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The Physics of Metals and Metallography

Correlation function of one-dimensional s = 1 Ising model

Alexey I. Proshkin, Tatyana Ponomareva, I. A. Menshikh, Alexander V. Zarubin, Felix A. Kassan-Ogly

Published Oct 1, 2017• 6 citations
Publisher

Abstract

The temperature behavior of the Fourier transform of the spin-correlation function has been studied in terms of the one-dimensional Ising model taking into account the interaction between the nearest neighbors in the cases of different signs of exchange interactions, magnetic field, and spin. It has been shown that, in the antiferromagnetic model, in the frustration field, the correlation function has a broad maximum and does not take on the form of a delta function as the temperature approaches zero, which indicates the absence of ordering in the system.

DOI
10.1134/s0031918x17100106

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The temperature behavior of the Fourier transform of the spin-correlation function has been studied in terms of the one-dimensional Ising model taking into account the interaction between the nearest neighbors in the cases of different signs of exchange interactions, magnetic field, and spin. It has been shown that, in the antiferromagnetic model, in the frustration field, the correlation function has a broad maximum and does not take on the form of a delta function as the temperature approaches zero, which indicates the absence of ordering in the system.
An antiferromagnetic classical XY model with the nearest-neighbor interaction 2 J and the next-nearest-neighbor interaction δ J on a one-dimensional lattice is investigated in the low temperature limit. We obtain the asymptotic behaviors at long distances, of the two-spin correlation function for δ>1/2 and of the chirality-chirality correlation function for δ=1. The behavior of the latter correlation function is shown to be similar to the one of the two-spin correlation function for the Ising model with the nearest-neighbor interaction J /2 on a one-dimensional lattice.
The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.
The phase transitions in two-dimensional antiferromagnetic Ising model are studied on a square lattice by taking the interactions of next-nearest neighbors into account. The model is investigated on basis of the replica Monte Carlo algorithm and the histogrammic analysis of data. The diagram of the critical temperature dependence on an interaction value of next-nearest neighbors is plotted. The studied model reveals the phase transition of second order and the frustration points depending on relative interaction values and magnetic field. A effect, heat capacity splitting near the frustration points, is found.
The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.