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Journal of Physics A Mathematical and General

A new real space renormalisation method and its Julia set

Bernard Derrida, Henrik Flyvbjerg

Published Apr 21, 1985• 8 citations
Publisher

Abstract

The authors introduce a new real space renormalisation method which is valid not only on the real axis but also everywhere in the complex plane of temperatures. The method requires the knowledge of the partition function on small lattices and can in principle be improved systematically. They test this method on the two-dimensional Ising model. They find that it gives a satisfactory picture for the set of zeros of the partition function. When restricted to the real axis, the estimates for the renormalisation transformation, the critical point and the thermal exponent are also very good.

DOI
10.1088/0305-4470/18/6/010

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The authors introduce a new real space renormalisation method which is valid not only on the real axis but also everywhere in the complex plane of temperatures. The method requires the knowledge of the partition function on small lattices and can in principle be improved systematically. They test this method on the two-dimensional Ising model. They find that it gives a satisfactory picture for the set of zeros of the partition function. When restricted to the real axis, the estimates for the renormalisation transformation, the critical point and the thermal exponent are also very good.
It is argued that since any real space renormalisation R is a transformation under which the partition function is invariant, those R which are based upon various real space approximation schemes can in principle be extended into the complex field of the parameter space. In this extension the Julia set of R (now extended to include R which are algebraic) is an approximation to the locus of the Yang-Lee zeros. Finite-size scaling using Onsager's solution of the two-dimensional Ising model is used to construct R which are highly accurate in real temperatures in the critical region; these are extended into the complex temperature plane and results are displayed for both the anisotropic and isotropic cases.
A real space renormalisation group method has been applied to find the critical temperature of the Ising model on cubic and octahedral lattices. The result obtained for a cubic lattice is in good agreement with that obtained from high-temperature series expansion and is better than the results of other papers. A modification of the transformation tested for a simple cubic lattice allowed the authors to find the critical temperature for an octahedral lattice. The latter has not so far been studied by any version of the real space renormalisation group method.
Using a continuous class of real-space renormalisation transformations the authors study the critical behaviour of the Ising model in a variety of two-dimensional lattices. They use the simplest possible cluster, with two cells, and a single renormalised coupling. Certain of these transformations-well separated from the majority spin rule-produce sharply more accurate critical properties than the rest. From this study they note an optimal set of conditions which characterise the successful transformations. The optimal transformations appear to have a certain 'locality' property: renormalised couplings beyond nearest neighbour are especially small.
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).
A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-12\frac{1}{2}21​ $X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.