Quantum Heisenberg Model

http://en.wikipedia.org/wiki/Heisenberg_model_%28quantum%29

The above is a link to a full description of the Quantum Heisenberg Model as we derived in class. Note that in class we considered the case of zero external field and unlimited neighbor interactions. Usually the Heisenberg model is referred to when considering a lattice with nearest-neighbor interactions only. The model is generally solvable for the ground-state, however as most of us are finding in Phys4040, it is not simple.

In general the Heisenberg model is much simpler when solved computationally, and in fact is much more efficient computationally than the classical Ising model. Using a suitable eigenvalue technique, for instance the Lanczos algorithm, it is possible to calculate the ground-state and lower excited states for an NxN lattice (and higher dimensions, although memory can become a problem for any lattice size which is a reasonable approximation to the thermodynamic limit). An important consequence one finds upon solving, is that the ferromagnet and anti-ferromagnet arrangements seen in class only occur for zero field. When a magnetic field is apply, a superposition of different spin states is found.