In this thesis, we consider two topics that involve strong spatial or temporal fluctuations in interacting quantum many-body systems. We first consider far-from equilibrium quantum hydrodynamics in light-driven helical edge states of a 2D topological insulator. We then consider the enhancement of superconducting and magnetic instabilities in fermion systems with random or structured inhomogeneity, driven by the nesting of fractal wave functions.

In the first part, we study quantum quenches of helical edge liquids with spin-flip inelastic scattering. Counterpropagating charge packets in helical edges can be created by an ultrashort electric pulse applied across a 2D topological insulator. Localized “hot spots” that form due to scattering enable two types of strongly nonlinear wave dynamics. First, propagating packets develop self-focusing shock fronts. Second, colliding packets with opposite charge can exhibit near-perfect retroreflection, despite strong dissipation. This leads to an interaction-mediated frequency doubling that could be detected experimentally from emitted terahertz radiation.

In the second part, we study the enhanced correlation effects in many-fermion systems induced by structured or random inhomogeneities. We consider the interplay of superconductivity and a wide spectrum of critical (multifractal) wave functions (“spectrum-wide quantum criticality,” SWQC) in the one-dimensional Aubry-Andr ́e and power-law random-banded matrix models (PRBMs) with attractive interactions, using self-consistent BCS theory. We find that SWQC survives the incorporation of attractive interactions at the Anderson localization transition, whereas the pairing amplitude is maximized near this transition in both models. Although increasing inhomogeneity always depletes the superfluid stiffness, we find that superconductivity is still controlled by the fractal-enhanced amplitude near the Anderson transition. Our results suggest that SWQC, recently discovered in two-dimensional topological surface-state and nodal superconductor models, can robustly enhance superconductivity.

We also study the interplay of magnetic ordering and the spectrum-wide quantum critical wave functions in PRBMs with repulsive Hubbard interaction. We employ self-consistent Hartree-Fock numerics and analytical field theory calculations using a Keldysh Finkel’stein nonlinear sigma model. We find that weak interactions delocalize the system, while strong interaction localizes the system. Ferromagnetism sets in near the Anderson transition, where the ferromagnetic spin susceptibility is also shown to peak. This indicates that ferromagnetic ordering is enhanced by the fractal wave functions. Using the sigma model, we calculate the quantum corrections to the conductance and spin susceptibility. We find that the analytical and numerical results are mutually consistent.