Confinement and the superfluid density in theories of the underdoped cuprates, and strongly commensurate dirty bosons in the large-N limit

In this thesis, we study various issues arising from the QED theory of underdoped, high temperature superconductors in 2+1 dimensions. The theory breaks up roughly into two sectors: fermionic and bosonic. With regard to the fermionic sector, we consider confinement of the emergent gauge field which we take to be compact. In the absence of fermions, the interaction between monopoles is Coulombic and the well known result is that the pure gauge theory is permanently confining. With the addition of fermions, the interaction becomes logarithmic, and an analogy with the usual Kosterlitz-Thouless transition suggests a deconfinement transition for the fermions. We show, however, that, when screening is taken into account, the deconfined phase is destabilized and fermions remain permanently confined. The bosonic sector models Cooper pair phase fluctuations, whose effect on the depletion of the superfluid density we examine in two separate studies. In the first of these, we study the quantum XY model, and show that the quasi-two dimensionality, low critical temperatures and large d-wave gap characteristic of underdoped cuprates severely constrain the form of the superfluid density. Under these assumptions, we find that phase fluctuations alone are insufficient to account for recent observations of deviations from Uemura scaling, and that the quasiparticle contribution is a necessity. We use our results to satisfactorily fit the recent data. In the second study, we model the cuprates by a layered system of interacting bosons and examine the collective excitations in this system. Depending on the anisotropy and the interaction strength, we find find four different regimes of temperature dependence of the superfluid density. We argue that interactions in the underdoped cuprates are effectively short-ranged and weak. Finally, we study the related issue of disordered, interacting bosons in the large-N limit and 2 strong commensuration. Perturbatively at weak disorder and numerically at strong, we show that th screening of the random potential due to interactions is insufficient to delocalize the single-particl states so that no superfluid transition occurs from the Mott insulator.