Model Spin Hamiltonians for Isotropic Interactions

Under the assumption of a common spatial part of the wave function, the lowest energy levels of the two-electron/two-orbital problem discussed in the previous section can be described with a model Hamiltonian that only contains spin operators. Starting from a general expression of the interaction of two spatially separated spin moments 5i and S2 of arbitrary strength (not limiting ourselves to the 5 = case discussed before), the spin Hamiltonian can be written in terms of local spin operators S and S2 

This expression is greatly simplified by orienting the system along the magnetic axis frame making all non-diagonal elements of the A-tensor equal to zero. 

It is common practice to divide the interaction in a part that does not depend on the spatial orientation of spin—the isotropic part, parametrized by the scalar J— and another part that models the anisotropy of the interaction parametrizing it with a diagonal tensor D. 

The minus sign in front of J is by convention, but be aware that other definitions are often used in the literature. Negative /-values indicate antiferromagnetic coupling and positive values are characteristic of ferromagnetic interactions in the definition that we use here. 

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The elimination of the anisotropic part in Eq. 3.22 leads to the Heisenberg Hamiltonian for isotropic magnetic interactions. The spins are considered as co-linear vectors whose principal quantization axis has no spatially preferred orientation. An even simpler model Hamiltonian can be obtained by putting Axx and Ayy to zero in Eq. 3.21. Then, the spin reduces to a classical vector whose orientation in space is not defined and the resulting model Hamiltonian describes the isotropic coupling of two (anti-)parallel spins. Replacing A z by -J, the following expression is obtained... 

Now we consider thermodynamic properties of the system described by the Hamiltonian (2.4.5) it is a generalized Hamiltonian of the isotropic Ashkin-Teller model100,101 expressed in terms of interactions between pairs of spins lattice site nm of a square lattice. Hamiltonian (2.4.5) differs from the known one in that it includes not only the contribution from the four-spin interaction (the term with the coefficient J3), but also the anisotropic contribution (the term with the coefficient J2) which accounts for cross interactions of spins a m and s m between neighboring lattice sites. This term is so structured that it vanishes if there are no fluctuation interactions between cr- and s-subsystems. As a result, with sufficiently small coefficients J2, we arrive at a typical phase diagram of the isotropic Ashkin-Teller model,101 102 limited by the plausible values of coefficients in Eq. (2.4.6). At J, > J3, the phase transition line... 

Symmetric anisotropy The basis of this Hamiltonian can no longer be restricted to determinants with the same Ms value as was done for the isotropic interactions. The inclusion of magnetic anisotropy in the model causes the removal of the degeneracy of the different Ms levels and eventually mixing of the wave functions with different spin moment. Here, we have to consider four CSFs the three components of the triplet plus the singlet. To facilitate the determination of the matrix elements of the model... 

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