Is quasiperiodic localization stable in two dimensions
In plain words
A non-random but irregular potential, built from waves whose wavelengths have an irrational ratio to the lattice spacing (quasiperiodic), has no rare weak spots, so the rare-region argument may not apply. Whether such systems localize in two dimensions is open.
Precise statement
Interacting bosons or fermions on a square lattice in the quasiperiodic potential $V_i = V [\operatorname{cos}(2 \pi \beta x_i + \phi_x) + \operatorname{cos}(2 \pi \beta y_i + \phi_y)]$ with irrational $\beta$: is there $V_c$ such that local memory persists at infinite time as $L \to \infty$? Answer yes or no.
What would settle it
A size-independent localization crossover over a wide range of L in experiment, together with a theory excluding resonant instabilities in the quasiperiodic case.
Status in the literature
Unverified note
Agrawal, Vasseur, Gopalakrishnan (PRB 106, 094206, 2022) argue it is stable in $d=2$ at strong modulation, and a 2025 optical-lattice experiment up to 24 x 24 sites saw no clear size drift of the quasiperiodic crossover.