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Journal of Physics C Solid State Physics

A complex short-range order phase diagram in a one-dimensional spin model

Andrzej Fleszar, J. Glazer, G. Baskaran

Published Sep 30, 1985• 2 citations
Publisher

Abstract

The authors study in detail the short-range order at low temperature in a one-dimensional Ising model with competing interactions by studying various staggered susceptibilities. From this they get a 'phase diagram' which divides regions of various dominant short-range orders. This phase diagram is shown to resemble qualitatively the low-temperature phase diagram of the anisotropic next-nearest-neighbour interaction Ising (ANNNI) model in higher dimensions. They argue that this method of studying a corresponding one-dimensional problem may be useful to understand the phase structure of complex higher-dimensional systems.

DOI
10.1088/0022-3719/18/27/020

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The authors study in detail the short-range order at low temperature in a one-dimensional Ising model with competing interactions by studying various staggered susceptibilities. From this they get a 'phase diagram' which divides regions of various dominant short-range orders. This phase diagram is shown to resemble qualitatively the low-temperature phase diagram of the anisotropic next-nearest-neighbour interaction Ising (ANNNI) model in higher dimensions. They argue that this method of studying a corresponding one-dimensional problem may be useful to understand the phase structure of complex higher-dimensional systems.
As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature. Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply. Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.
We have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant D, in which equivalent-neighbor ferromagnetic interactions of strength -J are superimposed on nearest-neighbor antiferromagnetic interactions of strength K. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the D/J-K/J plane, namely the ferromagnetic (F) and the antiferromagnetic one (AF). For lower values of D/J(D/J<0.25) these two ordered phases are separated by the point K_{c}=0.25J. For 0.25<D/J≤0.50, the paramagnetic phase P emerges in a region separated between the lines determined by D/J=0.5-K/J and D/J=K/J. For D/J>0.5, only phases AF and F exist and are separated by a line given by D/J=K/J. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the k_{B}T/J-D/J plane is enriched by another ferromagnetic phase F^{^{'}} above a first-order line for 0.195<K/J<0.250. This first-order line, which separates phases F and F^{^{'}}, begins at a coexistence point, where phases F,F^{^{'}}, and P coexist, and ends at an ordered critical point. Similarly, we found that the phase F^{^{'}} is present in the phase diagram in the k_{B}T/J-K/J plane for 0.228<D/J<0.286.
The analysis of phase diagrams of one-dimensional quantum versions of axial next-nearest-neighbor Ising (ANNNI) and competing two- and four-spin interaction models in two dimensions was made. The zero-temperature wave-vector-dependent spectra for finite chains with up to L≤16 spins indicate a qualitative difference between the two models: while the existence of an incommensurate phase is confirmed for the ANNNI case, our results do not suggest that such a phase exists in the two- and four-spin model.
We study the dependence of complex-temperature phase diagrams on details of the Hamiltonian, focusing on the effect of non-nearest-neighbor spin-spin couplings. For this purpose, we consider a simple exactly solvable model, the one-dimensional (1D) Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. We work out the exact phase diagrams for various values of JNNN{\mathrm{J}}_{\mathrm{NNN}}JNNN​/JNN{\mathrm{J}}_{\mathrm{NN}}JNN​ and compare these with the case of pure NN couplings. We also give some similar results for the 1D Potts model with NN and NNN couplings.
Due to intrinsic frustrations of interaction, the 2d Ising model with competing ferromagnetic short-range nearest-neighbour and antiferromagnetic long-range dipole interactions possesses a rich phase diagram. The order of the phase transition from the striped h = 1 phase to the tetragonal phase that is observed in this model has been a subject of recent controversy. We address this question by using the partition function density analysis in the complex temperature plane. Our results support the second-order phase transition scenario. To measure the strength of the phase transition, we calculate the values of specific heat critical exponent a. Along with the space dimension, it appears to depend on the ratio of strengths of the short-range and long-range interactions.