General approach to the critical phase with coupled quasiperiodic chains

In disordered systems, wave functions in the Schr\"odinger equation may exhibit a transition from the extended phase to the localized phase, in which the states at the boundaries or mobility edges may exhibit multifractal structures. Meanwhile, the critical phase (CP) in a phase diagram, separating the localized phase and extended phase, in which all states exhibit multifractal structures, has also attracted much attention in the past decades in some particularly designed models with quasiperiodic potentials. However, a generic way to construct the CP on demand still remains elusive. Here, a general approach for this phase is presented using two coupled quasiperiodic chains, where the chains are chosen in such a way that before coupling, one of them has extended states while the other one has localized states. We demonstrate the existence of the CP in the overlapped spectra when interchain coupling is presented, using fractal dimension and minimal scaling index based on multifractal analysis. Then we examine the generality of this physics by changing the forms of interchain coupling and quasiperiodic potential, where the CP also emerges in the overlapped spectra. We account for the emergence of this phase as a result of the effective unbounded potential, which yields singular continuous spectra and excludes the extended states in the overlapped regimes. Finally, the realization of this CP in the continuous model using ultracold atoms with a bichromatic quasiperiodic optical lattice is also discussed. Due to the tunability of the two chains, this work provides a general approach to realizing the CP in a tunable way. This approach may have wide applications in the experimental detection of CP and can be generalized to much more intriguing physics in the presence of interactions for the many-body CP.