The Heisenberg Model

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 

Finally, for purposes of comparison, it is useful to have a pi electron model which makes neither the Hiickel nor the VB assumptions about the strength of the electron-electron interaction. Such a model can provide a smooth bridge between the Hiickel and Heisenberg models as the relative strength of the chemical bonding and electron repulsion interactions between electrons are changed. Such a role is played by the Pariser-Parr-Pople (PPP) model [21,22], and its close relative, die Hubbard model [23], The PPP model can be 

The PPP Hamiltonian can then be written in second quantized form as 

The exact FCI (frill configuration interaction) solution of the PPP or Hubbard model is possible for molecules with up to about 16 atoms in the pi system. Any of the standard methods for performing approximate ab initio calculations, such as limited configuration interaction, Moeller-Plesset perturbation theory, or coupled cluster theory, may be applied to these models as well. All are expected to be very accurate at low order when U is small, but all will have to be pushed to higher order as U increases. 

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It would be important to figure out the low energy excitation modes (Nambu-Goldstone modes) built on the ferromagnetic phase. The spin waves are well known in the Heisenberg model [10]. Then, how about our case [32] ... 

The coefficients a and b (see Table 3.2) take into account the restrictions in spin dimensionality. For a = b= 1, the Heisenberg model with isotropic exchange interaction and isotropic susceptibility results. The combination of a = 1 and b = 0 yields the strictly anisotropic Ising model, in which the orientation of the spins is restricted to the z-axis. Consequently, the susceptibility is strongly orientation dependent and one needs to differentiate between x" in the direction of the z-axis ( easy axis ) and x perpendicular to z. The molar susceptibilities are then related as... 

An instructive view of the relation between the spectra of the Htickel model (U=0), the Heisenberg model (U large) and the Hubbard model is provided in Fig. 1. Those states described by the Heisenberg model are all of covalent... 

Full Cl ground state energy of the Heisenberg model (Eq. I-) in units of J... 

In the approach of micromagnetism the quantum mechanical spin operators are substituted by classical vectors, S. Thus, the exchange energy in the Heisenberg model [34]... 

In the case of the Ising model, only the 1-D curve does not show any ordering temperature, whereas in the case of the Heisenberg model neither the 1-D nor the 2-D curves have an ordering temperature. 

Three cases are identified, (1) fx Jyy with spin dimensionality three, called the Heisenberg model, (2) ... 

The uniform Heisenberg magnetic chain has been an object of study for three quarters of a century, dating back to the initial work of Bethe. Unhke the equivalent Ising and XY-chains, the Heisenberg model cannot be solved analytically. The first calculations of the susceptibihty and specific heat... 

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