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Journal of Physics A Mathematical Nuclear and General

Orderings of one-dimensional Ising systems with an arbitrary interaction of finite range

T. Morita

Published Jan 21, 1974• 22 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/7/2/014

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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 .
For pt. I see abstr. A11864 of 1974. A study is made of the ground state orderings of the one-dimensional Ising system, under the assumption that the interaction is of finite range r and has inversion symmetry. It is shown that the ground state energy is effected by the regular chain either of an irreducible block without or with one symmetry point of r or (r+1) sites, or of a symmetric irreducible block with two symmetry points. For the case of the one-dimensional Ising magnet under a uniform external field, it is effected by two, four, seven and twenty regular orderings, when the range r of the interaction is one, two, three and four, respectively.
We studied regular chains of one-dimensional Ising system of spin s = ½, with two types of interaction at nearest neighbors, at once we extend the analysis up to second neighbors with two types of interaction. The ground states in the diatomic model it is established exploring regular chains of blocks of spins with shorter length, named irreducible blocks. We construct the phase diagrams according to the parameters of interaction. Also there does a simulation of the decrease of the energy for random finite chains of spins, finding the regions of the configurations of lower energy of the chains.
We consider a linear spin 1/2 Ising chain with pair interactions extending up to the nth neighbor. The following general theorem is proved: 'The energy of any arbitrary spin sequence may be written as a linear sum of cluster energies, the upper limit of the number of Ising spins in a cluster being 2n'. The term cluster energy here is used to mean the energy of a group of spins in a certain configuration, evaluated as if the same configuration repeated itself throughout the chain. The structure of the ground state of the system is investigated using the above theorem. It is shown that the ground state is a repetition of a certain cluster, and that the upper limit of the cluster size is 2n, except for some specific combinations of the interaction strengths when the ground state may admit mixing of different clusters. The possible ground state configurations are worked out explicitly for n = 2, 3, and 4 in the absence of the magnetic field using the above theorem. Previous attempts were confined only up to n = 3.
We generalize Morita's works [J. Phys. A 7, 289 (1974); J. Phys. A 7, 1613 (1974)] on ground states of Ising chains, for chains with a periodic structure and different spins, to any interaction order. The main assumption is translational invariance. The length of the irreducible blocks is a multiple of the period of the chain. If there is parity invariance, it restricts the length in general only in the diatomic case. There are degenerated states and under certain circumstances there could be nonregular ground states. We illustrate the results and give the ground state diagrams in several cases.
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.
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.