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Physical review. B, Condensed matter

Random-anisotropy-axis magnet with infinite anisotropy

A. Brooks Harris, Robert G. Caflisch, Jayanth R. Banavar

Published Apr 1, 1987• 22 citations

Abstract

We have studied the random-axis magnet with infinite anisotropy by three methods: Cayley-tree approximation, Migdal-Kadanoff renormalization group (MKRG), and Imry-Ma scaling. In the Cayley-tree approximation, by an examination of susceptibilities, it is shown that there exists a competition between the coordination number z and the number of components n of the spins which leads to either ferromagnetic or spin-glass order. Using the MKRG at very low temperature we map out approximately the regimes of the ferromagnetic, spin-glass, and disordered phases as a function of n and the spatial dimension, d. The Imry-Ma arguments are made as an additional method for obtaining information on the critical dimension. Comparisons of these results with the previous literature are made.

DOI: 10.1103/physrevb.35.4929

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We have studied the random-axis magnet with infinite anisotropy by three methods: Cayley-tree approximation, Migdal-Kadanoff renormalization group (MKRG), and Imry-Ma scaling. In the Cayley-tree approximation, by an examination of susceptibilities, it is shown that there exists a competition between the coordination number z and the number of components n of the spins which leads to either ferromagnetic or spin-glass order. Using the MKRG at very low temperature we map out approximately the regimes of the ferromagnetic, spin-glass, and disordered phases as a function of n and the spatial dimension, d. The Imry-Ma arguments are made as an additional method for obtaining information on the critical dimension. Comparisons of these results with the previous literature are made.

The critical behavior of the Random Anisotropy Axis model (RAM) is studied by the Monte Carlo simulation method in the limit of infinite anisotropy. Finite size scaling analysis of the data for equilibrium averages strongly suggests a zero temperature critical point in two dimensions and a finite temperature phase transition in three dimensions. The values of the critical exponents in both two and three dimensions indicate that RAM belongs to the same universality class as that of short-range Ising spin-glass models.

The random anisotropy axis model (RAM) in two dimensions (d=2) is studied in the limit of infinite anisotropy strength using the large-cell renormalisation group method. It is found that the model exhibits a 'zero-temperature phase transition', to a nonferromagnetic ground state, with critical exponents consistent with those of the d=2 Ising spin glass model. It is concluded that the RAM belongs to the universality class of the Ising spin glass, at least for d=2 and probably for all d<or=4.

The influence of a local anisotropy of random orientation on a ferromagnetic phase transition is studied for two cases of anisotropy axis distribution. To this end a model of a random anisotropy magnet is analyzed by means of the field theoretical renormalization group approach in two loop approximation refined by a resummation of the asymptotic series. The one-loop result of Aharony indicating the absence of a second-order phase transition for an isotropic distribution of random anisotropy axis at space dimension

d<4

is corroborated. For a cubic distribution the accessible stable fixed point leads to disordered Ising-like critical exponents.

The large-distance behaviors of the random field and random anisotropy

O(N)

models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg

(N=3)

model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field

O(N)

model the correlation length is found to be finite at the arbitrarily weak disorder for any

N>3.

The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.

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.