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Journal of Physics C Solid State Physics

Anisotropy effects in the 2D transverse Ising model

Rita Maria Zorzenon dos Santos, Raimundo R. dos Santos

Published Apr 10, 1988• 0 citations
Publisher

Abstract

The authors study both the pure anisotropic and the anisotropically diluted transverse Ising model at zero temperature on the square lattice. In the former the coupling along each Cartesian direction are allowed to be different, whereas in the latter it is the bond concentration which may be different along the Cartesian directions. By means of a position space renormalisation group scheme, they find that in both cases the isotropic systems are stable with respect to anisotropy perturbations. They also discuss their results in view of a recently proposed criterion for the stability of pure systems of fixed points with respect to correlated dilution.

DOI
10.1088/0022-3719/21/10/016

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The authors study both the pure anisotropic and the anisotropically diluted transverse Ising model at zero temperature on the square lattice. In the former the coupling along each Cartesian direction are allowed to be different, whereas in the latter it is the bond concentration which may be different along the Cartesian directions. By means of a position space renormalisation group scheme, they find that in both cases the isotropic systems are stable with respect to anisotropy perturbations. They also discuss their results in view of a recently proposed criterion for the stability of pure systems of fixed points with respect to correlated dilution.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.
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.
The ferromagnetic-phase stability limit and magnetization of a quenched bond-mixed spin-1/2 transverse Ising model in an anisotropic simple square lattice are studied within the framework of an effective-field theory considering both competing and noncompeting interactions. In the bond-disordered case, where bonds J1{J}_{1}J1​ and J2{J}_{2}J2​ occur with probabilities 1-p and p, respectively, critical curves are obtained for various \ensuremath{\alpha}'s (\ensuremath{\alpha}\ensuremath{\equiv}J1{J}_{1}J1​/J2{J}_{2}J2​), and the critical bond concentrations pc{p}_{c}pc​ at which the ferromagnetic phase breaks down are determined. As \ensuremath{\alpha}\ensuremath{\rightarrow}0, the mixed model reduces to the diluted Ising model and the results obtained are compared to those available in the literature. The effect of the transverse field (for different \ensuremath{\alpha}'s) in the thermal behavior of the magnetization as a function of the concentration is also discussed.
The critical properties of the bond-diluted mixed spin-1/2 and spin-1 transverse Ising system with single-ion anisotropy are investigated by means of the effective field theory with correlations. Particular emphasis is given to the square lattice for which phase diagrams are obtained. If transverse field varies in the certain ranges we find that the tricritical point is obtained for the value of the bond concentration in a restricted region. We also observe that the reentrant phase transition may occur in the present system if single-ion anisotropy parameter is not large and if transverse field is small. On the other hand, for certain values of the system parameters, new induced magnetic ordering can be obtained. We find a number of interesting phenomena that are not predicted by previous literatures. The influence of the transverse field on the behaviours of the reentrant phase transition and induced magnetic ordering is discussed. A detailed description of the phase transition is presented.