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Journal of Mathematical Physics

Perpendicular susceptibility of two one-dimensional Ising chains

Gary S. Grest, A. K. Rajagopal

Published May 1, 1974• 12 citations
Publisher

Abstract

Exact, closed expressions are obtained for the perpendicular and parallel susceptibilities of the one-dimensional spin-1/2 Ising model (i) with nearest neighbor interactions in a uniform magnetic field and (ii) with next-nearest neighbor interactions but without the magnetic field, in the thermodynamic limit. Graphs of the susceptibilities as a function of temperature for various magnetic field strengths in (i), and as a function of different ratios of the interaction strengths in (ii), are discussed.

DOI
10.1063/1.1666689

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Journal of the Physical Society of Japan•Apr 15, 1993•3 citations
Exact, closed expressions are obtained for the perpendicular and parallel susceptibilities of the one-dimensional spin-1/2 Ising model (i) with nearest neighbor interactions in a uniform magnetic field and (ii) with next-nearest neighbor interactions but without the magnetic field, in the thermodynamic limit. Graphs of the susceptibilities as a function of temperature for various magnetic field strengths in (i), and as a function of different ratios of the interaction strengths in (ii), are discussed.
The zero-field isothermal susceptibility of the one-dimensional Ising model with nearest-neighbor interactions and a finite number of spins is shown to have a relatively simple singularity as the temperature approaches zero, proportional only to the inverse temperature. This is in contrast to what is seen throughout the literature for the inifinite chain: an essential singularity that includes an exponential dependence on the inverse temperature. Assuming an arbitrary (but finite) number of spins and retaining terms that are usually considered ignorable in the thermodynamic limit, the analysis involves nothing beyond straightforward series expansions, starting either from the partition function for a closed chain in a magnetic field, obtained using the transfer-matrix approach; or from the expression for the zero-field susceptibility found via the fluctuation-dissipation theorem. In both cases, the exponential singularity is exactly removed. In addition, the susceptibility per spin is found to increase with the number of spins (except in the case of noninteracting spins), a result which is also at variance with what is normally reported for an infinite chain.
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.
In the Ising model on a helical lattice with period M (in the limit M →∞ of this lattice, the model corresponds to the two-dimensional Ising model), the exact simultaneous equations to calculate physical quantities are derived. The zero field magnetic susceptibility χ 0 , and also the specific heat C are obtained numerically for the system of M ≤11. Furthermore, in the Ising model with up to the next nearest neighbor interactions on the helical lattice, χ 0 and C are calculated for the system of M ≤9. The results obtained are an excellent approximation to the exact χ 0 and C of the two-dimensional systems.