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Physics Letters A

An algorithm to obtain exact eigenvalues and eigenstates of the arbitrary spin Ising Hamiltonian in d dimensions

J.M. Dixon, Jack A. Tuszyński

Published May 1, 2001• 3 citations
Publisher
DOI
10.1016/s0375-9601(01)00263-8

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The Hamiltonian of the Ising model in d dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian in an arbitrary number of dimensions. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We have made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one and two-dimensional cases and found complete agreement. (C) 2001 Elsevier Science B.V. All rights reserved.

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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. (C) 2001 Elsevier Science B.V. All rights reserved.

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Disorder solutions for the free energy of the Ising-like models

Pavel Khrapov

For arbitrary Ising-like models of any dimension and Hamiltonians with a finite support with all possible multispin interactions and boundary conditions with a shift, the exact value of the free energy in the thermodynamic limit is obtained at some parametrically specified set of multispin interaction coefficients. In this case, half of the multispin interaction coefficients and the coordinates of the special eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix are parameters, and the second half of the multispin coefficients is calculated using simple explicit formulas. For models with Hamiltonians invariant under the reversal of signs of all spins, the formulas are simplified. As examples of independent interest, solutions are written for the cases when the support of the Hamiltonian is a simplex, a cube, the support of the ANNNI model in spaces of 2, 3 and arbitrary dimensions.

arXiv:2103.08425•Mar 15, 2021•0 citations
The Hamiltonian of the Ising model in d dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian in an arbitrary number of dimensions. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We have made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one and two-dimensional cases and found complete agreement. (C) 2001 Elsevier Science B.V. All rights reserved.
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. (C) 2001 Elsevier Science B.V. All rights reserved.
For arbitrary Ising-like models of any dimension and Hamiltonians with a finite support with all possible multispin interactions and boundary conditions with a shift, the exact value of the free energy in the thermodynamic limit is obtained at some parametrically specified set of multispin interaction coefficients. In this case, half of the multispin interaction coefficients and the coordinates of the special eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix are parameters, and the second half of the multispin coefficients is calculated using simple explicit formulas. For models with Hamiltonians invariant under the reversal of signs of all spins, the formulas are simplified. As examples of independent interest, solutions are written for the cases when the support of the Hamiltonian is a simplex, a cube, the support of the ANNNI model in spaces of 2, 3 and arbitrary dimensions.