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 Tc 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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