite value of the anisotropy parameter, namely
Δ0=Δc∼0.59. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for
Δc≤Δ0<1, belongs to the same universality class as the transition for the Ising spin glass.