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Journal of Physics C Solid State Physics

A position-space study of the anisotropic quantum Ising model

L. S. Campana, Alvaro Caramico D’Auria, Ugo Esposito, G. Kamieniarz

Published Jan 30, 1988• 0 citations
Publisher

Abstract

A global real-space renormalisation group method is applied to the transverse Ising model on an anisotropic square lattice. Recursion relations with non-trivial isotropic fixed points at zero and finite temperature are found in agreement with the universality hypothesis. The critical field as a function of the anisotropy at T=0 and the critical temperature as a function of the field for different values of the anisotropy parameters are calculated.

DOI
10.1088/0022-3719/21/3/013

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Physical review. B, Condensed matter•Nov 1, 1979•16 citations
A global real-space renormalisation group method is applied to the transverse Ising model on an anisotropic square lattice. Recursion relations with non-trivial isotropic fixed points at zero and finite temperature are found in agreement with the universality hypothesis. The critical field as a function of the anisotropy at T=0 and the critical temperature as a function of the field for different values of the anisotropy parameters are calculated.
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.
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 newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.
The mean-field renormalisation group approach is applied to the anisotropic Ising model by considering clusters consisting of finite blocks of spins and clusters consisting of infinite strips of spins with the finite widths. The critical temperature and estimates of the critical exponent nu are obtained as functions of the anisotropy. A two-step renormalisation group procedure using a phenomenological renormalisation transformation is also presented. In general, the results for nu are consistent with universality arguments.
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.
We apply a recently proposed real-space renormalization-group method to the two-dimensional square Ising model in a transverse magnetic field at zero temperature. We do the calculation to first and second order in the intercell coupling; in both cases we find a nontrivial fixed point. We compare the obtained critical exponents with accepted values for the three-dimensional Ising model from high-temperature series expansions. In first order the agreement is poor, in second order we obtain good agreement with the expected values.