## Quantum lattice models and strongly correlated Fermi systems

In this field several quantum lattice models
of physical interest have been studied.
In the papers
[42-44,
47]
a method to study bosonic systems on the
lattice which in the continuum limit reduce to the nonlinear
Schrödinger equation (NLS) (or its generalizations), was considered.
In [50], this method was used to construct basis
functions for bosonic hamiltonians which are invariant both
under the permutation group and under the group U(1) and in
[52] applied to a system of q-bosons.
For this system, it was possible
to show the existence, in the mean field approximation, of a
Bose-Einstein condensation. This result represents a first example
of Bose-Einstein condensation in a system of q-bosons.
A group theoretical method for exact
diagonalizations of strongly correlated Fermi systems of interest for
high T_{c} superconductivity, such as the Hubbard model,
was also developed.
In the set of
papers [57-59,
P11],
basis functions for Fermi systems invariant under
the permutation group and under SU(2), were constructed.
Using these functions, it was possible to give an exact solution
of the Hubbard model with unconstrained hopping for arbitrary number
of lattice sites.
The ground state of this system was characterized as a
function of the electronic filling.
In particular, it was shown in
[57] that below half filling
the ground state is
always a singlet or a doublet (depending on the number of sites),
while above half filling (work [59])
ferromagnetic ground states
can exist for arbitrary
values of the Coulomb repulsion. These ferromagnetic ground states,
except for the
case immediately above half filling which is always non degenerate
and maximally ferromagnetic,
are degenerate with respect to the total spin of the
system. In the work [60]
this method has been generalized to electronic
systems with
next neighbor hopping on bipartite lattices with SO(4) symmetry, allowing
an exact diagonalization of the Hubbard chain with finite number of sites.
In the papers [67,
68] the link
between the SO(4) symmetry of the
Hubbard model in one dimension and the Bethe states constructed with the
Bethe ansatz, has been studied. For Hubbard chains of finite length a
unitary transformation between the highest-lowest weight vectors of SO(4)
and regular Bethe states, has been found. This transformation implies a
shifting relation among the matrix elements of the Hamiltonianan which
simplify the study of the spectral properties as a function of the
electronic filling.

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