Exact mobility edges in quasiperiodic network models with slowly varying potentials

Abstract: Quasiperiodic models are important physical platforms to explore Anderson transitions in low dimensional systems, yet the exact mobility edges (MEs) are generally hard to be determined analytically. To date, the MEs in only a few models can be determined exactly. In this manuscript, we propose a new class of network models characterized by quasiperiodic slowly varying potentials and the absence of hidden self-duality, and exactly determine their MEs. We take the mosaic models with slowly varying potentials as examples to illustrate this result and derive its MEs from the effective Hamiltonian. In this method, we can integrate out the periodic sites to obtain an effective Hamiltonian with energy-dependent potentials $g(E)V$ and effective eigenenergy $f(E)$, which directly yields the MEs at $f(E) = \pm(2t^\kappa \pm g(E)V)$, where $\kappa \in \mathbb{Z}^+$. With this idea in hand, we then generalize our method to more quasiperiodic network models, including those with much more complicated geometries and non-Hermitian features. Finally, we propose the realization of these models using optical waveguides and show that the Anderson transition can be observed even in small physical systems (with lattice sites about $L = 50 - 100$). Our results provide some key insights into the understanding and realization of exact MEs in experiments.