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Physical review. B, Condensed matter

Finite-size rescaling transformations

Raimundo R. dos Santos, Leigh Sneddon

Published Apr 1, 1981• 48 citations
Publisher

Abstract

A theory of finite-size rescaling transformations (FSRT) is derived. It yields accurate estimates of the critical properties of statistical-mechanical systems of infinite extent. The arguments used are analogous to, but different from, conventional renormalization-group theory. The FSRT do not involve block spins and do not introduce any new couplings beyond those already in the problem. A technique which was suggested phenomenologically by Nightingale is derived as a special case of the FSRT theory. The optimization of convergence rates, and the relationship between FSRT and the finite-size scaling hypotheses of Fisher, are examined. Some new numerical results are presented.

DOI
10.1103/physrevb.23.3541

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A theory of finite-size rescaling transformations (FSRT) is derived. It yields accurate estimates of the critical properties of statistical-mechanical systems of infinite extent. The arguments used are analogous to, but different from, conventional renormalization-group theory. The FSRT do not involve block spins and do not introduce any new couplings beyond those already in the problem. A technique which was suggested phenomenologically by Nightingale is derived as a special case of the FSRT theory. The optimization of convergence rates, and the relationship between FSRT and the finite-size scaling hypotheses of Fisher, are examined. Some new numerical results are presented.
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 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.
Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.
An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent. PACS: 05.10.Cc, 64.60.Ak, 05.50.+q, 75.10.Hk 1 1
Abstract A number of numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the infinite system, it is thus necessary to understand how the infinite size limit is reached. In particular in a system in which the forces are short range, no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite-size extrapolation is somewhat non-trivial. In this chapter, the problem is analysed in the case of second-order phase transitions, in the framework of the N-vector model. The existence of a finite-size scaling is established, extending renormalization group (RG) arguments to this new situation. Then, finite volume geometry and cylindrical geometry, in which the size is finite in all dimensions except one, are distinguished. It is explained how to adapt the methods used in the case of infinite systems to calculate the new universal quantities appearing in finite-size effects, for example, in d = 4−ϵ or d = 2+ϵ dimensions. Special properties of the commonly used periodic boundary conditions are emphasized.