Higher‐Order Topological Band Structures

Abstract

The interplay of topology and symmetry in a material's band structure may result in various patterns of topological states of different dimensionality on the boundary of a crystal. The protection of these “higher‐order” boundary states comes from topology, with constraints imposed by symmetry. Herein, the bulk–boundary correspondence of topological crystalline band structures, which relates the topology of the bulk band structure to the pattern of the boundary states, is reviewed. Furthermore, recent advances in the K‐theoretic classification of topological crystalline band structures are discussed.

Topological Edge States of a Majorana BBH Model

“…Indeed, for systems with spatial symmetries, the simultaneous presence of standard symmetries (C, P, T ) and crystalline symmetries is essential to realize a second-order topology. In analogy with the Aharonov-Bohm (AB) effect for a magnetic field [23], one observes the presence of a quantized charge polarization in the absence of the Berry curvature; this phenomenon gives rise to the higher order topological superconductivity [5][6][7][8][9]22]. In Figure 2 panel (a), we report the topological phase diagram (PD) of our system, which falls in a non-trivial regime for |w| < |v|.…”

“…In particular, the 2D Zak phase takes the value P = (1/2, 1/2) in a non-trivial phase and P = (0, 0) in the trivial one. The presence of C 4 symmetry ensures that the model described is a second-order topological system with intrinsic Majorana corner states associated to the fractional quantized values of P, which cannot be removed by a change of boundary preserving the bulk [22]. Indeed, for systems with spatial symmetries, the simultaneous presence of standard symmetries (C, P, T ) and crystalline symmetries is essential to realize a second-order topology.…”

Quadrupole spin polarization as signature of second-order topological superconductors

“…1(b). We note that such a SOTSC phase remains stable against arbitrary types of disorder as long as additional perturbations do not close the surface gap [32][33][34].…”

“…SOTSCs with two corner states. To begin with, we consider a class of SOTSCs which supports a single pair of corner states [31][32][33][34][35][36][37][38]. The starting point of our consideration is a simple 2D model for a helical TSC [36], described by the following Hamiltonian in momentum representation: The Pauli matrices η j act on the particle-hole space, σ j on the spin space, and τ j on a generic local degree of freedom (e.g., an electron orbital).…”

Bulk-and-edge to corner correspondence

“…Notable examples include quantum (spin) Hall effect where Chern number (Kane-Mele invariant [13]) predicts quantized (spin) Hall conductance [14][15][16] and Z 2 invariant predicting quantized zero-energy conductance of the Kitaev chain [17][18][19][20]. In this work, such correspondence is referred to as topological bulk-boundary correspondence [12,[21][22][23][24]. In certain cases, bulk-boundary correspondence can be enriched by the attribute "topological" in the presence of certain symmetries: The bulk polarization and a fractional part of the end charge become quantized in the presence of inversion symmetry, the bulk geometric orbital magnetization and a fractional part of the time-averaged edge current are quantized in the presence of inversion or fourfold rotation symmetry [8], and similarly, the magnetoelectric polarizability and the associated boundary quantity are quantized in the presence of time-reversal or inversion symmetry [9].…”