Eigenvalues of Ising connection matrix with long-range interaction

Abstract

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of it eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions.

Inverse Problem for Ising Connection Matrix with Long-Range Interaction

“…All these functions are crucially dependent on the connection matrix whose main characteristics are its eigenvalues and eigenvectors. In papers [1][2][3], we obtained the expressions for the eigenvalues of the Ising connection matrix A = A ij N i,j=1 with an arbitrary long-range interaction. In the present paper, we solve the inverse problem: We suppose that we know the matrix spectrum and we have to determine the interaction constants providing this spectrum.…”

“…In papers [1][2][3], we calculated eigenvalues of Ising connection matrices defined on d-dimensional hypercube lattices (d = 1, 2, 3 . .…”

“…In our calculations, we accounted for interactions not only with the nearest spins but with distant spins, as well. In papers [1,2], we analyzed isotropic interactions, while in paper [3], we discussed the general case of anisotropic interactions. We succeeded to obtain analytical expressions for the eigenvalues of the above-described Ising connection matrices.…”

“…We succeeded to obtain analytical expressions for the eigenvalues of the above-described Ising connection matrices. For the d-dimensional system, the eigenvalues are polynomials of the degree d in the eigenvalues for the one-dimensional system with long-range interaction (see [2,3]). The coefficients of these polynomials are the constants of interaction between spins.…”

Converting Lattices into Networks: The Heisenberg Model and Its Generalizations with Long-Range Interactions

“…For example, one-dimensional models with long-range interactions is considered in [15]. Ising models on the hypercube lattices with long-range interaction is considered in [16]. In this paper, the model on lattice configuration with interaction between all particle-pairs is with long-range interactions.…”