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Journal of Mathematical Physics

Energy extremes and spin configurations for the one-dimensional antiferromagnetic Ising model with arbitrary-range interaction

Jack Douthett, Richard J. Krantz

Published Jul 1, 1996• 13 citations
Publisher

Abstract

The one-dimensional antiferromagnetic spin-1/2 Ising model is investigated using the formalism of Maximally/Minimally Even sets. The salient features of Maximally/Minimally Even set theory are introduced. Energy and spin content vectors are defined to facilitate the use of interval spectra used in Maximally/Minimally Even set theory. It is shown that Maximally Even sets of up- and down-spins minimize the configurational energy per spin and that Minimally Even sets maximize configurational energy per spin. An exponentially decreasing antiferromagnetic pairwise interaction of arbitrary range is used as an example interaction. The asymptotic (N→∞) configurational energy per spin and the energy per spin calculated for seven-near neighbors are compared.

DOI
10.1063/1.531568

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The one-dimensional antiferromagnetic spin-1/2 Ising model is investigated using the formalism of Maximally/Minimally Even sets. The salient features of Maximally/Minimally Even set theory are introduced. Energy and spin content vectors are defined to facilitate the use of interval spectra used in Maximally/Minimally Even set theory. It is shown that Maximally Even sets of up- and down-spins minimize the configurational energy per spin and that Minimally Even sets maximize configurational energy per spin. An exponentially decreasing antiferromagnetic pairwise interaction of arbitrary range is used as an example interaction. The asymptotic (N→∞) configurational energy per spin and the energy per spin calculated for seven-near neighbors are compared.
A new one-dimensional antiferromagnetic Ising model with exponential pair interactions is proposed. This model is composed of 2N Ising spins, half of which belong to an α-sublattice and the others to a β-sublattice. The interaction between spins in the same sublattice is long-rang ferromagnetic and that between spins in the different sublattices is long-rang antiferromagnetic. Thermodynamic properties of the model are shown to be the same as those obtained by using the molecular field theory for the Ising model with nearest-neighbor interactions.
The concept of the irreducible block is introduced for the one-dimensional Ising system with an arbitrary interaction of finite range. It is proved that the ground state energy of the system occurs for a regular chain of one of the irreducible blocks or for coexistence of the regular chains of the irreducible blocks. It is shown that the ground state energy of the linear Ising magnet with pair interactions up to third neighbours is effected by the seven spin orderings recently given for the case of spin 1 / 2 by Katsura and Narita (1973), and that the ground state energy of the Ising magnet of spin larger than 1 / 2 is effected by the same set of orderings as for the corresponding Ising magnet of spin 1 / 2 .
The mapping between the fermion and spinon compositions of eigenstates in the one-dimensional spin-1/2 XX model on a lattice with N sites is used to describe the spinon interaction from two different perspectives: (i) For finite N the energy of all eigenstates is expressed as a function of spinon momenta and spinon spins, which, in turn, are solutions of a set of Bethe ansatz equations. The latter are the basis of an exact thermodynamic analysis in the spinon representation of the XX model. (ii) For N -> infinity the energy per site of spinon configurations involving any number of spinon orbitals is expressed as a function of reduced variables representing momentum, filling, and magnetization of each orbital. The spins of spinons in a single orbital are found to be coupled in a manner well described by an Ising-like equivalent-neighbor interaction, switching from ferromagnetic to antiferromagnetic as the filling exceeds a critical level. Comparisons are made with results for the Haldane-Shastry model.
We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.
The one-dimensional s=1/2 quantum spin model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions is considered. The behavior of the system in the vicinity of the transition point from the antiferromagnetic ground state to the ferromagnetic one is studied. The consideration is based on the classical approximation and the regular procedure is proposed to find the corrections to the classical energy. The dependence of the energy on spin S is calculated. We obtain evidence of a phase separation for the state with intermediate S. \textcopyright{} 1996 The American Physical Society.
The critical values of the ratio of interaction energies at which the ordering of spins in the ground state is indeterminate (and probably Tc=0{T}_{c}=0Tc​=0) are calculated exactly for a variety of Ising lattices with antiferromagnetic next-nearest-neighbor coupling, and for anisotropic nearest-neighbor coupling. Using these results and the exact two-dimensional solutions, the general dependence of the critical point on interaction ratio is sketched. The complex case of the antiferromagnetic fcc is reconsidered.