Theory of spin Bott index for quantum spin Hall states in nonperiodic systems

Quantum spin Hall effects (QSHE) arising from electron band topology are usually limited to crystals. Here, the authors extend the concept of QSHE to nonperiodic systems. They derive a spin Bott index to characterize the electronic topology in aperiodic and amorphous systems, and confirm its applicability by realizing the QSHE in quasicrystals. As in crystals, the QSHE in quasicrystals manifests with robust metallic edge states and quantized conductance. Their findings not only provide a better understanding of electronic properties of nonperiodic systems, but also extend the search of the QSH phase to aperiodic and amorphous materials that are experimentally feasible.