We investigate the conditions under which the ground state of a low-density quasi-twodimensional
electron (or hole) system is a Bose-Einstein condensate of mobile dimers.
Such a ground state would require an effective attraction between electrons but an
effective repulsion between dimers to prevent clustering. A UV model is assumed;
this is not specific to the pairing mechanism but can be obtained from a Fr¨ohlich-
Coulomb model by the Lang-Firsov transformation. We survey the parameter space for
each lattice restricting the dimer Hilbert spaces to the low-energy sector since we are
interested in low-lying states and low densities. Singlet dimers are mobile on a triangular
lattice; in the simplest case the effective Hamiltonian for dimer hopping is that of a
kagome lattice. However, a dimer condensate is never the ground state in the triangular
lattice, as dimers will either cluster or dissociate. For a square lattice with nearest- and
next-nearest-neighbour hopping we find a substantial region in which dimers form a
ground state. These dimers turn out to be very light since they can propagate by a
“crab-like” motion without requiring virtual transitions. For a perovskite layer we find
a substantial region in which dimers, which are also light and mobile due to crab-like
motion, form a ground state. Our findings indicate that the existence of stable small
mobile bipolarons is very sensitive to the lattice structure.
We secondly identify circumstances under which triplet dimers are strictly localised
by interference in certain one- and two-dimensional lattices. We find that strict localisation
is possible for the square ladder and some two-dimensional bilayers. We thirdly
investigate the electronic properties of Graphene. We identify the origin of Graphene’s
Dirac points and subsequently identify Dirac points in other two- and three-dimensional
lattices. We finally investigate the dynamics of electrons and dimers on various oneand
two-dimensional lattices by the use of Green’s functions.
This thesis is confidential until 31/12/2021. A Doctoral Thesis. Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University.