UCLA

An analysis of three kinds of physical systems exhibiting superconducting properties is presented: the pseudogap regime of the cuprates, chiral $p$-wave superconductors, and chiral $d$-wave superconductors, following a brief overview of superconductivity. The effect of disorder on a $\mathbb{Z}_2$ symmetry breaking order parameter is examined in the aforementioned psuedogap regime. Majorana modes in $p$-wave superconductors are examined. Finally, currents in both $p$ and $d$-superconductors are calculated numerically.

In the pseudogap regime of the cuprates, commensurate charge order breaks a $\mathbb{Z}_{2}$ symmetry, reflecting a broken translational symmetry. Therefore, the interaction of charge order and quenched disorder due to potential scattering, can, in principle, be treated as a random field Ising model. A numerical analysis of the ground state of such a random field Ising model reveals local, glassy dynamics in both $2D$ and $3D$. The dynamics are treated in the glassy limit as a heat bath which couple to the itinerant electrons, leading to an unusual electronic non-Fermi liquid. If the dynamics are strong enough, the electron spectral function has no quasiparticle peak and the effective mass diverges at the Fermi surface, precluding quantum oscillations. In contrast to charge density, $d$-density wave order (reflecting staggered circulating currents) does not directly couple to potential disorder, allowing it to support quantum oscillations. At fourth order in Landau theory, there is a term consisting of the square of the $d$-density wave order parameter, and the square of the charge order. This coupling could induce parasitic charge order, which may be weak enough for the Fermi liquid behavior to remain uncorrupted. This distinction must be made clear, as one interprets quantum oscillations in cuprates.

A chiral $p_x+ip_y$ superconductor on a square lattice with nearest and next-nearest hopping and pairing terms is considered. Gap closures, as various parameters of the system are varied, are found analytically and used to identify the topological phases. The phases are characterized by Chern numbers (ranging from $-3$ to $3$), and (numerically) by response to introduction of weak disorder, edges,

and magnetic fields in an extreme type-II limit, focusing on the low-energy modes (which presumably become zero-energy Majorana modes for large lattices and separations). Several phases are found, including a phase with Chern number $3$ that cannot be thought of in terms of a single range of interaction, and phase with Chern number $2$ that may host an additional, disorder resistant, Majorana mode. The energies of the vortex quasiparticle modes were found to oscillate as vortex position varied. The spatial length scale of these oscillations was found for various points in the Chern number $3$ phase which increased as criticality was approached.

Finally, currents in chiral $p_x+ip_y$ and $d_{x^2-y^2}+id_{xy}$ superconductors are examined in a cylindrical configuration, self-consistently. Edge currents, while localized at the boundary as expected, are highly parameter dependent but generically non-vanishing. Preliminary results are presented, and the techniques employed are explained. In particular, the calculation of the correlators and currents in both the $p$ and $d$ wave cases, and the numerical techniques used to find the self-consistent Hamiltonian are overviewed.