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Europhysics Letters (EPL)

Finite-Size Scaling Renormalization Group

P.M.C. de Oliveira

Published Dec 1, 1992• 39 citations
Publisher

Abstract

A renormalization group approach based only on the finite-size scaling hypothesis, with no further assumptions, is presented. It is applied, as an example, to the 2D and 3D Ising ferromagnet in a uniform field. The critical behaviour of this system is discussed in detail, not only around the critical point ( J = J 0 and H = 0), but also around both the ferromagnetic ( J → ∞ and H = 0) and paramagnetic ( J = 0 and H = 0) points. All physical features of the system are successfully described, both qualitatively as well as quantitatively.

DOI
10.1209/0295-5075/20/7/008

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Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.
Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more used to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also presented.
The structure of finite-size scaling within the framework of the field-theoretic renormalization-group approach is examined. New size-dependent ultraviolet divergences are introduced into the theory. It is shown, nonetheless, that ``two-scale-factor'' universality reflects itself in the finite-size free energy and produces a scaling form such as that discussed by Privman and Fisher. A consequence is that the singular part of the finite-system free energy at the bulk critical point is universal. Extensions to surface and interfacial free energies, first proposed from the results of Ising simulations, are supported qualitatively.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
We propose a new picture of the renormalization group (RG) approach in the presence of disorder, which considers the RG trajectories of each random sample (realization) separately instead of the usual renormalization of the averaged free energy. The main consequence of the theory is that the average over randomness has to be taken after finding the critical point of each realization. To demonstrate these concepts, we study the finite-size scaling properties of the two-dimensional random-bond Ising model. We find that most of the previously observed finite-size corrections are due to the sample-to-sample fluctuation of the critical temperature and scaling predictions are fulfilled only by the new average.