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International Journal of Modern Physics B

Variational Migdal–Kadanoff Approach to Molecular Ordering on a Lattice

Fabio Siringo

Published Jul 20, 1997• 2 citations
Publisher

Abstract

A variational extension of the Migdal–Kadanoff real space renormalization group method is applied to a recently proposed d-dimensional lattice model describing some aspects of molecular orientation in solids. As expected the method yields the exact fixed point for d=2, given the formal equivalence to the two-dimensional Ising model. For d=3 such equivalence breaks down and an approximate estimate of the fixed point is explicitly recovered and compared to the mean-field prediction. The possible existence of an intermediate symmetric phase is discussed.

DOI
10.1142/s021797929700112x

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A variational extension of the Migdal–Kadanoff real space renormalization group method is applied to a recently proposed d-dimensional lattice model describing some aspects of molecular orientation in solids. As expected the method yields the exact fixed point for d=2, given the formal equivalence to the two-dimensional Ising model. For d=3 such equivalence breaks down and an approximate estimate of the fixed point is explicitly recovered and compared to the mean-field prediction. The possible existence of an intermediate symmetric phase is discussed.
A recently proposed molecular model is discussed as a nontrivial extension of the Ising model. For d=2 the two models are shown to be equivalent, while for d>2 the molecular model describes a peculiar second order transition from an isotropic high-temperature phase to a low-dimensional anisotropic low-temperature state. The general mean-field analysis is compared with the results achieved by a variational Migdal-Kadanoff real space renormalization group method and by standard Monte Carlo sampling for d=3. By finite size scaling the critical exponent has been found to be nu=0.44+/-0.02, thus establishing that the molecular model does not belong to the universality class of the Ising model for d>2.
The use of variational approximations in the study and application of renormalization groups is discussed. In particular, a simple approximation is derived which yields an upper bound to the free energy of Ising models on d-dimensional lattices. The optimal transformation, which yields the least upper bound, is determined analytically. The criterion proposed by Kadanoff (1975) to determine the 'best' approximation to the fixed point is found to fail in this case. The reasons for this failure and several of the basic problems posed by variational approximations are discussed.
The spin-3/2 Ising model, with nearest-neighbor interactions only, is the prototypical system with two different ordering species, with concentrations regulated by a chemical potential. Its global phase diagram, obtained in d=3 by renormalization-group theory in the Migdal-Kadanoff approximation or equivalently as an exact solution of a d=3 hierarchical lattice, with flows subtended by 40 different fixed points, presents a very rich structure containing eight different ordered and disordered phases, with more than 14 different types of phase diagrams in temperature and chemical potential. It exhibits phases with orientational and/or positional order. It also exhibits quintuple phase transition reentrances. Universality of critical exponents is conserved across different renormalization-group flow basins via redundant fixed points. One of the phase diagrams contains a plastic crystal sequence, with positional and orientational ordering encountered consecutively as temperature is lowered. The global phase diagram also contains double critical points, first-order and critical lines between two ordered phases, critical end points, usual and unusual (inverted) bicritical points, tricritical points, multiple tetracritical points, and zero-temperature criticality and bicriticality. The four-state Potts permutation-symmetric subspace is contained in this model.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
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.