Large random arrowhead matrices: Multifractality, semilocalization, and protected transport in disordered quantum spins coupled to a cavity

Large arrowhead matrices with randomly distributed entries describe a variety of important phenomena where a degree of freedom is nonlocally coupled to a disordered continuum of modes, including central-spin problems in condensed matter, molecular junctions, and quantum emitters in cavity QED. Here we provide an exact solution of random arrowhead Hamiltonians with diagonal disorder in the thermodynamic limit. For concreteness, we focus on the problem of N emitters homogeneously coupled to a nonlocal cavity mode, corresponding to the disordered Tavis-Cummings model of cavity QED in the single-excitation limit, for which we provide asymptotically exact formulas for static and dynamical quantities of interest. By varying the coupling strength, 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 statistics being usually associated with the critical point of Anderson localization-delocalization transitions. We show that the system has a peculiar diffusivelike 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, g -independent value for sufficiently large N in the collective strong-coupling limit, a cavity protection effect. We investigate the system in a two-terminal configuration and show that the steady-state excitation current exhibits features similar to the escape probability, thereby extending our cavity-protected transport scenario to out-of-equilibrium situations. We finally demonstrate that dark states can provide a major contribution to long-distance transport in disordered systems.