Reflection-Symmetric Second-Order Topological Insulators and Superconductors

Abstract

Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of e 2 /h.Introduction.-After the discovery of topological insulators and superconductors and their classification for the ten Altland-Zirnbauer symmetry classes [1][2][3], the concept of nontrivial topological band structures has been extended to materials in which the crystal structure is essential for the protection of topological phases [4]. This includes weak topological insulators [5], which rely on the discrete translation symmetry of the crystal lattice, and topological crystalline insulators [6], for which other crystal symmetries are invoked to protect a topological phase. Whereas the original strong topological insulators always have topologically protected boundary states, weak topological insulators or topological crystalline insulators have protected boundary states for selected surfaces/edges only.In a recent publication, Schindler et al. [7] proposed another extension of the topological insulator (TI) family: a higher-order topological insulator. Being crystalline insulators, these have well-defined faces and well-defined edges or corners at the intersections between the faces. An nth order topological insulator has topologically protected gapless states at the intersection of n crystal faces, but is gapped otherwise [7]. For example, a second-order topological insulator in two dimensions (d = 2) has zeroenergy states at corners, but a gapped bulk and no gapless edge states. Earlier examples of higher-order topological insulators and superconductors avant la lettre appeared in works by (see also [11,12]), who considered insulators and superconductors with protected corner states in d = 2 and d = 3 [13]. Sitte et al. showed that a threedimensional topological insulator in a magnetic field of generic direction also acquires the characteristics of a second-order topological Chern insulator, with chiral states moving along the sample edges [14].Since a second-order TI has a topologically trivial d-dimensional bulk, from a topological point of view its boundaries are essentially stand-alone (d − 1)-dimensional insulators, so that topologically protected states at corners (for d = 2) or edges (for d = 3) arise naturally as "domain walls" at the intersection of two boundaries if these ar...

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.…”

“…[28,35] Such an achievement was realized using the higher-order topological insulators. [27][28][29][30][31][32][33][34][35][36][37] 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. Through the concept of higher-order topological insulators, it has been demonstrated that topological 0D corner states can emerge in mechanical metamaterials, [27] microwave circuits with lumped components, [28] and coupled optical waveguides.…”

Deep‐Subwavelength Holey Acoustic Second‐Order Topological Insulators

“…Also, the above reported acoustic implementations have, for the most part, been implemented inside waveguides or were designed in an acoustically rigid enclosure, which hinders their capabilities from external insonification. [23][24][25][26] Most recently, the concept of higher-order TIs (HOTIs), [27][28][29][30][31][32][33] which is a special class of TIs with unconventional bulk-boundary correspondence, was proposed to support the existence of the lower-dimensional boundary states. The advantage in using holey metamaterials resides in their high levels of integration and miniaturization at scales much smaller than the sound wavelength.…”

“…Concerning this matter, the findings will thus have the capability to push forward exciting applications for robust acoustic imaging way beyond the diffraction limit.The discovery of topological insulators (TIs) [1,2] and the strikingly robust transmission of reflectionless edge states have boosted intense research in classical systems such as photonics, [3][4][5][6][7][8][9] acoustics, [10][11][12][13][14][15][16][17][18][19][20][21][22] and mechanics. [23][24][25][26] Most recently, the concept of higher-order TIs (HOTIs), [27][28][29][30][31][32][33] which is a special class of TIs with unconventional bulk-boundary correspondence, was proposed to support the existence of the lower-dimensional boundary states. For example, in d-dimensional (dD) systems, a second-order TI (SOTI) has dD gapped bulk states, (d − 1)D gapped boundary states, and (d − 2)D gapless boundary states, which are 0D corner states…”