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Journal of Physics C Solid State Physics

A position space renormalisation group approach to the compressible Ising model

Mustansir Barma, Deepak Kumar

Published Apr 14, 1979• 2 citations
Publisher

Abstract

A position space renormalisation group procedure is developed for the compressible Ising model. The procedure is essentially a generalisation of the cell variable approach of Niemeijer and Van Leeuwen (1974) in that vibrational cell variables are introduced in addition to spin cell variables. The procedure is applied to the two-dimensional triangular lattice, the shearless square lattice. In both cases the rigid Ising fixed point is stable to spin-lattice coupling as long as the latter is small. Beyond a critical strength of the spin-lattice coupling, the system develops instability leading to, presumably, a first-order transition. For the shearless square lattice the conclusion is at variance with the exact result of Baker and Essam (1971), so the approximation is regarded as unreliable for larger values of the spin-lattice coupling (1971).

DOI
10.1088/0022-3719/12/7/022

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A position space renormalisation group procedure is developed for the compressible Ising model. The procedure is essentially a generalisation of the cell variable approach of Niemeijer and Van Leeuwen (1974) in that vibrational cell variables are introduced in addition to spin cell variables. The procedure is applied to the two-dimensional triangular lattice, the shearless square lattice. In both cases the rigid Ising fixed point is stable to spin-lattice coupling as long as the latter is small. Beyond a critical strength of the spin-lattice coupling, the system develops instability leading to, presumably, a first-order transition. For the shearless square lattice the conclusion is at variance with the exact result of Baker and Essam (1971), so the approximation is regarded as unreliable for larger values of the spin-lattice coupling (1971).
The Niemeijer-van Leeuwen formulation of the renormalisation group has been used to discuss the quenched dilute Ising model and the Ising spin-glass in two dimensions. The authors chose a five-spin cell which, in the pure case, treats the Ising antiferromagnet equally as well as the ferromagnet. This choice of cell leads to the existence of a spin-glass ground state which was found to be stable for the whole composition range. The spin-glass transition was found to be sharp and the corresponding critical exponents were calculated. For the dilute Ising model the behaviour of the reduced critical temperature T c (p)/T c (1), the percolation concentration and the correlation length exponent were found to be in good agreement with the exact results and series calculations.
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.
In evaluating the critical properties of lattice spin systems in the real-space renormalization-group theory we use the cluster variation method. A configuration in the transformed system is constrained and the probability of occurrence of this configuration is calculated both in the transformed system and in the original system. By equating the two probabilities and forming ratios of two such equalities (for two or more constrained configurations) the fixed point of the renormalization transformation is evaluated. The method can avoid the trouble due to different singularities in the original and transformed systems, and hence can obviate the possible development of spurious singularities in the transformation at low temperatures. The two-dimensional triangular Ising model is treated with numerical results comparable with those obtained by the cluster treatment of Niemeijer and van Leeuwen who used more and larger cluster types than those we introduce.
Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).
The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.
A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T , nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.