Second-order topological insulators and superconductors with an order-two crystalline symmetry

Second-order topological insulators and superconductors have a gapped excitation spectrum in bulk and along boundaries, but protected zero modes at corners of a two-dimensional crystal or protected gapless modes at hinges of a three-dimensional crystal. A second-order topological phase can be induced by the presence of a bulk crystalline symmetry. Building on Shiozaki and Sato's complete classification of bulk crystalline phases with an order-two crystalline symmetry [Phys. Rev. B 90, 165114 (2014)], such as mirror reflection, twofold rotation, or inversion symmetry, we classify all corresponding second-order topological insulators and superconductors. The classification also includes antiunitary symmetries and antisymmetries.

Figure 1

Schematic picture of an “extrinsic” second-order topological insulator consisting of a three-dimensional topological insulator placed in a magnetic field in a generic direction, as proposed by Sitte et al. [10] (a). Each surface has a finite flux and there are chiral modes along hinges that touch two faces with opposite sign of the magnetic flux. The gapless hinge modes may be removed by exchange-coupling some of the crystal faces to a two-dimensional ferromagnetic insulator (b).

Figure 2

(a) Schematic picture of a generic corner of a two-dimensional crystal. A generic corner may host a protected zero-energy state if and only if the corresponding Altland-Zirnbauer class in $d-1$ dimensions is nontrivial. (b) Zero-energy corner states in a generic corner can always be moved to a different corner by a suitable change of the lattice termination. For the example shown here, a one-dimensional topological insulator or superconductor with two end states is “glued” to one of the crystal faces adjacent to the top corner, such that its end state and the original zero-energy corner state mutually gap out. As a result, the corner state has moved to the corner on the left.

Figure 3

Schematic picture of a lattice model for a crystalline insulator in a two-terminal scattering geometry. The reflection matrix $r_d$ relates the amplitudes $a_{\mathrm{in}}$ and $a_{\mathrm{out}}$ of incident and reflected waves as shown in the figure. The ideal leads are modeled as a grid of parallel one-dimensional chains, which endows the reflection matrix $r_d$ with a real-space structure.

Figure 4

Comparison of the reflection-matrix-based dimensional reduction scheme applied to a two-dimensional Chern insulator (left) and a three-dimensional second-order Chern insulator. In each case a lower-dimensional Hamiltonian can be constructed out of the reflection matrix $r_d$ describing scattering from a half-infinite crystal coupled to an ideal lead. Upon constructing the lower-dimensional Hamiltonian $H_{d-1}$, the chiral edge states (left) and hinge states (right) map to protected zero-energy eigenstates localized near ends (left) or corners (right).

Figure 5

Support of the zero-energy eigenstates (a) and 30 lowest energies of the spectrum (b) of the mapped Hamiltonian $H_1$ for a two-dimensional Chern insulator with Hamiltonian $H_2$ given in Eq. (11), following the reflection-matrix-based dimensional reduction scheme. (c) and (d) show the same for the mapped Hamiltonian $H_2$ for the three-dimensional second-order Chern insulator with Hamiltonian $H_3$ of Eq. (12) with $b=0.4$.

Figure 6

Schematic picture of a generic corner (a), a mirror-symmetric corner with locally broken mirror symmetry (b), and a mirror-symmetric corner in a crystal with a bulk mirror symmetry. (d)–(f) represent the possibility to add zero-energy corner states by changing the lattice termination. Effectively, this amounts to the addition of one-dimensional topological insulators or superconductors to the boundaries. At a generic corner, it is possible to change the termination of only one boundary, as shown in (d). In a symmetric corner, such a change in termination needs to be applied to both symmetry-related boundaries, shown schematically in (e) and (f) for a corner with and without a perturbation that locally breaks the mirror symmetry.

Figure 7

The extrinsic classification of corner states in a mirror-symmetric corner of a two-dimensional crystal is the same as that of end states of a one-dimensional crystal with a transverse mirror symmetry with $d_{\parallel }=0$. The vertical dashed line is the mirror axis.

Figure 8

Pairs of corner states may be created by “glueing” one-dimensional topologically nontrivial chain to the crystal edges. Mirror symmetry requires that the chains added to mirror-related edges are mirror images of each other.

Figure 9

A mirror-symmetric edge, with coordinate $x$ running along the edge (a) can be deformed into a corner joining two mirror-related edges (b). The situation shown in (a) has mirror symmetry acting everywhere along the edge; in (b) mirror symmetry exists only for a mirror reflection axis going through the corner at $x=0$.

Figure 10

(a) A two-dimensional crystal with a pair of mirror-related edges, but without a mirror-symmetric corner. (b) The crystal may be smoothly deformed into a crystal with a mirror-symmetric corner. The parity of the number of zero-energy states between the two mirror-related edges in (a) is the same as the parity of the number of zero-energy states at the mirror-symmetric corner in (b).

Figure 11

(a) A surface perpendicular to the twofold rotation axis hosts a gapless mode in a nontrivial topological crystalline phase. The surface Hamiltonian acquires a mass term $m(\theta ,\varphi )$ upon tilting the surface away from the normal direction, which depends on the tilt angle $\theta$ and the azimuthal angle $\varphi$. The mass term is odd under the twofold rotation operation, $m(\theta ,\varphi )=-m(\theta ,\varphi +\pi )$. (b) A generic rotation symmetric surface. Surfaces related by twofold rotation have opposite mass terms. As a result, a protected gapless hinge mode (thick black line) forms at the intersection of surfaces with masses of different sign. The situation shown in the figure corresponds to $\mathrm{sign}\left(m_1\right)=-\mathrm{sign}\left(m_2\right)$.

Figure 12

Dimensional reduction scheme from a three-dimensional second-order topological insulator with twofold rotation symmetry to a two-dimensional second-order topological insulator with inversion symmetry. Upon dimensional reduction, the Altland-Zirnbauer class changes from $s$ to $s-1$ (modulo 2 for the complex classes, modulo 8 for the real classes), see the discussion in the main text.

Figure 13

Support of the zero-energy corner state obtained from exact diagonalization of the two-dimensional time-reversal invariant superconductor in class DIII with Hamiltonian (26) with $m=-1$ and $\mathbf{\Gamma }$ and $\mathbf{B}$ given by Eq. (37) with $\mathbf{b}=(0.3,0)$ (a), $\mathbf{b}=(0.3,0.1)$ (b), and $\mathbf{b}=(0,0.3)$ (c).

Figure 14

Support of the zero-energy eigenstates (if present, left) and the lowest 30 eigenvalues [right (a)–(c)] of the model discussed in Sec. 6a4. (a) is for the case that mirror antisymmetry is present, $\mathbf{b}=(0.4,0)$ and $\mathbf{c}=(0,0)$, which has a Kramers pair of zero-energy states localized at the mirror-symmetric top and bottom corners. Breaking the mirror antisymmetry locally at the top corner removes one zero-energy Kramers pair, as shown in (b). No zero-energy Kramers pairs remain after removing the mirror antisymmetry at both the top and the bottom corner, as shown in (c).

Figure 15

Support of the zero-energy hinge modes for a three-dimensional crystal with tight-binding Hamiltonian specified by Eqs. (26) and (40) for $\mathbf{b}=(0.8,0.8,0.8)/\sqrt{3}$ (a) and $\mathbf{b}=(0.8,0.8,0)/\sqrt{2}$ (b). The example shown in (a) has mirror-reflection symmetry, twofold rotation symmetry, and inversion symmetry; the example in (b) has inversion symmetry only. (c) shows the support of the zero-energy corner modes obtained for the two-dimensional tight-binding model specified by Eqs. (26) and (41) with $\mathbf{b}=(0.4,0)$.

Figure 16

Support of helical hinge modes of the tight-binding Hamiltonian (26) with $\mathbf{\Gamma }$ and $\mathbf{B}$ given by Eq. (42) and $\mathbf{b}=(0.4,0.4,0)$ (a) and $\mathbf{b}=(0.4,0.4,0.4)$ (b). For the example shown in (a) the top and bottom surfaces are invariant with respect to the twofold rotation symmetry, which explains the presence of gapless surface modes at the top and bottom surface. The twofold rotation symmetry is broken in (b), which has inversion symmetry only.

Figure 17

Support of the zero-energy states of the tight-binding Hamiltonian (26) with $\mathbf{\Gamma }$ and $\mathbf{B}$ given by Eq. (44) and $\mathbf{b}=(0.4,-0.4,0)$ (a) and $\mathbf{b}=(0.4,-0.4,0.4)$ (b). For the example shown in (a), the top and bottom surfaces are invariant with respect to the twofold rotation symmetry, which explains the presence of gapless surface modes at the top and bottom surface. The twofold rotation symmetry is broken in (b), which only has inversion symmetry. (c) shows the support of the zero-energy corner modes of the two-dimensional Hamiltonian in class ${\mathrm{AIII}}^{{\mathcal{T}}^{+}{\mathcal{R}}_{-}}$ obtained by dimensional reduction of the three-dimensional model, with parameter $\mathbf{b}=(0.4,-0.4,0)$. For comparison, (d) shows the support of the zero-energy corner modes obtained for the two-dimensional tight-binding model specified by Eqs. (26) and (45) with $\mathbf{b}=(0.4,-0.4)$.

Figure 18

Schematic picture of a $d$-dimensional gapped crystalline insulator occupying the half space $x_d>0$, with periodic boundary conditions applied along the remaining $(d-1)$ dimension, coupled to an ideal lead with a $(d-1)$-dimensional cross section. The reflection matrix $r_d\left({\mathbf{k}}_{\perp }\right)$ relates the amplitudes $a_{\mathrm{out}}\left({\mathbf{k}}_{\perp }\right)$ and $a_{\mathrm{in}}\left({\mathbf{k}}_{\perp }\right)$ of outgoing and incoming modes in the lead.