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Eigenvalues of Ising connection matrix with long-range interaction

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  • Litinskii, L.B.
  • Kryzhanovsky, B.V.

Abstract

We examine multidimensional Ising systems on hypercube lattices and calculate analytically the eigenvalues of their connection matrices. We express the eigenvalues in terms of spin–spin​ interaction constants and the eigenvalues of the one-dimensional Ising connection matrix (the latter are well known). To do this we present the eigenvectors as Kronecker products of the eigenvectors of the one-dimensional Ising connection matrix. For periodic boundary conditions, it is possible to obtain exact results for interactions with an arbitrary large number of neighboring spins. We present exact expressions for the eigenvalues for two- and three-dimensional Ising connection matrices accounting for the first five coordination spheres (that is interactions up to next-next-next-next nearest neighbors). In the case of free-boundary systems, we show that in the two and three dimensions the exact expressions could be obtained only if we account for interactions with spins of not more than first three coordination spheres.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:phsmap:v:558:y:2020:i:c:s0378437120304805
    DOI: 10.1016/j.physa.2020.124929
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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. 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. Shen, Yao & Zhou, Chi-Chun & Chen, Yu-Zhu, 2022. "The elementary excitation of spin lattice models: The quasiparticles of Gentile statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).

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

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