This thesis is devoted to two frontier research topics in condensed matter physics: (i) novel probes for topological superconductors, and (ii) transport and superconductivity in disordered quantum critical systems.

In the first part, we present new electromagnetic probes to detect Majorana surface states in three dimensional topological superconductors (TSCs). We start by studying the temperature dependence of the magnetic penetration depth by incorporating the paramagnetic current due to the surface states. In addition to the bulk-dominated London response, we identify a T3 power-law-in-temperature contribution from the surface, valid in the low-temperature limit. Our system is fully gapped in the bulk, and should be compared to bulk nodal superconductivity, which also exhibits power-law behavior. Power-law temperature dependence of the penetration depth can be one indicator of topological superconductivity.

We then explore the optical absorption in a topological Weyl superconductor due to a novel surface-to-bulk mechanism, which we dub the topological anomalous skin effect. This occurs even in the absence of disorder for a single-band superconductor, and is facilitated by the topological splitting of the Hilbert space into bulk and chiral surface Majorana states. In the clean limit, the effect manifests as a characteristic absorption peak due to surface-bulk transitions. We also consider the effects of bulk disorder, using the Keldysh response theory. For weak disorder, the bulk response is reminiscent of the Mattis-Bardeen result for s-wave superconductors, with strongly suppressed spectral weight below twice the pairing energy, despite the presence of gapless Weyl points. For stronger disorder, the bulk response becomes more Drude-like and the p-wave features disappear.
We show that the surface-bulk signal survives when combined with the bulk in the presence of weak disorder. The topological anomalous skin effect can therefore serve as a fingerprint for Weyl superconductivity. We also compute the Meissner response in the slab geometry, incorporating the effect of the surface states.

In the second part, we explore the electric transport and superconductivity in a dirty quantum critical system. We first study the electrical transport of a two-dimensional non-Fermi liquid with disorder, and we determine the first quantum correction to the semiclassical dc conductivity due to quantum interference. We consider a system with N flavors of fermions coupled to SU(N) critical matrix bosons.
%Motivated by the Sachdev–Ye–Kitaev (SYK) model, we employ the bilocal field formalism and derive a set of finite-temperature saddle-point equations governing the fermionic and bosonic self-energies in the large-N limit.
Interestingly, disorder smearing induces a marginal Fermi liquid (MFL) self-energy for the fermions.
We next consider fluctuations around the saddle points and derive a MFL-Finkel'stein nonlinear sigma model. We find that the Altshuler-Aronov quantum conductance correction gives linear-T resistivity that can dominate over the Drude result at low temperature. The strong temperature dependence of the quantum correction arises due to rapid relaxation of the mediating quantum-critical bosons. We verify that our calculations explicitly satisfy the Ward identity at the semiclassical and quantum levels.

We then move on to study superconductivity of the disordered MFL. At the semiclassical level, the transition temperature Tc is strongly suppressed because marginal Fermi liquid effects destroy well-defined quasiparticles. However, we show that interference between quantum-critical collective modes must be included, and these \emph{enhance} Tc, violating Anderson's theorem.

Both of our results establish that quantum interference persists in two-particle hydrodynamic modes, even when quasiparticles are subject to strong (Planckian) dissipation.