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Higher-order band topology

Abstract

A conventional topological insulator (TI) has gapped bulk states but gapless edge states. The emergence of the gapless edge states is dictated by the bulk topological invariant of the insulator and the preservation of relevant symmetries. Over the past four years, a new type of TI has been found, which hosts gapless hinge or corner states, rather than edge states. These unconventional TIs, termed higher-order TIs (HOTIs), are common among crystalline and quasi-crystalline materials. Higher-order band topology expands our previous understanding of topological phases and provides unprecedented lower-dimensional boundary states for devices. Here, we review the principles, theories and experimental realizations of HOTIs for both electrons and classical waves. There is an emphasis on the development of HOTIs in photonic, phononic and circuit systems owing to their special contributions to these fields. From these discussions, we remark on trends and challenges in the field and the impact of higher-order band topology on other scientific disciplines.

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Fig. 1: Classification of higher-order topological insulators with multidimensional topological boundary states.
Fig. 2: Higher-order topological insulators with quantized multipole moments.
Fig. 3: Higher-order topological insulators without quantized multipole moments.
Fig. 4: Experimental realizations of higher-order topological insulators.

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Acknowledgements

This work was supported by the National Key R & D Program of China (grant numbers 2017YFA0303702, 2018YFA0306200 and 2016YFB0700301), the National Natural Science Foundation of China (grant numbers 11625418, 11474158, 11890700, 51732006, 11904060 and 12074281) and the Jiangsu specially appointed professor funding.

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Glossary

Ammann−Beenker tiling

A tiling generated by a set of 45−135-degree rhombuses and 45−45−90 degree triangles, which has a non-periodic and hierarchical structure.

Atiyah−Hirzebruch spectral sequence

A type of spectral sequence for calculating generalized cohomology.

Bulk-boundary correspondence

The correspondence between the topological phases of bulk materials and the emergence of gapless boundary states.

Codimensional

The dimension of bulk states minus the dimension of some specific states. Therefore, higher-codimensional states have lower dimensionality in one system.

K-homology

A homology theory, dual to K-theory and used to classify the elliptic pseudo-differential operators acting on the vector bundles over a space.

K-theory

A branch of mathematics that studies a ring generated by vector bundles over a topological space or scheme; it can be applied to classify topological phases with respect to different symmetries.

Multistep driving

Spatial control of the tunnelling amplitudes between lattice sites in multiple steps within each driving period in a Floquet system.

Non-Hermitian skin effect

A unique feature of non-Hermitian systems, in which an extensive number of eigenstates are exponentially localized at the boundary of the system under the open boundary conditions.

Transmission line networks

Transmission lines are coaxial cables that can be regarded as 1D waveguides; they can be connected at nodes to form a network that can be used to mimic tight-binding models.

Zak phase

Berry phase gained by a particle during the adiabatic motion across the Brillouin zone, which is quantized and can be regarded as the topological invariant of the band.

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Xie, B., Wang, HX., Zhang, X. et al. Higher-order band topology. Nat Rev Phys 3, 520–532 (2021). https://doi.org/10.1038/s42254-021-00323-4

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