There are two prominent applications of the mathematical concept of topology to the physics of materials: band topology, which classifies different topological insulators and semimetals, and topological defects that represent immutable deviations of a solid lattice from its ideal crystalline form. Although these two classes of topological phenomena have generally been treated separately, recent experimental advancements have begun to probe their intricate and surprising interactions, in solid-state materials as well as in metamaterials. Topological lattice defects in topological materials offer a platform to explore a diverse range of phenomena, such as robust topological-bound states, fractional charges, topological Wannier cycles, chiral and gravitational anomalies, topological lasers and topological-defect-induced pumping and non-Hermitian skin effects. In this Perspective article, we survey the developments in this rapidly moving field from both theoretical and experimental perspectives, with an emphasis on the latter. We also give an outlook on the potential impact of these phenomena on condensed matter physics, photonics, acoustics and materials science.
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Z.K.L., Y.L. and J.H.J. were supported by the National Key R&D Program of China (2022YFA1404400), the National Natural Science Foundation of China (grant no. 12125504), the “Hundred Talents Program” of the Chinese Academy of Sciences, and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. Q.W., H.X., Y.C. and B.Z. are supported by the Singapore Ministry of Education Academic Research Fund Tier 3 grant MOE2016-T3-1-006, and Tier 2 grant MOE2019-T2-2-085.
The authors declare no competing interests.
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- Altland–Zirnbauer ‘tenfold way’
By considering the time-reversal, particle-hole and chiral symmetries and their combinations, there are 10 fundamental symmetry classes that could yield nontrivial topology.
- Chern number
A quantized topological invariant of Bloch bands in the Brillouin zone for 2D and higher-dimensional systems, which are connected to the quantized Hall effect in 2D and other effects in higher dimensions.
- Fractional charges
Local fractional charges bound to a TD when all the valence bands are filled.
- Jackiw–Rebbi solution
Jackiw and Rebbi proposed a solution for Dirac equation with a mass domain wall, which predicts a soliton solution localized at the domain wall.
- Quantum anomaly
The breakdown of effective field theory whereby the behaviour of a material contradicts its classical symmetries. For instance, chiral anomaly is that the quantum system on the edge does not have the chiral symmetry, despite that the hosting material has it.
- Synthetic dimensions
Artificial dimensions created by additional degrees of freedom instead of the real-space dimensions.
- Topological band theory
Theory of topological invariants of energy bands that predict topological properties from the band perspective.
- Topological-bound states
States that are localized around the TDs in the system or the boundary of the system with a topological origin.
- Topological bulk-edge correspondence
The relationship between the properties of the in-gap edge states and the topological properties of all the bulk bands below the gap.
- Topological insulators
(TIs). Insulators with nontrivial band topology characterized by their topological invariants and the protecting symmetries.
- Topological Wannier cycles
Cyclic evolution of eigenstates across bandgaps in higher-order topological insulators owing to screw dislocations or quantized gauge flux insertion.
- Wannier bands
Bands formed by the evolution of Wannier centres with the wavevector along a symmetric direction in the Brillouin zone.
- Zak phase
Berry phase across the Brillouin zone of 1D systems. With inversion symmetry, Zak phase is quantized to 0 or π.
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Lin, ZK., Wang, Q., Liu, Y. et al. Topological phenomena at defects in acoustic, photonic and solid-state lattices. Nat Rev Phys 5, 483–495 (2023). https://doi.org/10.1038/s42254-023-00602-2