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Physica A Statistical Mechanics and its Applications

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

J.M. Dixon, Jack A. Tuszyński, M.L.A. Nip

Published Jan 1, 2001• 15 citations
Publisher
DOI
10.1016/s0378-4371(00)00318-6

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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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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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Estudo da densidade espectral de uma cadeia de Ising com campo transverso

Conrado Luiz da Silveira Folly

In this work, we study the frequency spectrum of a spins chain in the presence of a transverse magnetic field B using the Ising model for the interaction between spins in one dimension, considering only the magnetic interactions, and assuming that the system is all the time at high temperatures. We show that for a chain of 11 spins in the presence of an external transverse magnetic field B, the dynamics of the system closely approximate the asymptotic limit of a system of infinite particles. The methodology consists of the diagonalization of the Hamiltonian of the system, by taking into account the interaction J between neighboring spins in a chain, together with the interaction of each spin with an external field transverse field B. We exactly determine the eigenvectors and eigenvalues of the system. Then we calculate the time-dependent autocorrelation function. Finally, we perform a Fourier transform of the autocorrelation function to find the frequency dependent spectral density. We obtained exact analytic results for the case of two spins, and numerically exact results for systems with up to 11 spins. Our results for the spectral densities covering three significative values of the energy parameters J and B are already very close to the thermodynamic limit. For B > J the dynamics is of collective-mode type where the spins tend to align with the external field.

•Jan 1, 2019•0 citations
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.
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.
In this work, we study the frequency spectrum of a spins chain in the presence of a transverse magnetic field B using the Ising model for the interaction between spins in one dimension, considering only the magnetic interactions, and assuming that the system is all the time at high temperatures. We show that for a chain of 11 spins in the presence of an external transverse magnetic field B, the dynamics of the system closely approximate the asymptotic limit of a system of infinite particles. The methodology consists of the diagonalization of the Hamiltonian of the system, by taking into account the interaction J between neighboring spins in a chain, together with the interaction of each spin with an external field transverse field B. We exactly determine the eigenvectors and eigenvalues of the system. Then we calculate the time-dependent autocorrelation function. Finally, we perform a Fourier transform of the autocorrelation function to find the frequency dependent spectral density. We obtained exact analytic results for the case of two spins, and numerically exact results for systems with up to 11 spins. Our results for the spectral densities covering three significative values of the energy parameters J and B are already very close to the thermodynamic limit. For B > J the dynamics is of collective-mode type where the spins tend to align with the external field.