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Physical review. B, Condensed matter

Exact duality-decimation transformations and real-space renormalization for the Ising model on a square lattice

Felix Lee, H. H. Chen, Fa Yueh Wu

Published Sep 1, 1989• 29 citations
Publisher

Abstract

Critical properties of the Ising model with nearest-neighbor, next-nearest-neighbor, and four-spin interactions on the square lattice are studied by a real-space renormalization-group method. The recurrence relations of the couplings are obtained by a bond-moving operation followed by an exact duality-decimation transformation. Both the ferromagnetic and the superantiferromagnetic phase transitions are considered.

DOI
10.1103/physrevb.40.4871

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Diwen wuli xuebao•Jan 1, 2010•0 citations
Critical properties of the Ising model with nearest-neighbor, next-nearest-neighbor, and four-spin interactions on the square lattice are studied by a real-space renormalization-group method. The recurrence relations of the couplings are obtained by a bond-moving operation followed by an exact duality-decimation transformation. Both the ferromagnetic and the superantiferromagnetic phase transitions are considered.
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 FFF 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.
Real Space Renormalization Group (RSRG) treatment of Ising model for square and simple cubic lattice is investigated and critical coupling strengths of these lattices are obtained. The mathematical complications, which appear inevitable in the decimated partition function due to Block-spin transformation, is treated with a relevant approximation. The approximation is based on the approximate equivalence of [Formula: see text] for small [Formula: see text], where K is the nearest neighbor coupling strength and [Formula: see text] is the nearest neighbor spins degrees of freedom around a central spin. The values of the critical coupling strengths are obtained as 0.4830 for square lattice and 0.2225 for simple cubic (SC) lattice. The corresponding critical exponents values [Formula: see text] and [Formula: see text] are also calculated within very acceptable agreement with those values obtained from numerical works.
We investigate a fully frustrated Ising model with four nearest-neighbor interaction constants on a square lattice. We obtain an exact form of the partition function by using a decimation transformation and a dual transformation. The partition function is then expressed as a product of two partition functions each of which is the one of the Ising model with a nearest-neighbor interaction constant on a square lattice. Special systems with two interaction constants are studied in detail and phase transitions for these systems are 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.
Using the decimation technique,a decorated square Ising lattice with two kinds of interactions K1and K2 was transformed into a regular square Ising lattice with nearest-neighbor(NN),next-nearest-neighbor(NNN),and four-spin interactions.The critical point of the approximate solution in this system was at K′C=0.4406868.The result indicated that the phase transition occurred only when the NN interaction K1 and the NNN interaction K2 satisfied a certain relation.If we only calculated the NN interaction K1,the critical point of the exact solution was at K1C=0.7635.It was also found that if we decorated two spins between the two square lattice points,the answer could be solved exactly on condition that the NN interaction existed on the doubly decorated square Ising lattice.