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Nonlinear Renormalization Group Equation for Matrix Models

Saburo Higuchi, Chigak Itoi, Shinsuke M. Nishigaki, Norisuke Sakai

Published Jan 1, 1993• 15 citations
OpenAlex

Abstract

An exact renormalization group equation is derived for the free energy of matrix models. The renormalization group equation turns out to be nonlinear for matrix models, as opposed to linear for vector models. An algorithm for determining the critical coupling constant and the critical exponent is obtained. As concrete examples, one-matrix models with one and two coupling constants are analyzed and the exact values of the critical coupling constant and the associated critical exponent are found.

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An exact renormalization group equation is derived for the free energy of matrix models. The renormalization group equation turns out to be nonlinear for matrix models, as opposed to linear for vector models. An algorithm for determining the critical coupling constant and the critical exponent is obtained. As concrete examples, one-matrix models with one and two coupling constants are analyzed and the exact values of the critical coupling constant and the associated critical exponent are found.
An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and the limit $n=\ensuremath{\infty}$ of the nnn-vector model are calculated. We obtain agreement with the results of Wilson and Fisher, and with the spherical model.
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.
The nonperturbative renormalization group has been considered as a solid framework to investigate fixed point and critical exponents for matrix and tensor models, expected to correspond with the so-called double scaling limit. In this paper, we focus on matrix models and address the question of the compatibility between the approximations used to solve the exact renormalization group equation and the modified Ward identities coming from the regulator. We show in particular that standard local potential approximation strongly violates the Ward identities, especially in the vicinity of the interacting fixed point. Extending the theory space including derivative couplings, we recover an interacting fixed point with a critical exponent not so far from the exact result, but with a nonzero value for derivative couplings, evoking a strong dependence concerning the regulator. Finally, we consider a modified regulator, allowing to keep the flow not so far from the ultralocal region and recover the results of the literature up to a slight improvement.
According to the available publications, the field theoretical renormalization group approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This property is associated with the existence of nonanalytic contributions in the renormalization group functions. The situation is analyzed in this work using a new algorithm for summing divergent series that makes it possible to determine the dependence of the results for the critical exponents on the expansion coefficients for the renormalization group functions. It has been shown that the exact values of all the exponents can be obtained with a reasonable form of the coefficient functions. These functions have small nonmonotonic sections or inflections, which are poorly reproduced in natural interpolations. It is not necessary to assume the existence of singular contributions in the renormalization group functions.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter bbb. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of bbb around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of bbb used in the calculation. We study the problem of identifying the correct value of bbb.