The multifractal X-ray edge problem

We will discuss the nature of the X-ray edge problem when the single particle eigenstates are multifractal. We focus on the case where the many-body wavefunction is composed of single particle eigenstates that are generated from the Aubry-Andre model at its critical point, and introduce a local impurity via an instantaneous quench at time t. We find that the orthogonality catastrophe remains between the pre and post quench wavefunctions where the average wavefunction overlap vanishes in a power law fashion as a function of the system size. This power law is markedly distinct from the plane wave limit, with a much slower decay. This behavior is also manifested in the distribution of wavefunction overlaps, where the overlap distribution becomes significantly broad at the critical point of the Aubry-Andre model. We also focus on the core-hole Green function, which effectively translates this phenomena to the time domain and find a long-time power law decay on average, with its distribution approaching the distribution of wavefunction overlaps in the long time limit. Despite focusing on the Aubry-Andre model, we invoke the universality of the single particle wavefunctions to argue that our results are generally applicable to other problems with multifractal wavefunctions.