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Static critical properties of the pure and diluted Heisenberg or Ising models

Mathew Raymond. Davies, Davies, Mathew Raymond

Published Jan 1, 1982• 0 citations
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Abstract

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.

DOI
10.5287/ora-bpvwxxq1k

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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 bond-diluted classical Heisenberg ferromagnet is treated by a position space renormalisation group method valid when the valid behaviour is at low temperatures. The method is outlined and used to obtain exact (one-dimensional) and approximate (two- and three-dimensional) results for the critical behaviour of the correlation length near the percolation limit. The procedure agrees in the pure limit with exact results of Polyakov and Migdal for the two-dimensional case. At general concentrations an exact correspondence with related treatments of random resistor networks is exhibited. This implies that the percolation-to-thermal crossover exponents of the Ising and Heisenberg systems are the same. The generalisations required for a proper comparison with neutron scattering experiments are briefly discussed.
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.
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).
For pt.III see ibid., vol.20, p.1841 (1987). The global real-space renormalisation group approach is applied in two and three dimensions to study the pure and diluted transverse Ising model at finite temperatures. A phase diagram in the temperature-field plane is presented for the pure quantum model on the honeycomb, triangular and square lattices in the first-order cumulant expansion. An improvement of the classical Ising fixed-point data in two and three dimensions is considered by having recourse to larger cells and the second-order cumulant expansion. The zero-temperature percolative behaviour of the bond-diluted system is recovered from the fixed-point data analysis. Some shortcomings and relations to other approaches are also discussed.
A scaling transformation is developed for the pure square-lattice-spin- 1 / 2 Ising model with different exchange couplings J 1 , J 2 in the two lattice directions. The transformation preserves Onsager's exact duality relationship and so can treat cases of strong lattice anisotropy. It is generalised to the dilute case and used to obtain critical exponents and critical curves (transition temperature T c as a function of concentration p of magnetic bonds for various exchange ratios gamma =J 2 /J 1 ). For gamma small, corresponding to weakly coupled chains, the transition temperature falls rapidly with dilution, then levels off, finally vanishing at the two-dimensional percolation concentration. Extensions are made to three-dimensional dilute magnets, including weakly coupled layer and chain magnets as special cases. Critical curves and exponents are calculated, and a quantitative description is obtained of various crosscovers between low- and high-dimensional behaviour which occur in the quasi-low-dimensional magnets as the transition temperature falls with dilution and the weak exchanges become significant.