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Advanced materials research

Magnetic Phase Transitions in the Lattice Ising Model

Petr Dmitrievich Andriushchenko, Konstantin Valentinovich Nefedev

Published Jul 1, 2013• 5 citations
Publisher

Abstract

In this paper we consider an approach, which allows the research of order-disorder transitionin lattice systems (with any distribution of the exchange integrals) in the frame of Ising model. Anew order parameters, which can give a description of a phase transitions, are found. The commondefinition of such order parameter is the mean value of percolation cluster size. Percolation clusterincludes spins in ground state. The transition from absolute disorder to correlated phase could bestudied with using of percolation theory methods.

DOI
10.4028/www.scientific.net/amr.718-720.166

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In this paper we consider an approach, which allows the research of order-disorder transitionin lattice systems (with any distribution of the exchange integrals) in the frame of Ising model. Anew order parameters, which can give a description of a phase transitions, are found. The commondefinition of such order parameter is the mean value of percolation cluster size. Percolation clusterincludes spins in ground state. The transition from absolute disorder to correlated phase could bestudied with using of percolation theory methods.
In this paper we consider an approach, which allows researching a processes of order-disorder transition in various systems (with any distribution of the exchange integrals signs) in the frame of Ising model. A new order parameters, which can give a description of a phase transitions, are found. The common definition of these order parameters is the mean value of percolation cluster size. Percolation cluster includes spins with given energy. The transition from absolute disorder to correlated phase could be studied with using of percolation theory methods.
The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability ppp depending on the NN coupling constant JJJ and the temperature TTT, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.
We discuss the phase transition in an Ising model with correlated disorder. Two parameters describe the disorder: its variance and its finite correlation lengthscale. We show that in this model, depending on the disorder parameters, one of two qualitatively different scenarios for the transition applies. The first is a transition driven by thermal fluctuations around a spatially homogeneous ground state. This is also found in systems with uncorrelated disorder. The second scenario is a percolative one: locally ordered regions grow in the paramagnetic phase and form an infinite cluster at the critical temperature. In contrast to the first scenario, thermal fluctuations now occur around an inhomogeneous ground state. The dominating lengthscale is not the correlation length of thermal fluctuations but the connectivity length of ordered regions. Based on a discussion of the role of thermal fluctuations in the percolative scenario we identify the parameter ranges in which the different scenarios apply.
The research of one-dimensional Ising model with the different number of neighbors and the different sign of the exchange integral were performed. The clustered order parameter is proposed for the description of the magnetic transition in spin systems regardless of the sign of the exchange integral and the number of neighbors. The method of calculation of critical field to change the magnetic states was presented. For diluted 1D antiferromagnetic model two plateaus in magnetization behavior are observed, and explained. Influence of length of interaction plays critical role for phase transition of dipoles system. Dipoles on hexagonal lattice show the phase transition only in model of long range interaction, and there isn't critical behavior for short range. The approach to calculation of spin system in Heisenberg model is proposed.
The approach to determination of the order parameter in the finite 1D Ising model was proposed. For such systems a root mean square magnetization and an average size of the maximal ground state cluster which is formed by neighboring spins with an energy {-2J} does not disappear at T!=0. The temperature dependence of heat capacity and its derivative with respect to temperature from the average size of the maximum quasi ground state cluster, formed by the neighboring spins with an energy {-2J,0} in the 1D Ising chain were considered. The percolation approach in Curie-Weiss does not work due the internal structure of the model.
The research of one-dimensional Ising model with the different number of neighbors and the different sign of the exchange integral were performed. Magnetization shows the transition from disorder to order (in finite systems) only at positive exchange integral JJJ but clustered order parameter describes the transition regardless of the sign of the exchange integral and the number of neighbors.