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Analytical expressions for energies, degeneracies and critical temperatures of the 2D square and 3D cubic Ising models

Author

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  • Dixon, J.M.
  • Tuszyński, J.A.
  • Carpenter, E.J.

Abstract

This paper revisits the fundamental statistical properties of the crucial model in critical phenomena i.e., the Ising model, guided by our knowledge of the energy values of the Ising Hamiltonian and aided by numerical estimation techniques. We have obtained exact energies in 2D and 3D and nearly exact analytical forms for the degeneracies of the distinct eigenvalues. The formulae we obtained, both for energies and degeneracies, have an exceedingly simple analytical form and are easy to use. The resultant partition functions were utilised to determine the critical behaviour of the Ising system on cubic lattices in 2D and 3D. We obtained a logarithmic divergence of the specific heat in the 2D case and the critical temperature estimates provided additional confirmation of the correctness of our approach.

Suggested Citation

  • Dixon, J.M. & Tuszyński, J.A. & Carpenter, E.J., 2005. "Analytical expressions for energies, degeneracies and critical temperatures of the 2D square and 3D cubic Ising models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(3), pages 487-510.
    • Handle: RePEc:eee:phsmap:v:349:y:2005:i:3:p:487-510
    DOI: 10.1016/j.physa.2004.10.029
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    References listed on IDEAS

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    1. Dixon, J.M. & Tuszynski, J.A. & Nip, M.L.A., 2001. "Exact eigenvalues of the Ising Hamiltonian in one-, two- and three-dimensions in the absence of a magnetic field," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 289(1), pages 137-156.
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    Cited by:

    1. Kryzhanovsky, Boris & Litinskii, Leonid, 2017. "Applicability of n-vicinity method for calculation of free energy of Ising model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 493-507.
    2. Litinskii, Leonid & Kryzhanovsky, Boris, 2018. "Spectral density and calculation of free energy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 510(C), pages 702-712.

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    2. Litinskii, L.B. & Kryzhanovsky, B.V., 2020. "Eigenvalues of Ising connection matrix with long-range interaction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 558(C).

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