We present a theoretical framework for understanding the behavior of the normal and superconducting states of overdoped cuprate high temperature superconductors in the vicinity of the doping-tuned quantum superconductor-to-metal transition. The key i
ngredients on which we focus are d-wave pairing, a flat antinodal dispersion, and disorder. Even for homogeneous disorder, these lead to effectively granular superconducting correlations and a superconducting transition temperature determined in large part by the superfluid stiffness rather than the pairing scale.
Lattices with a basis can host crystallographic defects which share the same topological charge (e.g.~the Burgers vector $vec b$ of a dislocation) but differ in their microscopic structure of the core. We demonstrate that in insulators with particle-
hole symmetry and an odd number of orbitals per site, the microscopic details drastically affect the electronic structure: modifications can create or annihilate non-trivial bound states with an associated fractional charge. We show that this observation is related to the behavior of end modes of a dimerized chain and discuss how the end or defect states are predicted from topological invariants in these more complicated cases. Furthermore, using explicit examples on the honeycomb lattice, we explain how bound states in vacancies, dislocations and disclinations are related to each other and to edge modes and how similar features arise in nodal semimetals such as graphene.
The origin of magnetic flux noise in Superconducting Quantum Interference Devices with a power spectrum scaling as $1/f$ ($f$ is frequency) has been a puzzle for over 20 years. This noise limits the decoherence time of superconducting qubits. A conse
nsus has emerged that the noise arises from fluctuating spins of localized electrons with an areal density of $5times10^{17}$m$^{-2}$. We show that, in the presence of potential disorder at the metal-insulator interface, some of the metal-induced gap states become localized and produce local moments. A modest level of disorder yields the observed areal density.
We argue that by inducing superconductivity in graphene via the proximity effect, it is possible to observes the quantum valley Hall effect. In the presence of magnetic field, supercurrent causes valley pseudospin to accumulate at the edges of the su
perconducting strip. This, and the structure of the superconducting vortex core, provide possibilities to experimentally observe aspects of the deconfined quantum criticality.
