Symmetry representation approach to topological invariants in C2zT -symmetric systems

Abstract

We study the homotopy classification of symmetry representations to describe the bulk topological invariants protected by the combined operation of a two-fold rotation C2z and time-reversal T symmetries. We define topological invariants as obstructions to having smooth Bloch wave functions compatible with a momentumindependent symmetry representation. When the Bloch wave functions are required to be smooth, the information on the band topology is contained in the symmetry representation. This implies that the d-dimensional homotopy class of the unitary matrix representation of the symmetry operator corresponds to the d-dimensional topological invariants. Here, we prove that the second Stiefel-Whitney number, a two-dimensional topological invariant protected by C2zT , is the homotopy invariant that characterizes the second homotopy class of the matrix representation of C2zT . As an application of our result, we show that the three-dimensional bulk topological invariant for the C2zT -protected topological crystalline insulator proposed by C. Fang and L. Fu in Phys. Rev. B 91, 161105(R) (2015), which we call the 3D strong Stiefel Whitney insulator, is identical to the quantized magnetoelectric polarizability. The bulk-boundary correspondence associated with the quantized magnetoelectric polarizability shows that the 3D strong Stiefel-Whitney insulator has chiral hinges states as well as 2D massless surface Dirac fermions. This shows that the 3D strong Stiefel Whitney insulator has the characteristics of both the first order and the second order topological insulators, simultaneously, which is consistent with the recent classification of higher-order topological insulators protected by an order-two symmetry.arXiv:1810.05363v3 [cond-mat.mes-hall]

Geometric approach to fragile topology beyond symmetry indicators

“…The Euler class is computed as the integral of the Pfaffian of the two-band Berry-Wilczek-Zee curvature [52,75,76] over the Brilouin zone. It can also be conveniently computed as a two-band Wilson loop winding [36,49].…”

“…gives a two-dimensional example of a nontrivial topology in momentum space which also admits an atomic limit. This is readily indicated by the fact that the Wilson loop spectrum of a rank-p 3 subspace is generically gapped [76], i.e., it does not wind.…”

Axionic band topology in inversion-symmetric Weyl-charge-density waves

“…In 3D TIs and AXIs, the combination of θ = π and ν x,y,z = 0 leads to unusual response properties, including low-energy excitations resembling magnetic monopoles (the Witten effect [40,41]) and quantized Faraday and Kerr rotations [4,42]. AXIs have recently been recognized as "higher-order" TIs (HOTIs) [28][29][30][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] featuring gapped surfaces and odd numbers of sample-encircling chiral hinge modes [ Fig. 1(a)].…”

Braiding Majorana corner modes in a second-order topological superconductor

“…The analysis of noninteracting electrons in materials possessing fundamental symmetries-time-reversal (T ), particle-hole (P), and/or chiral symmetry (C)-led to the initial classifications of such topological phases [9][10][11]. Later on, possible topological phases protected by discrete crystalline symmetries were studied [12][13][14][15][16][17][18][19][20][21][22][23][24] and, lastly, another class of exotic noninteracting topological phases was discovered, namely the second-order topological insulators (SOTIs) and superconductors (SOTSs) [25][26][27][28][29][30][31].…”