Symmetry representation approach to topological invariants in $C_{2z}T$-symmetric systems

We study the homotopy classification of symmetry representations to describe the bulk topological invariants protected by the combined operation of a twofold rotation $C_{2z}$ and time-reversal $T$ symmetries. We define topological invariants as obstructions to having smooth Bloch wave functions compatible with a momentum-independent 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 (2D) topological invariant protected by $C_{2z}T$, is the homotopy invariant that characterizes the second homotopy class of the matrix representation of $C_{2z}T$. As an application of our result, we show that the three-dimensional (3D) bulk topological invariant for the $C_{2z}T$-protected topological crystalline insulator proposed by C. Fang and L. Fu in Phys. Rev. B 91, 161105 (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 hinge 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- and the second-order topological insulators, simultaneously, which is in consistence with the recent classification of higher-order topological insulators protected by an order-two symmetry.

Figure 1

3D strong Stiefel-Whitney insulator (SWI) protected by $C_{2z}T$ symmetry. (a) Schematic figure describing the second Stiefel-Whitney numbers on the $C_{2z}T$-invariant planes in momentum space. In a 3D strong SWI, $w_2(k_z=\pi )-w(k_z=0)=1$ modulo two. (b) Schematic figure describing the gapless states on the surface and hinges in real space. An odd number of 2D Dirac fermions appear on each of the top and bottom surfaces. 1D chiral fermions appear on the side hinges.

Figure 2

Effective domain for the sewing matrix. (a) A plane representing the 3D Brillouin zone. The yellow region shows the effective Brillouin zone, and the red region with $k_z=0$ is a $C_{2z}T$-invariant plane. The $k_z=\pi$ plane is assumed to be topologically trivial. (b) A 3-sphere equivalent to the 3D Brillouin zone. (c) The sewing matrix $G$ in the yellow region and on its boundary (red).

Figure 3

Gauge transformation from a real to a smooth complex gauge in a $C_{2z}T$-invariant plane. (a) $C_{2z}T$-invariant 2D Brillouin zone covered by two patches $A$ and $B$ in a real gauge. (b) The patch $A$ whose $k_x=\pi$ line is contracted to a point. (c) The gauge transformation matrix $U$ on the patch $A$.

Figure 4

Dehn twist of the Brillouin zone. A Brillouin zone defined by $0\le k_x,k_y\le 2\pi$ is Denn twisted to a tilted Brillouin zone shown as a yellow (shaded) region. When both 1D cycles along $k_x$ and $k_y$ have nontrivial 1D topological invariants, i.e., $w_{1x}=w_{1y}=1$, we have a trivial 1D cycle $k_y^{\prime }$ by a Denn twist because $w_{1y^{\prime }}=w_{1x}+w_{1y}=0mod2$.

Figure 5

Numerical calculation using the model in Eq. (31). $v=0.5,m_{14}=0.5$, and $m_{24}=0.3$. (a), (b) Wilson loop calculation on (a) $k_z=0$ and (b) $k_z=\pi$ planes. (c)–(g) Finite-size calculation with $14\times 14\times 14$ unit cells. (c) Energy eigenvalues in an increasing order. Near the half-filling ($n=2\times {14}^3=5488$), anomalous states appear within the bulk gap $|E_g|\approx 0.37$. (d)–(g) Density profile of in-gap states computed by using the (d) first, (e) second (f), third (g), and fouth highest occupied states below Fermi level at half-filling. (h) Band structure of a model which is periodic in the $(x,y)$ direction, and has 20 unit cells in the $z$ direction. The Dirac point in the bulk band gap originates from the top and bottom surfaces. (i) Band structure of the model which has $20\times 20$ unit cells in the $(x,y)$ direction but periodic in the $z$ direction. Hinge states appear within the bulk band gap.

Figure 6

Wilson line operator in a $C_{2z}T$-invariant Brillouin zone. (a) $W(k_x,k_y)$ is the Wilson line operator $W_{(k_x,-\pi )\to (k_x,k_y)}$. (b) Deformation of (a) after $k_x=-\pi ,k_x=\pi$, and $k_y=-\pi$ lines are contracted to a point.

Figure 7

Anomalous boundary states of axion insulators. (a)–(g) Symmorphic symmetries: (a) $P$, (b) $C_{2z}T$, (c) $C_{4z}T$, (d) $C_{6z}T$, (e) $M=C_{2}P$, (f) $C_{3}P$, (g) $C_{4}P$, (h) $C_{6}P$. (i)–(l) Nonsymmorphic symmetries: (i) glide $M$, (j) screw $C_{2z}T$, (k) screw $C_{4z}T$, (l) screw $C_{6z}T$. Shaded gray regions are the (glide) mirror-invariant planes.