Conduction through materials crucially depends on how ordered the materials are. Periodically ordered systems exhibit extended Bloch waves that generate metallic bands, whereas disorder is known to limit conduction and localize the motion of particles in a medium^1,2. In this context, quasiperiodic systems, which are neither periodic nor disordered, demonstrate exotic conduction properties, self-similar wavefunctions and critical phenomena³. Here, we explore the localization properties of waves in a novel family of quasiperiodic chains obtained when continuously interpolating between two paradigmatic limits⁴: the Aubry–André model^5,6, famous for its metal-to-insulator transition, and the Fibonacci chain^7,8, known for its critical nature. We discover that the Aubry–André model evolves into criticality through a cascade of band-selective localization/delocalization transitions that iteratively shape the self-similar critical wavefunctions of the Fibonacci chain. Using experiments on cavity-polariton devices, we observe the first transition and reveal the microscopic origin of the cascade. Our findings offer (1) a unique new insight into understanding the criticality of quasiperiodic chains, (2) a controllable knob by which to engineer band-selective pass filters and (3) a versatile experimental platform with which to further study the interplay of many-body interactions and dissipation in a wide range of quasiperiodic models.

通过材料的传导关键取决于材料的排列方式。周期性有序的系统表现出扩展的布洛赫波，产生金属带，而已知无序限制了传导，并限制了介质^1,2中粒子的运动。在这种情况下，既不是周期性也不是无序的准周期性系统表现出奇异的传导特性，自相似波函数和临界现象³。在这里，我们探讨了在两个范式极限⁴之间连续内插时获得的新颖准周期链族中的波的定位特性：以金属到绝缘体过渡而闻名的奥布里-安德烈模型^5,6和斐波那契链^{7 ，8}，以其关键特性而闻名。我们发现，Aubry-André模型通过一系列能选择性地塑造斐波纳契链的自相似临界波函数的能带选择定位/离域转变转变为临界状态。通过对腔极化设备的实验，我们观察到了第一个跃迁并揭示了级联的微观起源。我们的发现提供了（1）对理解准周期链的重要性的独特的新见解；（2）可控旋钮，用于设计带选择通滤波器；（3）通用的实验平台，可用来进一步研究许多互斥关系拟周期模型中的人体相互作用和耗散。