In this thesis we consider a variational wave function approach as a possible
route to describe the competition between disorder and strong electron-electron
interaction in two dimensions. In particular we aim to obtain a transparent
and physically intuitive understanding of the competition between these two
localizing forces within the simplest model where they both are active, namely
the disordered Hubbard model at half filling and in a square lattice. Our approach
is based on an approximate form of the ground-state wave function,
which we believe contains the physically relevant ingredients for a correct description
of both the Mott and the Anderson insulators, where electrons are
localized by the Coulomb repulsion and by disorder, respectively. For strongly
interacting fermionic systems, a standard variational wave function is constructed
by a correlation term acting on a Slater determinant, the latter being
an uncorrelated metallic state. Previous variational calculations showed that a
long-range density-density correlation factor, so called Jastrow factor, is needed
to correctly describe the Mott insulator [9]. This term, which is collective by
definition, correlates spatially charge
uctuations, thus preventing their free
motion that would otherwise imply metallic conductance. For this reason, our
variational wave function does include such a term. Anderson localization is
instead mostly a matter of single-particle wave functions, hence it pertains to
the uncorrelated Slater determinant which the Jastrow factor acts onto. We
consider both the case in which we enforce paramagnetism in the wave function
and the case in which we allow for magnetic ordering.
Summarizing briefly our results, we find that, when the variational wave
function is forced to be paramagnetic, the Anderson insulator to Mott insulator
transition is continuous. This transition can be captured by studying several
quantities. In particular, a novel one that we have identified and that is easily
accessible variationally is the disconnected density-density
fluctuation at long wavelength, defined by
lim
where ^nq is the Fourier transform of the charge density at momentum q, (...)
denotes quantum average at fixed disorder and the overbar represents the average
over disorder configurations. We find that Ndisc
q!0 is everywhere finite in
the Anderson insulator and vanishes critically at the Mott transition, staying
zero in the Mott insulator.
When magnetism is allowed and the hopping only connects nearest neighbor
sites, upon increasing interaction the paramagnetic Anderson insulator
first turns antiferromagnetic and finally the magnetic and compressible Anderson
insulator gives way to an incompressible antiferromagnetic Mott insulator.
The optimized uncorrelated Slater determinant is always found to be the eigenstate
of a disordered non-interacting effective Hamiltonian, which suggests that
the model is never metallic. Finally, when magnetism is frustrated by a next
to nearest neighbor hopping, the overall sequence of phases does not change.
However, the paramagnetic to magnetic transition within the Anderson insulator
basin of stability turns first order. Indeed, within the magnetically ordered
phase, we find many almost degenerate paramagnetic states with well defined
local moments. This is suggestive of an emerging glassy behavior when the
competition between disorder and strong correlation is maximum.
