The normal state of unconventional superconductors has long challenged our conceptions of traditional metals or the mechanisms of superconductivity. The anomalous, or "strange", transport properties of these materials imply that these systems lack single-particle excitations. Understanding the nature of this parent state stands as a critical challenge in modern condensed matter physics. This talk will discuss the question of transport in such quantum critical systems governed by quenched disorder. This work is conducted through the lens of the Sachdev-Ye-Kitaev (SYK) model and its critical Yukawa extensions. These systems serve as paradigmatic examples of non-Fermi liquid behavior, quantum chaos, and strange metallicity.

The research employs a combination of analytical techniques in the large-N limit and numerical simulations using Extended Dynamical Mean Field Theory coupled with continuous-time Quantum Monte Carlo. The first part of the talk explores the emergence of nematicity in multi-orbital lattice extensions of the SYK model. In the two-orbital model, we identify a phase diagram featuring a tricritical point that separates a nematic insulator from a nematic metal, characterized by a peak in the d.c. elastoresistivity. Extensions to three-orbital systems reveal spontaneous orbital-selective transitions driven by Z3 Potts nematic order. The richness of the Z3 order allows for distinct states where the orbital degeneracy can be lifted at different temperatures, depending on the hierarchy of scales.

The second part of the talk investigates the effects of localization by studying a model of fermions with both Hubbard-U Coulomb repulsion and random Yukawa coupling to critical bosonic fluctuations. This model provides a genuine realization of a strange metal with marginal Fermi liquid behavior down to zero temperature and a vanishing quasiparticle residue, in contrast to the non-Fermi liquid behavior of the lattice SYK model. These localization effects significantly enhance the effects of quantum critical bosonic fluctuations, stabilizing a strange metal regime with Planckian quasiparticle decay rates in the proximity of a Mott transition. Time permitting, I will conclude with an experimental collaboration on an iron-based superconductor, where the role of quenched disorder and criticality becomes manifest.