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Physical review. B, Condensed matter

Real-space renormalization-group study of the exchange-interaction model

H. H. Chen, Felix Lee

Published Dec 1, 1990• 6 citations
Publisher

Abstract

Through use of an approximation introduced by Suzuki and Takano, the Migdal-Kadanoff (MK) renormalization-group transformation with the rescaling length b=2 is derived for the exchange-interaction (EI) model. Both the standard MK method and its modification, which preserves the free energy in the renormalization transformation, are used to determine critical temperatures and thermal exponents of the EI model on cubic lattices for various spins.

DOI
10.1103/physrevb.42.10540

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Through use of an approximation introduced by Suzuki and Takano, the Migdal-Kadanoff (MK) renormalization-group transformation with the rescaling length b=2 is derived for the exchange-interaction (EI) model. Both the standard MK method and its modification, which preserves the free energy in the renormalization transformation, are used to determine critical temperatures and thermal exponents of the EI model on cubic lattices for various spins.
A modification of the Migdal-Kadanoff (MK) real-space renormalization technique is studied and applied to the q-state Potts model on the square and the simple cubic lattices. A parameter x which describes the boundary condition is introduced to the cluster-decimation (CD) approximation. When x=2, the present method is the same as the CD approximation, and in the limit x\ensuremath{\rightarrow}\ensuremath{\infty} this method reduces to the MK technique. Critical temperatures and exponents of the Potts model are calculated for 0<q\ensuremath{\le}4 by using various boundary conditions. Very good estimates of critical properties are obtained. The choice of the boundary condition is discussed.
The spin-(1/2 anisotropic Heisenberg model is studied by a modified Migdal-Kadanoff method, called the cluster-decimation approximation. The free energy for a small cluster of spins with an appropriate boundary condition is assumed to be preserved in the bond-moving step and the approximate decimation of Suzuki and Takano is adopted in the site-decimation step. Critical temperatures obtained by the present method are in general much better than those determined by preserving the high-temperature series expansion of the free energy. In particular the XY fixed point and the isotropic Heisenberg fixed point for the square lattice are not found in the present calculation.
The critical temperature, thermal and magnetic exponents, and the graphs of dTeffdl\frac{d{T}_{\mathrm{eff}}}{\mathrm{dl}}dldTeff​​ versus Teff{T}_{\mathrm{eff}}Teff​ (where Teff{T}_{\mathrm{eff}}Teff​ denotes "effective" temperature of the system and lll the renormalization-group parameter) for the d=3d=3d=3, n=2,3n=2, 3n=2,3 models are obtained using the Migdal-Kadanoff scheme. Results of finite-lattice approximations for the d=2d=2d=2, n=2n=2n=2 model are also presented.
This paper represents some general aspects of real-space renormalization group approach to critical dynamics. The main idea is to combine Kadanoff's cell analysis with coarse graining of time corresponding to Markoffian approximation, which has been proposed by Suzuki to derive the dynamic scaling law. In order to realize this idea explicitly, the Migdal approximation is extended to the two-dimensional kinetic Ising model, to obtain z=1.96 for the dynamical critical exponent z. This result agrees very well with the value z=2.0±0.05 obtained by Yahata and Suzuki in a high temperature expansion method. Other new methods for real-space RGA are also proposed.
The simple, but ad hoc, prescription for bond shifting in the Migdal-Kadanoff approximation is replaced by a systematic calculational procedure. In this method, the free energy is required to be preserved exactly by means of matching series expansions. The resulting renormalization group is characterized by a small parameter, and can be extended by working to higher order. A significant quantitative improvement over the standard Migdal-Kadanoff method is obtained with little increase in calculational effort. This approach is ultimately limited, however, by being restricted to a one-dimensional interaction parameter space.