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Physical review. B, Condensed matter

Applications of an exact duality-decimation transformation for two-dimensional spin systems on a square lattice

Theodore W. Burkhardt

Published Oct 1, 1979• 13 citations
Publisher

Abstract

An exact duality-decimation transformation is derived for a large class of two-dimensional spin systems on a square lattice. It maps a model with interactions in every second square (checkerboard interactions) onto a model with interactions in every square and vice versa. The transformation is applicable to Ising, Potts, solid-on-solid (SOS), and $x\ensuremath{-}y$ models. It may be used to reformulate the site percolation problem in terms of the $s\ensuremath{\rightarrow}1$ limit of a Potts model with four-spin interactions in every elementary square. The transformation maps the van Beijeren SOS model, which is equivalent to the $F$ model, onto an $x\ensuremath{-}y$ model with checkerboard interactions. Thus one has an exactly soluble $x\ensuremath{-}y$ model, which, however, bears little resemblance to the ordinary $x\ensuremath{-}y$ model. The transformation may be combined with a bond-shifting approximation to obtain a real-space renormalization scheme which in some respects represents an improvement over the Migdal-Kadanoff formula. No nearest-neighbor bonds are shifted. Better values of the exponent $\ensuremath{\nu}$ are obtained for the Ising and Potts models. Apart from exponentially vanishing corrections, the renormalization transformation for SOS and $x\ensuremath{-}y$ models has a surface of fixed points corresponding to generalizations of the usual discrete Gaussian and Villain models.

DOI
10.1103/physrevb.20.2905

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Physical review. B, Solid state•Apr 1, 1976•24 citations
An exact duality-decimation transformation is derived for a large class of two-dimensional spin systems on a square lattice. It maps a model with interactions in every second square (checkerboard interactions) onto a model with interactions in every square and vice versa. The transformation is applicable to Ising, Potts, solid-on-solid (SOS), and x\ensuremath−yx\ensuremath{-}yx\ensuremath−y models. It may be used to reformulate the site percolation problem in terms of the s\ensuremath→1s\ensuremath{\rightarrow}1s\ensuremath→1 limit of a Potts model with four-spin interactions in every elementary square. The transformation maps the van Beijeren SOS model, which is equivalent to the FFF model, onto an x\ensuremath−yx\ensuremath{-}yx\ensuremath−y model with checkerboard interactions. Thus one has an exactly soluble x\ensuremath−yx\ensuremath{-}yx\ensuremath−y model, which, however, bears little resemblance to the ordinary x\ensuremath−yx\ensuremath{-}yx\ensuremath−y model. The transformation may be combined with a bond-shifting approximation to obtain a real-space renormalization scheme which in some respects represents an improvement over the Migdal-Kadanoff formula. No nearest-neighbor bonds are shifted. Better values of the exponent \ensuremathν\ensuremath{\nu}\ensuremathν are obtained for the Ising and Potts models. Apart from exponentially vanishing corrections, the renormalization transformation for SOS and x\ensuremath−yx\ensuremath{-}yx\ensuremath−y models has a surface of fixed points corresponding to generalizations of the usual discrete Gaussian and Villain models.
Two-dimensional anisotropic NNN-vector models are discussed in three contexts. (i) A comprehensive approach to the description of phase transitions in two-dimensional physical systems is outlined. It involves the identification of discrete models for critical phenomena in two-dimensional systems (such as adsorbed thin films) and their investigation by symmetry, duality, and Migdal renormalization-group methods. The identification is based on the Landau-Ginzburg-Wilson Hamiltonian concept and universality arguments. (ii) Relations among anisotropic continuous-spin Hamiltonians and discrete models are established by the Hubbard transformation and the Migdal renormalization-group transformation. Discrete models are conjectured to be equivalent to NNN-component continuous-spin models with local anisotropies. For example, it is shown that the Migdal recursion relations map the continuous-spin, cubic Heisenberg Hamiltonian onto the discrete cubic model. (iii) Many of the anisotropic NNN-vector Hamiltonians can be associated with discrete models that have the form of a generalized Potts model. Such a model, termed (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) model, is defined in terms of two interacting Potts-like variables associated with each lattice site, and is analyzed by duality and renormalization-group methods. The (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) Hamiltonian provides a unified description for large classes of discrete models. The concepts are exemplified by a detailed discussion of the two-dimensional Heisenberg model with cubic anisotropy, which has applications to the magnetic $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$ phase transition in overlayers of molecular oxygen on graphite. New experiments for the study of this system are also discussed.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
The fixed-point topology of Kadanoff's lower-bound renormalization transformation is discussed for the d=2d=2d=2 square-lattice Ising model in the three-dimensional space of coupling constants for the nearest-neighbor, next-nearest neighbor, and four-spin interactions. An obstacle one encounters in trying to calculate a critical surface is pointed out. In the three-dimensional space of coupling constants Kadanoff's fixed point has two relevant vectors and can only be reached from a critical line, not from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. The critical exponents of this fixed point are close to the exact values, though not nearly so close as for Kadanoff's fixed point. The properties of both fixed points as a function of the variational parameter ppp are described. Critical values of the nearest-neighbor couplings leading to the fixed points are compared.