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Chinese Journal of Physics

An Exact Renormalization Group Transformation and Properties of Positive Symmetry Matrices

Chin‐Kun Hu

Published Dec 1, 1983• 1 citation
OpenAlex

Abstract

A positive symmetric m×m matrix P can be interpreted as the transfer matrix for an one-dimensional array of particles which have m-discrete energy levels and the most general nearest neighbor interactions. It is shown that P^1=P^2 can be considered as exact renormalization group (RG) transformation applied on such one dimensional system which does not have any phase transitions for T＞0. The exact RG transformation can be iterated to reach the high temperature fixed point. From the known asymptotic behaviour of RG transformations near the high temperature fixed point, we can derive and have a physical picture of properties of P^(2n) as n→∞, which can also be derived directly from the matrix theory. We also show that the fixed point of the RG transformations can be obtained directly from the eigenvector corresponding to the largest eigenvalue of P. The formulation is applied to the spin-j-Ising model as an illustration.

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Physical review. B, Condensed matter•Mar 1, 1981•1 citation
A positive symmetric m×m matrix P can be interpreted as the transfer matrix for an one-dimensional array of particles which have m-discrete energy levels and the most general nearest neighbor interactions. It is shown that P^1=P^2 can be considered as exact renormalization group (RG) transformation applied on such one dimensional system which does not have any phase transitions for T＞0. The exact RG transformation can be iterated to reach the high temperature fixed point. From the known asymptotic behaviour of RG transformations near the high temperature fixed point, we can derive and have a physical picture of properties of P^(2n) as n→∞, which can also be derived directly from the matrix theory. We also show that the fixed point of the RG transformations can be obtained directly from the eigenvector corresponding to the largest eigenvalue of P. The formulation is applied to the spin-j-Ising model as an illustration.
Exact renormalization-group recursion relations are derived in closed form for a one-dimensional spin-1 Ising model. The recursion relations possess tricritical and critical fixed points. We have studied the flow diagram corresponding to the renormalization-group transformations, and we have linearized about the fixed points. The predicted asymptotic homogeneity is shown to be satisfied by the true free energy, which has been computed using the transfer matrix. The pseudocritical singularities, existing in the zero-temperature limit, are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the pseudotricritical point is marked by a triple degeneracy. The increased instability of the tricritical fixed point, relative to the critical fixed points, is shown to be directly related to the eigenvalue degeneracy of the transfer matrix being greater at a tricritical point than at a critical point.
We present an exact differential renormalization-group method for the square Ising model. This method is analogous to that introduced by Hilhorst et al. for the triangular Ising model. Our approach is based upon the construction of such a renormalization transformation which remains confined to the space of four spatially dependent nearest-neighbor interactions. We derive the renormalization-group equations in the form of a set of four linear first-order partial differential equations for the interactions. We obtain a class of solutions of these equations and determine a nontrivial fixed point. The analysis of the flow around the fixed point shows that this fixed point is stable with respect to perturbations within the critical surface and is unstable in the temperaturelike direction. A unique eigenvalue, yT=1{y}_{T}=1yT​=1, from the eigenvalue equation for the temperature direction is found, in accordance with the exact Onsager solution.
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 real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.