  In two dimensions there is a direct superconductor-to-insulator quantum phase
transition driven by increasing disorder. We elucidate, using a combination of
inhomogeneous mean field theory and quantum Monte Carlo techniques, the nature
of the phases and the mechanism of the transition. We make several testable
predictions specifically for local spectroscopic probes. With increasing
disorder, the system forms superconducting blobs on the scale of the coherence
length embedded in an insulating matrix. In the superconducting state, the
phases on the different blobs are coherent across the system whereas in the
insulator long range phase coherence is disrupted by quantum fluctuations. As a
consequence of this emergent granularity, we show that the single-particle
energy gap in the density of states survives across the transition, but
coherence peaks exist only in the superconductor. A characteristic pseudogap
persists above the critical disorder and critical temperature, in contrast to
conventional theories. Surprisingly, the insulator has a two-particle gap scale
that vanishes at the SIT, despite a robust single-particle gap.
