Hidden localization transitions in canonically rotated Aubry-André models

Abstract

Anderson localization is a phase transition between a "metallic phase", where wavefunctions are extended and delocalized in space, and an "insulating phase", where wavefunctions are completely localized. These transitions are driven by uncorrelated or quasiperiodic disorder, e.g., in the case of the Aubry-André model. Here, I consider a family of Hamiltonians that generalizes the Aubry-André model, obtained by replacing the position and momentum operators with an arbitrary pair of canonically conjugate operators. These models exhibit a hidden localization transition. The system transitions between phases where wavefunctions are either localized or delocalized with respect to the new canonically conjugate operators, acting as an insulator or metal in this rotated space. These canonically conjugate operators can be taken as a linear combination of position and momentum, corresponding to a "rotation" in the abstract space of canonical operators. In this case, the hidden localization transition is signaled by the simultaneous vanishing of both the inverse participation ratio (IPR) and the normalized participation ratio (NPR) in the position and momentum space in the thermodynamic limit. This identifies the emergence of multifractal states that are neither fully extensive nor localized on the lattice. Hence, the states exhibit a multifractal dimension at the hidden phase transition, while remaining extended (i.e., one-dimensional) in both momentum and position everywhere else in the parameter space. Surprisingly, I found that at the phase transition, this model Hamiltonian coincides with the lattice Hamiltonian of a massless Dirac fermion in a curved spacetime background, indicating an unexpected relation between localization transitions and analog gravity.