Higher-order topological insulators

Abstract

A new class of materials that are insulating in the bulk and on surfaces but have conducting hinge channels is studied.

Topological Insulators Turn a Corner

“…Since this mass term is odd under reflection, we expect an integer number of one-dimensional helical states at the intersection of two surfaces related by R. An even number of helical states is unstable, however, to a local perturbation at the edge and can be gapped out without closing the gaps in the sample bulk or at the surfaces, consistent with the Z 2 classification in Table I. (At this point our classification differs from that of Ref. [7], which does not allow for reflection-symmetry breaking perturbations at the crystal edge, thus arriving at a Z [22] classification.) As a specific example, we consider the eight-band Hamiltonian…”

“…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.…”

“…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 ¼ 2) has zero-energy states at corners, but a gapped bulk and no gapless edge states.…”

“…However, crystal symmetries can be a key to ensure that a natural surface termination-, i.e., a surface termination that respects the crystal symmetries-automatically leads to a nontrivial higher-order topological phase. For example, Benalcazar et al employed a combination of multiple reflection symmetries [9], whereas Schindler et al considered C 4 T symmetry, the product of a π=2 rotation and time reversal, as well as a model with reflection symmetry [7].…”

Reflection-Symmetric Second-Order Topological Insulators and Superconductors

“…However, crystal symmetries can be a key to ensure that a natural surface termination -i.e., a surface termination that respects the crystal symmetries -automatically leads to a nontrivial higher-order topological phase. For example, Benalcazar et al employed a combination of multiple reflection symmetries [9], whereas Schindler et al considered C 4 T symmetry, the product of a π/2 rotation and time reversal, as well as a model with reflection symmetry [7].In this letter, we show that a single mirror symmetry is sufficient to construct models for secondorder topological insulators and superconductors in d = 2 and d = 3 for all five Altland-Zirnbauer classes for which second-order topological insulators are allowed. Reflection-symmetric topological crystalline insulators were the first to be realized experimentally [16,19,20] use of this classification.…”