Large Random Arrowhead Matrices: Multifractality, Semi-Localization, and Protected Transport in Disordered Quantum Spins Coupled to a Cavity

We provide an exact solution of large random arrowhead Hamiltonians with diagonal disorder, a minimal model for inhomogeneously broadened quantum emitters coupled to a cavity mode. We find that the distribution of energy spacing can be continuously tuned between Poisson statistics and a distribution that is very close to semi-Poisson statistics - the latter being usually associated to the critical point of "Anderson" localization-delocalization transitions. We demonstrate that all the eigenstates - including two polaritons and a continuum of dark states - are multifractal, which indicates the existence of a critical "semi-localized" phase for all values of the light-matter coupling strength, where dark states are localized over multiple, arbitrarily-distant sites. By computing the escape probability from an initial site, we find that the system has a peculiar diffusive-like behavior with an escape probability growing linearly with time for any finite coupling strength, and that the escape rate can be controlled by selecting the energy of the initial site. The escape rate averaged over the disorder configurations is found to exhibit a maximum for intermediate coupling strengths, before saturating at a lower value in the collective strong coupling limit - a "cavity protection" effect. Surprisingly, we show that the saturation value increases with the disorder, indicating that the cavity does not only protect transport against disorder but can also turn the latter into an ally improving transport. We finally investigate the system in a two-terminal configuration, and show that the steady-state excitation current exhibits similar features as the escape probability, thereby extending our cavity-protected transport scenario to out-of-equilibrium situations. We finally demonstrate that dark states can provide the major contribution to long-distance transport in disordered systems.