Phenomenological renormalization group methods

Abstract

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 nite size scaling hypothesis, the approximate critical behavior of model on in nite lattice is obtained through the exact computation of some thermal quantities of the model on nite clusters. In this work some of these methods are reviewed, namely the mean eld renormalization group, the e ective eld renormalization group and the nite size scaling renormalization group procedures. Although special emphasis is given to the mean eld renormalization group since it has been, up to now, much more used to study a wide variety of di erent systems a discussion of their potentialities and interrelations to other methods is also presented. I IntroductionThe renormalization group formalism introduced by Wilson in the early 70's 1, 2 is by n o w one of the basic strategies to solve fundamental problems in statistical mechanics. It is also a very useful tool to tackle problems in several elds of theoretical physics such as the study of nonlinear dynamics and transitions to chaos 3 , disorder surface growth 4 , earthquakes 5 , among others. The conceptual foundation of the method, rst laid by Kadano 6 to qualitatively predict scaling behavior at a second-order phase transition, is to reduce, in a step-by-step way, the degrees of freedom of the system leaving unchanged the underlying physics of the problem. This reduction, carried out repeatedly through a renormalization recursion relation, leads the original system with a large correlation length the range at which p h ysical microscopic operators are correlated to one with correlation length of unity order, where well-known methods as perturbation theory can, at least in principle, be used to treat the problem. Depending on the mathematical technique, such thinning of the degrees of freedom can be implemented directly in the reciprocal momentum space or in the real position space. The former approach makes use of mathematical tools from quantum eld theory with the crystalline system being replaced by its continuous limit. As a result, the so-called ,expansion proposed by Wilson and Fisher 7 and further developed by using techniques of renormalized perturbation theory 8, 9 provides analytical and quite well controlled asymptotically exact results for critical exponents despite being unable to predict values of critical points, critical lines and phase diagrams. On the other hand, the more intuitive real space version of the renormalization group works directly in the position space. It was introduced by Niemeijer and van Leeuwen 10 and several di erent techniques have been proposed and applied to a great variety of statistical models 11 . The real space renormalization group RSRG has since become an important apparatus in studying critical phenomena.The main feature of the renormalization group is to obtain, from the renormalization recursion relations, ow diagrams in the parameter space of the...

The Semi-Infinite Quantum Spin-1/2 XY Model

“…In this paper, we apply the mean field RG (MFRG) method [16,17] to treat the critical properties of the anisotropic XY model on a semi-infinite lattice. The MFRG approach is based on the comparison of two clusters of different sizes, each of them simulating the infinite system.…”

“…(6), (8) and (10) for the one-spin cluster with Eqs. (7), (9) and (11) for the two-spin cluster with the MFRG ideas [16,17] we obtain the following set of equations:…”

Real-space renormalization group study of the anisotropic antiferromagnetic Heisenberg model on a square lattice

“…This method was proposed some time ago [11] and has been successfully applied to classical systems (both static and dynamic properties have been studied), like Ising, Potts, or BlumeCapel models [12]. The second is just the generalization of the usual bond-moving Migdal-Kadanoff [13] approximation to antiferromagnetic quantum systems.…”

The magnetic properties of disordered Fe–Al alloy system

“…The Fe 1−q Al q system in the bcc structure shows an interesting magnetic behavior since its critical temperature decreases with q = 1 − c but shows a kind of plateau for low Al concentrations. Theoretical studies [5,6], using mean field renormalization group [7] and Bogoliubov inequality [8] approaches have been used to explain this behavior by taking a simple Ising Hamiltonian. Sato and Arrot [9] obtained the magnetization by assuming a ferromagnetic exchange between nearest-neighbor Fe atoms and an antiferromagnetic superexchange between two Fe atoms separated by an Al atom.…”