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.
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Funding
The work was performed within the State assignment of the Ministry of Science and Higher Education of the Russian Federation (theme “Quantum,” no. AAAA-A18-118020190095-4 and “Alloys,” no. AAAA-A19-119070890020-3) and under partial support of the Ural branch of the Russian Academy of Sciences (project no. 18-2-2-11).
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Translated by V. Astakhov
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Tsuvarev, E.S., Kassan-Ogly, F.A. Generalized Ising Model in a Magnetic Field. J. Exp. Theor. Phys. 133, 191–205 (2021). https://doi.org/10.1134/S1063776121080112
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DOI: https://doi.org/10.1134/S1063776121080112