Effective-field renormalization-group study for the diluted Ising model

Li, Z. Y.

doi:10.1103/physrevb.37.5744

Cited by 18 publications

(10 citation statements)

References 21 publications

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“…The MFRG in its several formulations computes exactly the magnetization (order parameter) for each finite cluster of spins according to the canonical distribution (19). An alternative way of obtaining this order parameter for a Hamiltonian model system H has been proposed by some authors [175][176][177] by employing an effective field theory based on the exact generalized Callen-Suzuki identity [178,179] O n = T r n O n exp −βH n T r n exp −βH n ,…”

Section: A Effective Field Renormalization Group (Efrg)mentioning

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: A Effective Field Renormalization Group (Efrg)mentioning

confidence: 99%

Phenomenological renormalization group methods

1999

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“…The principle of the phenomenological renormalization group [11] is based on the comparison of clusters of different sizes in the presence of symmetry-breaking fields. In order to obtain the phase diagrams and the critical concentrations for various lattices, we have used the simplest choice of finite clusters with N' = 1 (si) and N = 2 (si and sj) spins.…”

Section: Model and Formalismmentioning

confidence: 99%

“…In this way an important bridge between the classical and modern theories of critical phenomena was established. The approach is based on constructing classical effective-field equations of state, by using rigorous Ising spin identities as a starting point and has already been applied to a variety of physical problems, such as bond-random king ferromagnet [11,13,141, transverse Ising model [15], and Ashkin-Teller model [16].…”

Section: Introductionmentioning

confidence: 99%

Effective‐field renormalization‐group approach for quenched site‐dilute ising models

Bobák

Jaščur

Mockovčiak

1994

Physica Status Solidi (b)

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The quenched site-dilute Ising model in two and three dimensions is studied by employing the effective-field renormalization-group method. The concentration dependence of the critical temperature and values of the critical concentration are obtained and compared to other different methods. Quite good results for a honeycomb lattice are achieved even by considering the smallest possible clusters. In particular, it is shown that the present formalism is able to distinguish between lattices of the same coordination number but of different dimensions.

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“…The critical frontier line which separates the antiferromagnetic and the disordered phase is obtained by solving numerically cluster with one-spin, that constitutes the MFA-1, solving Eqs. (29) and (30). Analogously to clusters two-spins the Eqs.…”

Section: B Clusters In Mean-field Approximationmentioning

confidence: 96%

“…Something similar was applied by Slotte in his doctoral thesis for the Ising model [28] by mixing MF and RG, which is the Mean-Field Renormalization Group method (MFRG) (see also Reference [26]). Then, Li and Yang implemented for the first time the EFRG for the diluted Ising model, to the best of our knowledge [29].…”

Section: Introductionmentioning

confidence: 99%

Phase transition induced by an external field in a three-dimensional isotropic Heisenberg antiferromagnet

et al. 2018

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In this paper, we report mean-field and effective-field renormalization group calculations on the isotropic Heisenberg antiferromagnetic model under a longitudinal magnetic field. As is already known, these methods, denoted by MFRG and EFRG, are based on the comparison of two clusters of different sizes, each of them trying to mimic certain Bravais lattice. Our attention has been on the obtention of the critical frontier in the plane of temperature versus magnetic field, for the simple cubic and the body-centered cubic lattices. We used clusters with N = 1, 2, 4 spins so as to implement MFRG-12, EFRG-12 and EFRG-24 numerical equations. Consequently, the resulting frontier lines show that EFRG approach overcomes the MFRG problems when clusters of larger sizes are considered.

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Effective-field renormalization-group study for the diluted Ising model

Cited by 18 publications

References 21 publications

Phenomenological renormalization group methods

Phenomenological renormalization group methods

Effective‐field renormalization‐group approach for quenched site‐dilute ising models

Phase transition induced by an external field in a three-dimensional isotropic Heisenberg antiferromagnet

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