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Brazilian Journal of Physics

Spin pair correlation of the ANNNI chain in a field

Nelson A. Alves, Carlos S. O. Yokoi

Published Dec 1, 2000• 4 citations
Publisher

Abstract

We study the spin pair correlation function of the one-dimensional Ising model with competing nearest and next-nearest neighbor interactions, or ANNNI chain, in the presence of an external field. Of particular interest are the disorder lines where exponential decay of the spin pair correlation changes from monotonic to oscillatory. We extend previous studies for higher field values and obtain asymptotic expressions for disorder lines at low temperatures. We also observe reentrant disorder lines.

DOI
10.1590/s0103-97332000000400005

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Journal of Physics A Mathematical and General•Aug 21, 1991•3 citations
We study the spin pair correlation function of the one-dimensional Ising model with competing nearest and next-nearest neighbor interactions, or ANNNI chain, in the presence of an external field. Of particular interest are the disorder lines where exponential decay of the spin pair correlation changes from monotonic to oscillatory. We extend previous studies for higher field values and obtain asymptotic expressions for disorder lines at low temperatures. We also observe reentrant disorder lines.
The existence of a disorder point TD{T}_{D}TD​ is established for some one- and two-dimensional Ising lattices with antiferromagnetic next-nearest-neighbor interactions. Within the disordered phase, the decay of axial next-nearest-neighbor pair correlations with increasing spin separation is positive monotonic exponential below TD{T}_{D}TD​ and oscillatory with exponential envelope above TD{T}_{D}TD​. A general definition of a disorder point is formulated, and a method of estimating TD{T}_{D}TD​ described.
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.
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 study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent zzz, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the T=0T=0T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.
Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.