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Progress of Theoretical Physics

Scaling Structures of Free-Energy Fluctuations in a One-Dimensional Dilute Ising Model

Takuma Tanaka, Hirokazu Fujisaka, Masayoshi Inoue

Published Oct 1, 1990• 5 citations
Publisher

Abstract

Free-energy fluctuations in a one-dimensional dilute Ising system in the low dilution case are studied from the fluctuation-spectrum standpoint by discussing how the position-to-position fluctuation of free energy arises as magnetic bonds are gradually replaced by non-magnetic ones. The fluctuation-strength monotonically increases as the ratio of dilution is increased. This is found to be described as the scaling laws of relevant functions characteristic of the fluctuation-spectrum analysis.

DOI
10.1143/ptp.84.584

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Progress of Theoretical Physics Supplement•Jan 1, 2000•2 citations
Free-energy fluctuations in a one-dimensional dilute Ising system in the low dilution case are studied from the fluctuation-spectrum standpoint by discussing how the position-to-position fluctuation of free energy arises as magnetic bonds are gradually replaced by non-magnetic ones. The fluctuation-strength monotonically increases as the ratio of dilution is increased. This is found to be described as the scaling laws of relevant functions characteristic of the fluctuation-spectrum analysis.
The hypothesis of weak universality is tested for the moments of the zero-field correlation function and the magnetic-field derivatives of the free energy of dilute Ising models. The investigation, which is carried out for dilution of both the magnetic sites (site dilution) and interactions (bond dilution), is performed just above the critical temperature. In the case of bond dilution both the quenched and annealed distributions are considered. The quenched model is investigated by series methods for site and bond dilution on the face-centred cubic lattice and for bond dilution on the triangular lattice.
We consider a one-dimensional Ising model (chain) with the the nearest-neighbor interaction and with a random nonmagnetic dilution. We find the exact free energy of such a chain as a function of the impurity concentration, temperature, and the external magnetic field. In the case of antiferromagnetic interaction in the chain, we find the specific magnetization, the mean value of the product of neighboring spins, and the entropy as functions of these parameters. We study the residual system entropy.
The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe=1/2ν_e=1/2νe​=1/2 in contrast to the pure system where νe=1ν_e=1νe​=1. The calculation is based on a previously developed effective field theory for the energy density fluctuations.
In this letter we show strong numerical evidence that the four dimensional Diluted Ising Model for a large dilution is not described by the Mean Field exponents. These results suggest the existence of a new fixed point with non-gaussian exponents. PACS numbers: 75.10 Nr. 1 Introduction Random magnetic systems have been the subject of intensive studies over the last 20 years and much progress has been achieved. The simplest model for a random magnetic system is a ferromagnetic system in which the disorder induces fluctuations in the value of the coupling (or equivalently of the temperature). The simplest realization is a randomly diluted Ising system, where sites (site diluted) or bonds (bond diluted) are randomly removed. The equivalent Ginsburg Landau model has the following form Z J = Z d[OE] exp(\\GammaS J [OE]); (1) where S J [OE] = Z d D x ` 1 2 (@ ¯ OE(x)) 2 + 1 2 (m 2 + J(x))OE(x) 2 + g 4! OE(x) 4 ' ; (2) and the quenched random variables J are Gaussian d...
The non-equilibrium dynamics of the bond-diluted Ising model in two-dimensions (2d) is investigated numerically for a range of bond concentration 0≤p≤1. The fraction of spins which never flip, P(∞), is found to increase monotonically from zero for p = 0 with increasing bond concentration to a maximum value of about 0.46 for p = 0.5 and then decreases to zero for p = 1. For strong bond-dilution (0≤p≤0.5) we find that r(t) = P(t)-P(∞) decays exponentially to zero at large times. for weak dilution (0.975≤p ≪ 1.0), three distinct regimes are identified: a short time regime where the behaviour is purelike; an intermediate regime where the persistence probability decays nonalgebraically with time; and a long time \lqfrozen \rq regime where the domains cease to grow. Our results for strong dilution are consistent with recent work of Newman and Stein, and suggest that persistence in diluted and pure systems falls into different classes. Furthermore, its behaviour would also appear to depend crucially on the strength of the dilution present.