  Mobility edges, separating localized from extended states, are known to arise
in the single-particle energy spectrum of disordered systems in dimension
strictly higher than two and certain quasiperiodic models in one dimension.
Here we unveil a different class of mobility edges, dubbed anomalous mobility
edges, that separate bands of localized states from bands of critical states in
diagonal and off-diagonal quasiperiodic models. We first introduce an exactly
solvable quasi-periodic diagonal model and analytically demonstrate the
existence of anomalous mobility edges. Moreover, numerical multifractal
analysis of the corresponding wave functions confirms the emergence of a finite
band of critical states. We then extend the sudy to a quasiperiodic
off-diagonal Su-Schrieffer-Heeger model and show numerical evidence of
anomalous mobility edges. We finally discuss possible experimental realizations
of quasi-periodic models hosting anomalous mobility edges. These results shed
new light on the localization and critical properties of low-dimensional
systems with aperiodic order.
