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Thermodynamical Averages For The Ising Model And Spectral Invariants Of Toeplitz Matrices

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Abstract

We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of N particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large N, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable β (β is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as N → ∞

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Authors and Affiliations

  1. Institute of Mathematics, Mechanics, and Computer Sciences, Southern Federal University, Rostov-on-Don, Russia

    V. M. Kaplitsky

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  1. V. M. Kaplitsky
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Correspondence to V. M. Kaplitsky.

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The author declares no conflicts of interest.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 3, pp. 401–416, June, 2020.

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Kaplitsky, V.M. Thermodynamical Averages For The Ising Model And Spectral Invariants Of Toeplitz Matrices. Theor Math Phys 203, 780–793 (2020). https://doi.org/10.1134/S0040577920060069

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  • DOI: https://doi.org/10.1134/S0040577920060069

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