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Journal of Physics A Mathematical and General

Real space renormalization group approach to the random field Ising model

Ido Dayan, Moshe Schwartz, A. P. Young

Published Jul 7, 1993• 27 citations
Publisher

Abstract

The authors discuss results from two types of real space renormalization group (RSRG) calculations applied to the random field Ising model in three dimensions. Starting from a lattice of size L, the RSRG is used to reduce the lattice to a size L=2, on which the trace is done exactly. In this way, thermodynamic properties, such as the magnetization and susceptibility, can be determined approximately. They find that, for a given size, the susceptibility increases as the temperature, T, is reduced down to the transition temperature, T c , and becomes essentially independent of temperature below T c . Both in the vicinity of T c and at lower temperatures, there are large sample-to-sample fluctuations in the susceptibility which grow with increasing system size. They interpret these results in terms of the droplet theory of the transition.

DOI
10.1088/0305-4470/26/13/014

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Physical review. B, Condensed matter•May 1, 1979•23 citations
The authors discuss results from two types of real space renormalization group (RSRG) calculations applied to the random field Ising model in three dimensions. Starting from a lattice of size L, the RSRG is used to reduce the lattice to a size L=2, on which the trace is done exactly. In this way, thermodynamic properties, such as the magnetization and susceptibility, can be determined approximately. They find that, for a given size, the susceptibility increases as the temperature, T, is reduced down to the transition temperature, T c , and becomes essentially independent of temperature below T c . Both in the vicinity of T c and at lower temperatures, there are large sample-to-sample fluctuations in the susceptibility which grow with increasing system size. They interpret these results in terms of the droplet theory of the transition.
We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d , introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .
We present real-space renormalization-group (RG) calculations of the critical properties of the random-field Ising model on a cubic lattice in three dimensions. We calculate the RG flows in a two-parameter truncation of the Hamiltonian space. As predicted, the transition at finite randomness is controlled by a zero-temperature, disordered critical fixed point, and we exhibit the universal crossover trajectory from the pure Ising critical point. We extract scaling fields and critical exponents, and study the distribution of barrier heights between states as a function of length scale.
Here is the first part of the summary of my work on random Ising model using real-space renormalization group (RSRG), also known as a Migdal-Kadanoff one. This approximate renormalization scheme was applied to the analysis thermodynamic properties of the model, and of probabilistic properties of a pair correlator, which is a fluctuating object in disordered systems. PACS numbers: 02.50.-r, 05.20.-y, 05.70.Fh, 64.60.ae, 75.10.-b, 75.10.Hk, 87.10.+e Keywords: statistical physics, magnetism, spin glass, neural networks
We introduce and discuss a real-space renormalization group (RSRG) procedure on very small lattices, which in principle does not require any of the usual approximations, e.g. a cut-off in the expansion of the Hamiltonian in powers of the field. The procedure is carried out numerically on very small lattices (4x4 to 2x2) and implemented for the Ising Model and the q=3,4,5 Potts Models. Nevertheless, the resulting estimates of the correlation length exponent and the magnetization exponent are typically within 3% to 7% of the exact values. The 4-state Potts Model generates a third magnetic exponent which seems to be unknown in the literature. A number of questions about the meaning of certain exponents and the procedure itself arise from its use of symmetry principles and its application to the q=5 Potts Model.
We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain $\overline{\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉\ensuremath{-}〈\ensuremath{\sigma}(0)〉〈\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``connected'' correlation function, and $\overline{S(\stackrel{\ensuremath{\rightarrow}}{r})}=\overline{〈\ensuremath{\sigma}(0)\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{r})〉},$ the ``disconnected'' correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function $\overline{\ensuremath{\Gamma}(r)}$ along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such $\ensuremath{\nu}$ or $\ensuremath{\eta}).$ The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.
The mean-field approximation is used for the evaluation of the Niemeijer---Van Leeuwen real-space renormalization. Although this new method is very simple, it gives reasonable results for the critical properties of two-, three-, and four-dimensional Ising models, for the transition temperature Tc{T}_{c}Tc​ of an Ising model with clusters and for the two-dimensional Ising antiferromagnet in an external magnetic field. Using this new mean-field renormalization in connection with the real-space renormalization of the kinetic Ising model, which recently has been suggested by the author, we obtain the long-time relaxation of the magnetization for all temperatures above Tc{T}_{c}Tc​. In the last application the critical properties of nnn-dimensional spins in two and three space dimensions are calculated for arbitrary nnn. Here the mean-field approximation and Monte Carlo methods are used to evaluate the renormalization step.