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Exact eigenvalues of the Ising Hamiltonian in one-, two- and three-dimensions in the absence of a magnetic field

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  • Dixon, J.M.
  • Tuszynski, J.A.
  • Nip, M.L.A.

Abstract

The Hamiltonian of the Ising model in one-, two- and three-dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian for an arbitrary number of dimensions and sites. Although the solution provided assumes the absence of external magnetic fields an extension to include a magnetic field along the z-axis is readily extracted. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one- and two-dimensional cases. We find complete agreement with these studies.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:289:y:2001:i:1:p:137-156
    DOI: 10.1016/S0378-4371(00)00318-6
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    Cited by:

    1. Leonid Litinskii & Boris Kryzhanovsky, 2021. "Inverse Problem for Ising Connection Matrix with Long-Range Interaction," Mathematics, MDPI, vol. 9(14), pages 1-11, July.
    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).
    3. 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.

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