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arXiv:2212.12693 · Journal of the Physical Society of Japan

The Exact Susceptibility of the Spin-S Transverse Ising Chain with Next-nearest-neighbor Interactions

Kazuhiko Minami

Published Apr 14, 2023• 1 citation
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Abstract

The zero-field susceptibility of the spin-S transverse Ising chain with next-nearest-neighbor interactions is obtained exactly. The susceptibility is given in an explicit form for S=1/2, and expressed in terms of the eigenvectors of the transfer matrix for general spin S. It is found that the low-temperature limit is independent of spin S, and is divergent at the transition point.

DOI
10.7566/jpsj.92.054001

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The zero-field susceptibility of the spin-S transverse Ising chain with next-nearest-neighbor interactions is obtained exactly. The susceptibility is given in an explicit form for S=1/2, and expressed in terms of the eigenvectors of the transfer matrix for general spin S. It is found that the low-temperature limit is independent of spin S, and is divergent at the transition point.
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.
We study the transverse susceptibility of the fully frustrated antiferromagnetic Ising Δ-chain, extending Minami's transfer-matrix method to the transverse susceptibility of general-type Ising linear-chains [J. Phys. Soc. Jpn. 67 (1998), 2255]. For the transverse fields Γ1 in tip spin sites and Γ2 in bottom spin sites, we calculate zero-field transverse susceptibilities χxtip = lim Γ1 →0Mxtip/Γ1 and χxbottom = lim Γ2 →0Mxbottom/Γ2 while keeping the ratio Γ1/Γ2 fixed, where Mxtip (bottom) denotes the magnetization for tip (bottom) spin sites. Both the transverse susceptibilities follow Curie's law at low temperatures. We also calculate χxbottom(Γ1 > 0), the transverse susceptibility of the bottom spin chain under a finite tip-spin transverse field, to understand the Curie-type behavior in zero-field susceptibility. Using the second-order perturbation theory, we also discuss the Γ1 dependence of χxbottom (Γ1) at zero temperature.
The zero-field transverse susceptibility of a one-dimensional random bond Ising model with alternating spin 1/2 and S is exactly obtained. The integral form of the susceptibility is written. The thermal curve of the susceptibility becomes irregular when weak interactions appear with a small probability. The leading-order terms of an expansion, in terms of the moment of the distribution of a random variable, are obtained. The first-order term vanishes and the second-order term is written explicitly. The expansion up to infinite order is exactly obtained in the case of uniform spin S = 1/2. The classical limit S → ∞ is also considered. It is found that the low-temperature limit T → 0 of the susceptibility is independent of S .
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.