Analytical form of the Hamiltonian
in the invariant subspaces of the U-independent symmetries.
(the method of diagonalization is according to the paper:
[1] Thermodynamics of a 4-site-Hubbard model by analytical diagonalization )
Please give some attention also to a recent preprint by
E. A. Yuzbashyan, B. L. Altshuler, B. S. Shastry
from Princeton university, devoted to
The Origin of Degeneracies and Crossings in the 1d Hubbard Model ,
which deals
with the benzen also.
Some remarks:
- At the moment we show the most interesting case for Ne=6.
In this subspace (ie. the z-component of the pseudospin operator has the eigenvalue 0),
there are 924 states, which dispatch to 170 invariant subspaces differing in the eigenvalues
of the following set of commuting operators:
Sz, R(R+1), S(S+1), U6, X .
Here U6 and X are
the two commuting operators constructed from the multiplication table
of the point group C6v.
- Of course, we have also the basis vectors spanning the subspace in analytical form,
but they are so lengthy, that it is not very useful to display them here.
-
The postscript files can be viewed using
GhostView 3.5.8 (or higher), with
appropriate reduction.
- To get some systematics into the interesting subspaces we calculated the energy eigenvalues
for t=1, U=5. The chemical potential was chosen to be U/2, and the external magnetic field
(in energy units) was chosen to 0.01 to break the spin symmetry.
The table shows in the second column the energy eigenvalues in ascending order.
In
the third column we show to which subspace the eigenvalue belongs,
eg. the groundstate (and also
the state with maximum energy) belong to the subspace 74,
which is of dimension 14, and therefore
the groundstate
has the first place 74-1 and
the maximum energy state has 74-14.
See the