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Journal of the Physical Society of Japan

Spin-Lattice Coupling in an Ising Chain with Nearest Neighbour and Next Nearest Neighbour Exchange Interactions

Norikazu Ishimura

Published Apr 1, 1984• 5 citations
Publisher

Abstract

A one-dimensional Ising chain in which nearest neighbour (n.n.) and next nearest neighbour (n.n.n.) exchange interactions depend linearly on the distance between the spins is considered. The magnetic field and temperature dependence of the chain length is exactly calculated by use of a transfer matrix. It is shown that the chain length changes variously because of the competition among the n.n. and n.n.n. exchange interactions and the magnetic field. The zero-temperature phase diagram is also determined, which is composed of four spin states ↑, ↑↓, ↑↑↓ and ↑↑↓↓ (the smallest units of repetitions are shown). Dimerization and trimerization of the lattice are found in the ↑↑↓↓ and ↑↑↓ states respectively.

DOI
10.1143/jpsj.53.1472

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A one-dimensional Ising chain in which nearest neighbour (n.n.) and next nearest neighbour (n.n.n.) exchange interactions depend linearly on the distance between the spins is considered. The magnetic field and temperature dependence of the chain length is exactly calculated by use of a transfer matrix. It is shown that the chain length changes variously because of the competition among the n.n. and n.n.n. exchange interactions and the magnetic field. The zero-temperature phase diagram is also determined, which is composed of four spin states ↑, ↑↓, ↑↑↓ and ↑↑↓↓ (the smallest units of repetitions are shown). Dimerization and trimerization of the lattice are found in the ↑↑↓↓ and ↑↑↓ states respectively.
Abstract The frustration properties of the Ising model on a one-dimensional monoatomic equidistant lattice are investigated taking into account the exchange interactions of atomic spins at the sites of the first (nearest), second (next-nearest) and third neighbors. The exact solution of the model was obtained using the Kramers–Wannier transfer matrix method. The types of magnetic ordering of the ground state of the model are determined, and a magnetic phase diagram is constructed. Criteria are formulated for the occurrence of magnetic frustrations in the presence of competition between the energies of exchange interactions. Non-zero entropy values in the ground state of the frustrated system were found.
Phase transitions of the spin-one Ising model with nearest-neighbour and next-nearest-neighbour interactions are investigated for a square lattice using a molecular-field theory and a Monte Carlo method. Special attention is paid to the effects of the biquadratic next-nearest-neighbour interaction on the phase transitions of the model. Phase diagrams characterizing types of phase transition are obtained for several values of the coupling constants. The existence of the multi-critical points and the successive phase transitions are also discussed in connection with these phase diagrams.
A one-dimensional Ising chain with spin-lattice coupling is considered. If the lattice is harmonic but has no translational symmetry, there appears effectively through the spin-lattice coupling a four-spin interaction which decays exponentially with distance. It is shown rigorously that this spin system in the ground state exhibits a devil's staircase as a function of the exchange parameter J because of the long range interaction. The ground state phase diagram of the spin system under an external magnetic field is speculated.
A one-dimensional antiferromagnetic Ising chain with spin-lattice coupling is considered. If the underlying lattice is harmonic but has no translational symmetry, there appears at zero temperature an effective interaction between down spins, which is repulsive and decays exponentially with distance. It is shown rigorotusly that this system exhibits in the ground state a devil's staircase as a function of an external magnetic field on account of the effective long range interaction. The ground state phase diagram is also given.