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arXiv:1612.08494

Clustering in generalized 1D Ising models

Petr Dmitrievich Andriushchenko, Konstantin Valentinovich Nefedev

Published Dec 27, 2016• 0 citations
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Abstract

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 $J$ but clustered order parameter describes the transition regardless of the sign of the exchange integral and the number of neighbors.

DOI
10.48550/arxiv.1612.08494

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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.
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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 one-dimensional Ising model with random exchange energy is formulated in terms of a homogeneous integral equation. Assuming the input distribution has a narrow width proportional to ${N}^{\ensuremath{-}1}$, the integral equation is solved by perturbation in ${N}^{\ensuremath{-}1}$. The shift in the free energy of the system up to the order of ${N}^{\ensuremath{-}2}$ is given. It is found that for a symmetrical distribution, the shift due to the randomness is second order in ${N}^{\ensuremath{-}1}$ and negative, depending only upon the variance of the input distribution. The first-order shift for the asymmetric distribution comes entirely from the asymmetry. After a shift in energy is made to account for this asymmetry, the effective shift is identical to that for the symmetrical case. The shifts of all the thermodynamic properties of the system are also given. The randomness is found to decrease the magnetization for all temperature and applied field. However, shifts in magnetic susceptibility and specific heat are oscillatory in sign.
The statement that any phase transition is related to the appearance or disappearance of long-range spatial correlations and precludes a finite transition temperature in one-dimensional (1D) systems is rooted in condensed matter theory. In this paper, we demonstrate that the 1D Ising model with short-range exchange interactions exhibits a second-order phase transition at a finite temperature relying on the proper choice of the order parameter. We combined analytical calculations and high-precision entropic sampling simulations and chose a slightly different order parameter, namely, the modulus of the magnetization. Notably, we detected a phase transition with a corresponding critical temperature around 15 K, which is in excellent agreement with experimental results. Our study indicates that the incorrect choice of the order parameter may mask phase transitions in 1D systems.