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Physical review. B, Condensed matter

Preserving the free energy in a Migdal-Kadanoff approximation for the q -state Potts model

David Andelman, James S. Walker

Published Jan 1, 1983• 14 citations
Publisher

Abstract

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 $q$-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.

DOI
10.1103/physrevb.27.241

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Physical review. B, Condensed matter•Apr 1, 1988•5 citations
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.
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.
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.
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.
The changeover from first-order to second-order phase transitions in q-state Potts models is obtained at q_{c}=2 in spatial dimension d=3 and essentially at q_{c}=4 in d=2, using a physically intuited simple adaptation of the Migdal-Kadanoff renormalization-group transformation. This simple procedure yields the latent heats at the first-order phase transitions. In both d=2 and 3, the calculated phase transition temperatures, respectively compared with the exact self-duality and Monte Carlo results, are dramatically improved. The method, when applied to a slab of finite thickness, yields dimensional crossover.