Heisenberg spin model

The discussion of the previous section concerned non-interacting microscopic magnetic moments. Many aspects of magnetic behavior involve magnetic moments that are strongly coupled. A model that captures the physics of such systems is the so 

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When this result is substituted into the Heisenberg spin model, Eq. (7.21), taking into account that raising and lowering operators at different lattice sites commute with each other because they operate on different spins, we find... 

However, magnetic coupling behavior may be more complex to model than possible with a simple isotropic Heisenberg Spin Hamiltonian as defined in Eq. (79) and several recent studies set out to improve this description by modification of this Hamiltonian (86-89). 

Though the exact solution of the Pauhng-Wheland VB model (or the positive-./ Heisenberg spin Hamiltonian) is generally a nontrivial matter, there are a number... 

As illustrated above, the microscopic explanation of observed magnetic properties hinges on the construction of an appropriate model. In most instances, simplifications have to be weighed and phenomenological models can be employed, such as the Heisenberg spin Hamiltonian. 

For organic spin systems, one frequently assumes applicability of Heisenberg spin behavior, in which all interactions can be reasonably modeled by pairwise exchange interactions. A typical Heisenberg spin Hamiltonian for exchange Jy between various spin sites i and j, with spin quantum numbers S, and Sj, is given in the following equation ... 

An even more quantitative application of VB theory can be developed from the realization that the nearest-neighbor VB model as developed, for example, by Pauling [10], can be mapped exactly onto a Heisenberg spin Hamiltonian [17]. The Heisenberg spin Hamiltonian has long been used to study the interaction between magnetic atoms in transition metal compounds and other paramagnetic substances [18], and can be written most simply as... 

Figure 5 Diamagnetic rings currents, Nx z), of half-filled Pariser-Parr-Pople models for regular polygons with D h symmetry. The dashed line at 2 = U/4 t0 = 1.17 corresponds to standard parameters z = 0 is the Hiickel limit of free electrons, while z 1 is the strong-correlation limit of antiferromagnetic Heisenberg spin chain with vanishing ring currents [50].

To reasonably limit the focus here it the survey is primarily of many-body solution techniques as applied to a particular VB model, the covalent-space Pauling-Wheland VB model, represented by the Heisenberg spin Hamiltonian... 

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