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Journal of Physics C Solid State Physics

Criticality of the D=2 bond-dilute anisotropic Heisenberg ferromagnet

Ananias M. Mariz, Constantino Tsallis, A. O. Caride

Published Jul 30, 1985• 28 citations
Publisher

Abstract

The authors discuss the critical frontier and critical exponents associated with the quenched bond-dilute quantum anisotropic spin- 1 / 2 Heisenberg ferromagnet in square lattice. To perform the calculations, they extend (and analyse in some detail) an approximate real-space renormalisation group framework recently developed by some of the authors for the pure model. Whenever comparison with available exact results is possible, the agreement is either perfect or quite satisfactory. Some effort has been dedicated to the extraction of the main asymptotic behaviour of the critical frontier. Also several interesting quantum effects appearing in the composition laws of (Heisenberg) bond arrays are exhibited.

DOI
10.1088/0022-3719/18/21/014

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The authors discuss the critical frontier and critical exponents associated with the quenched bond-dilute quantum anisotropic spin- 1 / 2 Heisenberg ferromagnet in square lattice. To perform the calculations, they extend (and analyse in some detail) an approximate real-space renormalisation group framework recently developed by some of the authors for the pure model. Whenever comparison with available exact results is possible, the agreement is either perfect or quite satisfactory. Some effort has been dedicated to the extraction of the main asymptotic behaviour of the critical frontier. Also several interesting quantum effects appearing in the composition laws of (Heisenberg) bond arrays are exhibited.
We consider the square-lattice spin-(1/2) anisotropic Heisenberg ferromagnet with interactions whose symmetry can independently (quenched model) and randomly be of two competing types: namely, the isotropic Heisenberg type and the Ising type. Within a real-space renormalization-group framework, we perform a quite precise numerical calculation of the critical frontier, and establish its main asymptotic behaviors. We also characterize the relevant universality classes through the analysis of the correlation-length critical exponent.
A simple modification of the real-space renormalisation group method is used to discuss critical behaviour of site diluted spin- 1 / 2 Ising and Heisenberg ferromagnets in two and three dimensions. The critical parameters and critical exponents are evaluated, and are shown to be in good agreement with the exact results and series calculations for site-diluted Ising ferromagnets on the square and triangular lattices. Good values of the critical parameters have also been obtained for the site-diluted Heisenberg and Ising models in two and three dimensions using the Migdal-Kadanoff approximation (1976).
Within a real space renormalisation group framework (12 different procedures, all of them using star-triangle and duality-type transformations) the authors calculate accurate approximations for the critical frontiers associated with the quenched bond-diluted first-neighbour, spin-1/2Ising ferromagnet on triangular and honeycomb lattices. All of them provide, in both pure bond percolation and pure Ising limits, the exact critical points and exact or almost exact derivatives in the p-t space (p is the bond-independent occupancy probability and t identical to tanh(J/kBT)). The authors best numerical proposals lead to the exact derivative in the pure percolation limit (p=pc) and, in what concerns the pure Ising limit (p=1) derivative, to a 0.15% error for the triangular lattice and to a 0.96% error for the honeycomb one; in the intermediate region (pc<p<1), where the exact critical frontiers are still unknown, the worst error in the variable t (for fixed p) is estimated to be less than 0.27% for the triangular lattice and 0.14% for the honeycomb one.
Real space renormalisation group scaling techniques are used to investigate the static critical behaviour of the pure and dilute, classical, anisotropic Heisenberg model. Transfer matrix methods are employed to obtain asymptotically exact expressions for the correlation lengths and susceptibilities of the one-dimensional system. The resulting scaling relationships are combined with an approximate bond moving scheme to treat pure and dilute models in higher dimensionalities. Detailed discussions are given for the dependence of correlation lengths and susceptibilities on temperature, anisotropy and concentration, and fcr the critical temperature on anisotropy and concentration. Particular emphasis is given to the weakly anisotropic system near percolation threshold and comparisons are made between the results of the present analysis and those of neutron-scattering experiments on dilute quasi-two- and three-dimensional systems.
The choice of a convenient self-dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q-state Potts ferromagnet criticality. The exact critical frontier and dimensionality crossover exponent Hf as well as the expected universality behaviour (renormalisation flow sense) are recovered for any linear scaling factor b and all values of q(q<or=4). The b=2 and b=3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den Nijs conjecture. The same calculation is performed, for all values of b for the exponent nu (d=1) associated with the one-dimensional limit and the exact result nu (d=1)=1 is recovered in the limit b to infinity .