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Journal of Applied Physics

Critical behavior of the anisotropic Heisenberg model by effective-field renormalization group

J. Ricardo de Sousa, I. P. Fittipaldi

Published May 15, 1994• 15 citations
Publisher

Abstract

A real-space effective-field renormalization-group method (ERFG) recently derived for computing critical properties of Ising spins is extended to treat the quantum spin-1/2 anisotropic Heisenberg model. The formalism is based on a generalized but approximate Callen–Suzuki spin relation and utilizes a convenient differential operator expansion technique. The method is illustrated in several lattice structures by employing its simplest approximation version in which clusters with one (N′=1) and two (N=2) spins are used. The results are compared with those obtained from the standard mean-field (MFRG) and Migdal–Kadanoff (MKRG) renormalization-group treatments and it is shown that this technique leads to rather accurate results. It is shown that, in contrast with the MFRG and MKRG predictions, the EFRG, besides correctly distinguishing the geometries of different lattice structures, also provides a vanishing critical temperature for all two-dimensional lattices in the isotropic Heisenberg limit. For the simple cubic lattice, the dependence of the transition temperature Tc with the exchange anisotropy parameter Δ [i.e., Tc(Δ)], and the resulting value for the critical thermal crossover exponent φ [i.e., Tc≂Tc(0)+AΔ1/φ ] are in quite good agreement with results available in the literature in which more sophisticated treatments are used.

DOI
10.1063/1.355584

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A real-space effective-field renormalization-group method (ERFG) recently derived for computing critical properties of Ising spins is extended to treat the quantum spin-1/2 anisotropic Heisenberg model. The formalism is based on a generalized but approximate Callen–Suzuki spin relation and utilizes a convenient differential operator expansion technique. The method is illustrated in several lattice structures by employing its simplest approximation version in which clusters with one (N′=1) and two (N=2) spins are used. The results are compared with those obtained from the standard mean-field (MFRG) and Migdal–Kadanoff (MKRG) renormalization-group treatments and it is shown that this technique leads to rather accurate results. It is shown that, in contrast with the MFRG and MKRG predictions, the EFRG, besides correctly distinguishing the geometries of different lattice structures, also provides a vanishing critical temperature for all two-dimensional lattices in the isotropic Heisenberg limit. For the simple cubic lattice, the dependence of the transition temperature Tc with the exchange anisotropy parameter Δ [i.e., Tc(Δ)], and the resulting value for the critical thermal crossover exponent φ [i.e., Tc≂Tc(0)+AΔ1/φ ] are in quite good agreement with results available in the literature in which more sophisticated treatments are used.
A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.
A unified effective-field renormalization-group framework (EFRG) for both quenched bond- and site-diluted Ising models is herein developed by extending recent works. The method, as in the previous works, follows up the same strategy of the mean-field renormalization-group scheme (MFRG), and is achieved by introducing an alternative way for constructing classical effective-field equations of state, based on rigorous Ising spin identities. The concentration dependence of the critical temperature, Tc(p), and the critical concentrations of magnetic atoms, pc, at which the transition temperature goes to zero, are evaluated for several two- and three-dimensional lattice structures. The obtained values of Tc and pc and the resulting phase diagrams for both bond and site cases are much more accurate than those estimated by the standard MFRG approach. Although preserving the same level of simplicity as the MFRG, it is shown that the present EFRG method, even by considering its simplest size-cluster version, provides results that correctly distinguishes those lattices that have the same coordination number, but differ in dimensionality or geometry.
For the isotropic Heisenberg model with an arbitrary spin 5, the correlation exponent l/ defined by ~ ~ T-v near the critical point Tc=O is evaluated as l/=1 with the use of the real space renormalization group method.This yields y=2 for the susceptibility exponent y defined by Xo~ T-r , using the scaling relation y=l + (2-1]) l/= 1 + l/.Our general result y=2 for all values of spin 5 includes Fisher's result in the classical limit 5-->00.For the anisotropic Heisenberg model (XXZ model) with a spin 5=1/2, an anisotropy scaling law is derived.
The spin 1/2 anisotropic Heisenberg model is studied by means of the real space renormalization transformations.Thermodynamic properties of one-dimensional systems are calculated by using an approximate decimation method.These results agree well with those obtained by Bonner and Fisher and by Katsura.Two-and three-dimensional systems are studied by generalizing the Migdal-Kadanoff renormalization transformations to quantum spin systems.For the Ising and Heisenberg models in two and three dimensions and for the X-V model in three dimensions, the critical properties thus obtained are in qualitative agreement with those obtained from high temperature series expansions.For the two-dimensional X-V model, our results suggest that this model undergoes some kind of phase transition and that the critical behaviour of this model is similar to that expected for the classical X-V and planar models.Contents I. Hiroshi Takano 2-3.Three-dimensional systems III.Decimation method for one-dimensional systems 3-1.Transformation function and approximation 3-2.Renormalization equations for scale factor Q= 2 3-3.Free energy IV.The Migdal-Kadanoff transformations for two-and three-dimensional systems 4-1.Fixed points and critical lines 4-2.Critical properties 4-3.Thermodynamic properties V. Summary and conclusions Appendix A. Real space renormalization transformations A-I.Renormalization transformations A-2.Critical properties A-3.Free energy