Fellows, Jonathan Michael (2013)
Ph.D. thesis, University of Birmingham.
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 AbstractIn this thesis we develop a model of ultracold dipolar bosons in a highly anisotropic quasionedimensional optical lattice. We will see that the model is identical to one describing quasionedimensional superconductivity in condensed matter systems giving rise to the possibility
of using this ultracold atoms system as an analogue simulator of interesting electronic systems.
In investigating the properties of this model we find a rich phase diagram containing density wave, superfluid, and possibly supersolid phases, accessible by tuning the optical lattice parameters and the alignment of the dipole moments.
An important property of this model turns out to be the existence of an enhanced symmetry at the self dual point where the density wave and superfluid orders are maximally competing. At this point the BerezinskiiKosterlitzThouless transition temperature of either phase must necessarily vanish to zero due to the HohenbergMerminWagner theorem.
Inspired by this model we go on to study a more general system in two dimensions with O(M) x O(2) symmetry which has an enhanced symmetry point of O(M + 2) symmetry. The BKT transition in the O(2) sector is mediated by vortex excitations, but these must somehow disappear as the high symmetry point is approached. Using both a variational argument adapting the standard BKT argument, and a more rigorous RG analysis we show that the size of the vortex cores in such a system must diverge as 1/\(\sqrt{\Delta}\) where \(\Delta\) measures the distance from the high symmetry point, and further that the BKT transition temperature must vanish as 1/ln(1/\(\Delta\)).

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