Building on the results in arxiv:2103.03895, we develop a renormalization group theory to understand the localization phase diagrams of 1D quasiperiodic lattice models, using commensurate approximants. We define renormalized couplings that measure the dependence of single-particle energies on phase twists/Bloch momenta and real-space shifts for increasing unit cell size.
We show that for widely different models, the phase twist (real-space shift) coupling becomes irrelevant in the localized (extended) phase as the unit cell size is increased: the original model flows to an effective model with no hopping (potential) term. At a critical point/phase, both couplings are relevant at any scale.
We identify a special class of models for which the renormalized couplings may be computed analytically, which enables the exact analytical determination of the phase diagram. We also show that approximate analytical predictions can also be made if we add perturbations to such models respecting their original symmetries under shifting and twisting.
To demonstrate the wide applicability of our description, we apply it to a number of known models and also to new models that we introduce.
We finally show preliminary results supporting the possibility to generalize this theory to describe ground-state localization phase diagrams of 1D interacting many-body systems.
Our findings provide a deep and simple understanding of localization for generic 1D quasiperiodic systems; a way to fully characterize the localization phase diagram, including extended, localized and critical phases and the transitions between them; and clear insights on how to create models with analytically trackable phase diagrams.