(d−2) -Dimensional Edge States of Rotation Symmetry Protected Topological States

Abstract

We study fourfold rotation-invariant gapped topological systems with time-reversal symmetry in two and three dimensions (d=2, 3). We show that in both cases nontrivial topology is manifested by the presence of the (d-2)-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through the explicit construction of microscopic models having robust (d-2)-dimensional edge states.

Waves cornered

“…The experimental discovery of materials known as higher-order topological insulators corroborates theoretical predictions and expands the toolbox for examples [7][8][9] , such that ordinary topological insulators appear at the first order. A higher order insulator can be thought of as having a nested topological structure.…”

Higher‐Order Topological States in Surface‐Wave Photonic Crystals

“…The quest for topological 0D cavity modes in 2D electromagnetic‐wave systems, which could serve as an important ingredient to build‐up of robust electromagnetic‐wave/photonic devices, was unsuccessful until very recently . Such an achievement was realized using the higher‐order topological insulators . Unlike the conventional D ‐dimensional topological insulators which have ( D −1)‐dimensional topological gapless boundary states, a D ‐dimensional higher‐order topological insulator gives rise to ( D − 2)‐dimensional (or even lower‐dimensional) topological gappless boundary states, in addition to the ( D − 1)‐dimensional gapped boundary states, offering a paradigm beyond the conventional bulk‐boundary correspondence.…”

Second-order bulk-boundary correspondence in rotationally symmetric topological superconductors from stacked Dirac Hamiltonians

“…where Ξ is an antiunitary operator satisfying Ξ 2 = +1. The presence of an additional n-fold rotational symmetry C n allows for a richer topological classification [14,18,29] than if the only symmetry was PH symmetry. We now review the classification scheme devised by Benalcazar et al for classifying crystalline superconductors with rotational symmetry [14].…”