I am solving a Hamiltonian including a term \begin{equation}(x\cdot S)^2\end{equation}
The Hamiltonian is like this form:
\begin{equation}
H=L\cdot S+(x\cdot S)^2
\end{equation}
where L is angular momentum operator and S is spin operator. The eigenvalue for \begin{equation}L^2 ...
two papers
arXiv: 0710.1128 Supplementary material to Heavy electrons and the symplectic symmetry of spin
arXiv: 0810.5144 Symplectic N and time reversal in frustrated magnetism
the action S or hamiltonian invariant under transformations corresponds to the symmetry of the system. these transformations may be continuous( U(1), SU(2)) or discrete.
while the state may not retain these symmetry--symmetry breaking.
Hope the following link useful to you
http://docs.google.com/fileview?id=0B4k0T2TYaCC2YWZiNmRlMDYtMzUwOC00YWZlLThmMzgtNzk1ZDUxYTExNzAz&hl=en
Eq.(57)-Eq.(60)
i remember the <S_z(x)S_z(0)>=x^(1/2)
For these 1d spin problems, such as xxz model and the Heisenberg model in magnetic field, the standard method is bosonization.
Hope these help!
Jordan-wigner transformation is useful for 1/2 spin system
i don't remember the Hamiltonian of xx model, but for 2 d system, seems difficult to sovle it exactly.
some numerical methods, like lanczos algorithm ,VMC...
try some old archives in prl and prb at 1980s...
The 2-dimensional 1/2 spin Heisenberg model is the key problem in the copper oxide superconductor.
For the 1-D 1/2 spin antiferromagnetic Heisenberg model, it has been exactly solved by bethe ansatz. And the low-energy excitation can be analyzed be many methods, such as Bosonizaiton, RG...
In...