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Physical review. B, Condensed matter

Modified Migdal-Kadanoff renormalization for the Potts model

H. H. Chen, Felix Lee, H. C. Tseng

Published Nov 1, 1986• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.34.6448

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Modern Physics Letters B•Jul 20, 1994•0 citations
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 randomly bond-diluted two-dimensional nearest-neighbor q-state Potts ferromagnet and antiferromagnet on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. In the bond-diluted ferromagnet Potts model, differential recursion relations yield a phase diagram which is in quantitative agreement with all known results for q\ensuremath{\le}4. In the bond-diluted antiferromagnet Potts model, the phase diagram obtained shows that the threshold p∗{p}^{\mathrm{*}}p∗ at which the critical temperature goes to zero depends upon q.
We study a new modification of the Migdal-Kadanoff (MK) approximation, in which the bond-shifting step is generalized so that the resulting transformation preserves the free energy. This method is then utilized to study the qqq-state Potts model in two and three dimensions. While this method is just as easy to implement as the standard MK approximation, significant quantitative improvement is achieved by preserving the free energy even to very low order in a series expansion. We also discuss the limitations inherent in employing a single-parameter renormalization-group transformation.
Critical properties of the qqq-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for qc{q}_{c}qc​, the largest value of qqq for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
Based upon the known results of the critical properties of the q state Potts ferromagnets obtained by different methods, we proposed an empirical formula of the model. It gives the critical points and the critical exponents for general Potts spin value q in any d dimensions which are in good agreement with all the available results. It improves the results given by Migdal–Kadanoff renormalization group method. It gives more accurate results than Hajdukovic conjecture to the problem of interest. This empirical formula may serve as the primary reference to the problem of interest.