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In disordered systems, wave functions in the Schr\"{o}dinger equation mayexhibit a transition from the extended phase to the localized phase, in whichthe states at the boundaries or mobility edges may exhibit multifractality.Meanwhile, the Critical Phase (CP), where all states exhibit multifractalstructures, has also attracted much attention in the past decades. However, ageneric way to construct the CP on demand still remains elusive. Here, ageneral approach for this phase is presented using two coupled quasiperiodicchains, where the chains are chosen so that before coupling one of them hasextended states while the other one has localized states. We demonstrate theexistence of CP in the overlapped spectra in the presence of inter-chaincoupling using fractal dimension and minimal scaling index based onmultifractal analysis. Then we examine the generality of this physics bychanging the forms of inter-chain coupling and quasiperiodic potential, wherethe CP also emerges in the overlapped spectra. We account for the emergence ofthis phase as a result of effective unbounded potential, which yields singularcontinuous spectra and excludes the extended states in the overlapped regimes.Finally, the realization of this CP in the continuous model using ultracoldatoms with bichromatic incommensurate optical lattice is also discussed. Due tothe tunability of the two chains, this work provides a general approach torealizing the CP in a tunable way. This approach may have wide applications inthe experimental detection of CP and can be generalized to much more intriguingphysics in the presence of interaction for the many-body CP.

The general approach to the critical phase with coupled quasiperiodic chains