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Materials science forum

Exact Solution of 1D Ising Model on Linear Chain with Arbitrary Spin

Alexey I. Proshkin, Felix A. Kassan-Ogly

Published Mar 1, 2016• 9 citations
Publisher

Abstract

We investigated the Ising model on a linear chain with arbitrary spin including interactions between nearest and next-nearest neighbors in an applied magnetic field. A series of exact solutions and formulas for frustration fields, magnetizations and entropies at these fields at T→0 are found.
DOI
10.4028/www.scientific.net/msf.845.93

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The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.
Abstract Results on the generalization of the Ising model to an arbitrary number of translations of a linear chain in an external magnetic field, taking into account various interactions between spins, are presented. An exact analytical expression has been derived for the largest eigenvalue of the Kramers–Wannier transfer matrix with a translation per two chain periods in an external magnetic field when the nearest and second neighbors are taken into account. Exact expressions have been established for the zero-temperature entropies and magnetizations at various magnitudes and signs of the exchange interactions and the magnetic field. Many of the zero-temperature entropies and zero-temperature magnetizations found are represented via the so-called mathematical ratios long known in the wonderful world of number mathematics (the golden ratio φ, the silver ratio δ, the supergolden ratio ψ, the plastic number ρ, and new (nameless) ones). A result whereby the frustrating entropies and magnetizations can be expressed via the limit of the ratio of certain number sequences without invoking the formalism of the Kramers–Wannier transfer matrix has been obtained.
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.
Consider a closed, linear chain of NNN localized spins (each s=12s=\frac{1}{2}s=21​) with nearest-neighbor Ising interaction. Such a chain, exposed to a uniform external magnetic field, will be called the pure host. Replace a single spin in the pure host by an impurity spin which is also subjected to nearest-neighbor Ising interaction and to the external field. The magnitude of the impurity spin, the magnitude of its magnetic moment, and the magnitude and sign of its interaction with the host are allowed to differ from the corresponding values characterizing the host. For the bulk system ($N\ensuremath{\rightarrow}\ensuremath{\infty}$, for constant linear density of spins), the thermodynamic properties, such as the impurity magnetization, the (position-dependent) magnetization of the impurity-host system and the impurity-host spin correlation functions, are obtained exactly in terms of conventional, tabulated functions. Numerical results are presented for impurity spins of magnitude S=32,52S=\frac{3}{2},\frac{5}{2}S=23​,25​.