Topological states in quasicrystals

TL;DR

This review summarizes recent advances in understanding topological states in quasicrystals, emphasizing 1D and 2D systems, their unique symmetries, and topological invariants adapted to aperiodic structures.

Contribution

It provides a comprehensive overview of topological phases in quasicrystals, including new topological invariants and symmetry-protected states specific to quasicrystalline structures.

Findings

Topological states in 1D quasicrystals linked to synthetic dimensions.

Extension of crystalline topological states to 2D quasicrystals.

Identification of symmetry-protected topological states unique to quasicrystals.

Abstract

With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We first give a brief introduction to quasicrystalline structures. Then, we discuss topological phases in 1D quasicrystals where the topological nature is attributed to the synthetic dimensions associated with the quasiperiodic order of quasicrystals. We further present the generalization of various types of crystalline topological states to 2D quasicrystals, where real-space expressions of corresponding topological invariants are introduced due to the lack of translational symmetry in quasicrystals. Finally, since quasicrystals possess forbidden symmetries in crystals such as five-fold and eight-fold rotation, we provide an overview of unique quasicrystalline symmetry-protected topological states without crystalline counterpart.