Mean field renormalization group study of surface effects in the Ising ferromagnet

Marques, Marta; Santos, M.A.

doi:10.1016/0375-9601(86)90532-3

Cited by 15 publications

(1 citation statement)

References 11 publications

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“…The Ising model has been extensively studied and has also been a test system for all proposed procedures involving the MFRG approach [27,32,[49][50][51][76][77][78], including a combination with the decimation procedure [79,80], the effects of reaction [81] and symmetry dependent fields [82], and treatments of some extended models such as: antiferromagnetic model on triangular [57] and square lattices with external field [60]; anisotropic Ising model on a triangular lattice [68]; surface effects in pure and random semi-infinite models [83][84][85][86][87][88][89][90][91][92][93];…”

Section: Static Problems Applicationsmentioning

confidence: 99%

Phenomenological renormalization group methods

1999

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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...

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Section: Static Problems Applicationsmentioning

confidence: 99%

Phenomenological renormalization group methods

1999

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Mean Field Renormalization Group for Site‐Bond Diluted Ising Ferromagnets

Yang

1988

Physica Status Solidi (b)

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It is shown that the mean field renormalization group techniques provide a reasonable description of the critical behaviour of the site-bond diluted Ising model, both in the two-dimensional system and the surface of a semi-infinite three-dimensional system. Es wird gezeigt, drtn die Feldmittel-Renormierungsgruppentechnik eine verniinftige Beschreibung des kritischen Verhaltens des platzgebundenen verdunnten Ising-Modells sowohl im eweidimensionalen System als auch fur die Oberflache des halbunendlichen dreidimensionalen Systems liefert.

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Effective—Field Renormalization—Group Study of Surface Effects in the Ising Ferromagnet

Ilkovič

1991

Physica Status Solidi (b)

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Mean field renormalization group study of surface effects in the Ising ferromagnet

Cited by 15 publications

References 11 publications

Phenomenological renormalization group methods

Phenomenological renormalization group methods

Mean Field Renormalization Group for Site‐Bond Diluted Ising Ferromagnets

Effective—Field Renormalization—Group Study of Surface Effects in the Ising Ferromagnet

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