Quantum criticality in the disordered Aubry-Andr\'{e} model

In this paper, we explore quantum criticality in the disordered Aubry-Andr\'{e} (AA) model. For the pure AA model, it is well-known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $\Delta$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $\xi$ scales with $\Delta$ as $\xi\propto\Delta^{-\nu_\Delta}$ with $\nu_\Delta$ a new critical exponent, which is estimated to be $\nu_\Delta\approx0.46$. This value is remarkably different from the counterparts for both the pure AA model and the Anderson model. Moreover, rich critical phenomena are discovered in the critical region spanned by the quasiperiodic and the disordered potentials. In particular, in the extended phase side, we show that the scaling theory satisfy a hybrid scaling form as a result of the overlap between the critical regions of the AA model and the Anderson localization.