Topological Insulators Turn a Corner

Abstract

Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but with 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 secondorder topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of e 2 =h.

Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems

“…For closed systems, as described by Hermitian Hamiltonians, the so-called bulk-boundary correspondence represents a ubiquitous guiding principle to the phenomenology of topological insulators: bulk topological invariants characterizing a given phase are uniquely reflected in gapless (metallic) surface states. This general pattern is found throughout the recently established hierarchy of topological phases, where nth order phases in d spatial dimensions feature d−n dimensional generalized boundary states [4][5][6][7][8].In contrast to this systematic picture, quite basic questions have so far remained unanswered for open systems governed by non-Hermitian Hamiltonians [9] with applications ranging from various mechanical and optical meta-materials subject to gain and loss terms [10], to quasiparticles with finite lifetime in heavy-fermion systems [11,12]. The topological properties of such systems can crucially differ from their Hermitian counterparts [9,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], as exemplified by the prediction and experimental observation of unconventional topological boundary modes in certain parity time-reversal (PT ) symmetric systems [27][28][29][30][31][32].…”

Geometric orbital magnetization in adiabatic processes

“…However, only the fractional part of Q edge is independent of the detailed choice of the interpolation -the fractional part depends only on the initial and the final values of P 1 that can be individually computed by Eq. (20). What we described here can be straightforwardly translated to 2D systems.…”

Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence

“…Introduction. Topological phases of matter are at the forefront of condensed-matter research with a recent focus on higher-order topological phases [1][2][3][4][5][6][7][8][9][10][11][12][13], where a subtle interplay between topology and crystalline symmetry results in the appearance of boundary states on boundaries with a codimension higher than one, i.e., corners or hinges. Another increasingly popular direction of research revolves around studying topology in the context of non-Hermitian physics, which is a relevant approach for describing a wide range of dissipative systems [14?…”