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Canadian Journal of Physics

Decimation transformations in high-temperature renormalization-group methods

Nahomi Miyamoto Fujiki, Keith De'Bell

Published Mar 1, 1987• 0 citations
Publisher

Abstract

The high-temperature series renormalization group developed by Betts et al. is modified by using the decimation transformation. A conventional fixed-point analysis of the recursion relations is discussed and, in addition, an analysis based on scaling relations for correlation functions is considered. As test cases, we apply these methods to two-dimensional Ising systems with nearest neighbour interactions. The results for a triangular and a square lattice are presented.

DOI
10.1139/p87-033

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The high-temperature series renormalization group developed by Betts et al. is modified by using the decimation transformation. A conventional fixed-point analysis of the recursion relations is discussed and, in addition, an analysis based on scaling relations for correlation functions is considered. As test cases, we apply these methods to two-dimensional Ising systems with nearest neighbour interactions. The results for a triangular and a square lattice are presented.
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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.
Abstract In Chapter 8 we gave an introduction to the formalism of the renormalization group. A renormalization transformation changes the scale of a system by a factor b. Systems at criticality flow to fixed points under the transformation and their critical exponents and scaling behaviour are determined by linearizing the recursion equations about the fixed point. The aim of this chapter is two-fold: firstly to illustrate these ideas by giving several examples of renormalization group transformations and secondly to demonstrate the ways in which they can be used to calculate critical properties. Our examples are divided into three classes. In Section 9.1, we shall consider a one-dimensional model where the recursion relations can be written down exactly. The drawback of this as a canonical example is that any critical behaviour is at zero temperature. The decimation transformation presented in one dimension is an example of a real-space transformation and the second task, in Section 9.2, is to show how these may be used approximately to treat models in higher dimensions. Thirdly, in Section 9.5, we shall mention the E-expansion where the renormalization takes place in momentum space. This enables an understanding of the crossover to mean-field behaviour. However, the formalism is somewhat involved and the details are beyond the scope of this book.
A method for computing low--temperature series for renormalized operators in the two--dimensional Ising model is proposed. Series for the renormalized magnetization and nearest--neighbor correlation function are given for the majority rule transformation on 2 \\Theta 2 blocks and random tie--breaker. These series are applied to the study at very low temperature of the first--order phase transition undergone by this model. We analyze how truncation in the renormalized Hamiltonian leads to spurious discontinuities of the Renormalization Group transformation. Keywords: Renormalization group, position--space renormalization--group transformations, Ising model, low--temperature expansions. E-mail: salas@mafalda.physics.nyu.edu 1 Introduction The behavior of the Renormalization Group (RG) in the vicinity of first order phase transitions has been a very controversial matter for the last 20 years. In 1975 Nienhuis and Nauenberg [1] proposed that the RG transformations behave near first--or...