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Physical review. B, Condensed matter

Pair correlations for double-chain and triple-chain Ising models with competing interactions

Terufumi Yokota

Published Jun 1, 1989• 4 citations
Publisher

Abstract

Pair correlations for the Ising model on a triple-chain with a crossed second-neighbor interaction are obtained analytically and their asymptotic decay for the large spin separation is investigated. The change in the nature of the long-distance asymptotic decay is observed in small regions of the parameter space, where the interactions are competing. This change is caused by the interchange of the second largest eigenvalue of the transfer matrix in absolute value. The results are compared with the double-chain case to study the effect of the frustration in the triangle made of three nearest-neighbor spins on different chains when the interchain nearest-neighbor interaction is antiferromagnetic.

DOI
10.1103/physrevb.39.12312

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Pair correlations for the Ising model on a triple-chain with a crossed second-neighbor interaction are obtained analytically and their asymptotic decay for the large spin separation is investigated. The change in the nature of the long-distance asymptotic decay is observed in small regions of the parameter space, where the interactions are competing. This change is caused by the interchange of the second largest eigenvalue of the transfer matrix in absolute value. The results are compared with the double-chain case to study the effect of the frustration in the triangle made of three nearest-neighbor spins on different chains when the interchain nearest-neighbor interaction is antiferromagnetic.
We present exact four-spin- and string-correlation functions derived for an Ising chain with competing geometrical nearest-neighbor short-range and mirror-image-type long-range interactions. Unusual $T\ensuremath{\ne}0$ correlation effects were found, indicating qualitative changes in the behavior of the system in different T domains. These domains are not separated by a sharp, traditional phase transition, but are delimited by a temperature Ti{T}_{i}Ti​ at which the exact connected four-spin-correlation function vanishes. At the same Ti{T}_{i}Ti​ the pair-correlation function changes its character: The functional form of the correlation length and the nature of the long-range decay are modified.
Abstract We study the effect of competing interactions on ensemble inequivalence. We consider a one-dimensional Ising model with ferromagnetic mean-field interactions and short-range nearest-neighbor (NN) and next-NN couplings which can be either ferromagnetic or antiferromagnetic. Despite the relative simplicity of the model, our calculations in the microcanonical ensemble reveal a rich phase diagram. The comparison with the corresponding phase diagram in the canonical ensemble shows the presence of phase transition points and lines which are different in the two ensembles. As an example, in a region of the phase diagram where the canonical ensemble shows a critical point and a critical end point, the microcanonical ensemble has an additional critical point and also a triple point. The regions of ensemble inequivalence typically occur at lower temperatures and at larger absolute values of the competing couplings. The presence of two free parameters in the model allows us to obtain a fourth-order critical point, which can be fully characterized by deriving its Landau normal form.
The partition function and spin pair correlation functions have been calculated exactly for a classical linear chain model with alternate next-nearest-neighbor (nnn) interactions, in which the interaction energies between pairs of nearest (nn) and next-nearest (nnn) neighbor spins are arbitrary functions of the angles between the relevant spins. Of special interest is the cosine interaction model described by the Hamiltonian ℋ=−ΣNi=1[J1(cosϑ2i−1,2i +cosϑ2i,2i+1)+J2cosϑ2i−1,2i+1]. When the nnn interaction is antiferromagnetic (J2<0) it competes with the nn interaction J1, and there can be disorder point(s) at which nnn correlations change from monotonic to oscillatory. The ground state is ferromagnetic when the interaction ratio r≡J2/‖J1‖ ≳−1/2=rc, but is disordered for more negative values. The disorder point locus has been determined. It terminates at zero temperature at rD=−1/21/2, at which point the ground state energy is a maximum. The result that rD differs from rc is thought to be peculiar to one-dimensional models. Over a limited range of values of r there can be two disorder points. The low temperature asymptotic behavior of the partition and correlation functions is analyzed in detail. Also a novel summation formula for spherical Bessel functions is obtained.
We study the ground-state properties of the one-dimensional anisotropic spin-1/2 antiferromagnetic Heisenberg model with competing antiferromagnetic second-neighbor Ising interaction. Using the modified Lanczos method, we obtain the phase diagram in parameter space and different correlation functions, which we use to characterize the different phases. For small next-nearest-neighbor parameters we evaluate the critical exponent \ensuremath{\eta} of the spin-spin correlation function. We find good agreement between the numerical results and those obtained using the Luther and Peschel approach.
The square Ising lattice with nearest- and next-nearest-neighbour interactions is considered. On the basis of the simplest ansatz on the principal eigenvector of the transfer matrix of the model and of the CLE condition, developed in previous papers, the correlation length along a column is calculated for all T>or=T c , in the ferromagnetic and antiferromagnetic regions. A comparison is made with results obtained through other methods. For competing interactions, it is shown that there is a mechanism which leads to a correlation length made of two analytic pieces.