Classification of reflection-symmetry-protected topological semimetals and nodal superconductors

While the topological classification of insulators, semimetals, and superconductors in terms of nonspatial symmetries is well understood, less is known about topological states protected by crystalline symmetries, such as mirror reflections and rotations. In this work, we systematically classify topological semimetals and nodal superconductors that are protected, not only by nonspatial (i.e., global) symmetries, but also by a crystal reflection symmetry. We find that the classification crucially depends on (i) the codimension of the Fermi surface (nodal line or point) of the semimetal (superconductor), (ii) whether the mirror symmetry commutes or anticommutes with the nonspatial symmetries, and (iii) how the Fermi surfaces (nodal lines or points) transform under the mirror reflection and nonspatial symmetries. The classification is derived by examining all possible symmetry-allowed mass terms that can be added to the Bloch or Bogoliubov–de Gennes Hamiltonian in a given symmetry class and by explicitly deriving topological invariants. We discuss several examples of reflection-symmetry-protected topological semimetals and nodal superconductors, including topological crystalline semimetals with mirror ${\mathbb{Z}}_2$ numbers and topological crystalline nodal superconductors with mirror winding numbers.

Figure 1

The tenfold classification of gapless topological materials depends on the location of the Fermi surfaces in the Brillouin zone, which in turn determines how the Fermi surfaces transform under global antiunitary symmetries (see Table 1). (a) Each Fermi surface (red point/line) is left invariant under global (i.e., nonspatial) symmetries. The contour, on which the topological invariant is defined, is indicated by blue circles/spheres. Here, $d$ denotes the spatial dimension and $p=d-d_{\mathrm{FS}}$ is the codimension of the Fermi surface. (b) Different Fermi surfaces are pairwise related to each other by global symmetries $\left(\mathbf{k}\leftrightarrow -\mathbf{k}\right)$.

Figure 2

(a) Edge band structure of the nodal topological superconductor (4) (class DIII) for the (11) face as a function of edge momentum $k_{\parallel }=(k_x+k_y)/\sqrt{2}$. The flat-band edge states (red traces) are protected by time-reversal symmetry and particle-hole symmetry. (b) Band structure of the time-reversal-invariant semimetal (A9) (class AII) at the (11) edge as a function of edge momentum $k_{\parallel }$. Linearly dispersing edge states (red traces) connect the projected Fermi points in the edge BZ. (c) Edge spectrum of the sublattice-symmetric (chiral-symmetric) semimetal (14) with $A=B=0.7$ at the (01) face as a function of edge momentum $k_x$. The flat-band edge states (red trace) are protected by sublattice (chiral) symmetry.

Figure 3

The classification of reflection-symmetry-protected topological semimetals and nodal superconductors depends on the location of the Fermi surfaces with respect to the reflection plane (highlighted in green) in the Brillouin zone, which in turn determines how the Fermi surfaces transform under reflection and global antiunitary symmetries (see Tables 2 and 3). (a) Each Fermi surface (red points/lines) is left invariant under reflection and global antiunitary symmetries. (b) The Fermi surfaces are left invariant by reflection, but transform pairwise into each other by the global antiunitary symmetries. The contours on which the $M\mathbb{Z}$- and $M{\mathbb{Z}}_2$-type invariants are defined are indicated by blue points/circles in panels (a) and (b). (c) Different Fermi points are pairwise related to each other by both reflection and global symmetries. The contours on which the $C{\mathbb{Z}}_2$-type invariants are defined are indicated by blue lines/planes (see Table 3).

Figure 4

(a) The honeycomb lattice of graphene is a bipartite lattice composed of two interpenetrating triangular sublattices. The two sublattices are marked “$A$” (black dots) and “$B$” (blue dots). The nearest-neighbor bond vectors (green arrows) are given by ${\mathbf{s}}_1=(-1,0),{\mathbf{s}}_2=\frac{1}{2}(1,\sqrt{3})$, and ${\mathbf{s}}_3=\frac{1}{2}(1,-\sqrt{3})$. The second-neighbor bond vectors (red arrows) are ${\mathbf{d}}_{\mathbf{1}}=-{\mathbf{d}}_{\mathbf{4}}=\frac{1}{2}(3,\sqrt{3}),{\mathbf{d}}_{\mathbf{2}}=-{\mathbf{d}}_{\mathbf{5}}=\frac{1}{2}(3,-\sqrt{3})$, and ${\mathbf{d}}_{\mathbf{3}}=-{\mathbf{d}}_{\mathbf{6}}=(0,-\sqrt{3})$. The mirror line $x\to -x$ is indicated by the green line. (b) Energy spectrum of a graphene ribbon with (10) edges (i.e., zigzag edges) and $(t_1,t_2)=(1.0,0.1)$. A linearly dispersing edge state (red trace) connects the Dirac points, which are located at $k_{\parallel }=2\pi /3$ and $4\pi /3$ in the edge BZ and are projected from the bulk Dirac points at $(0,\pm k_0)$.

Figure 5

(a) Surface band structure of semimetal (54) for the (100) face with ${\mu }_s=0,k_z=0,m_1=2.5$, and $m_2=0.2$ as a function of surface momentum $k_y$. Note that the (100) surface is not symmetric under $k_x\to -k_x$. Zero-energy surface flat bands (red traces) appear within regions of the surface BZ that are bounded by the projected bulk Fermi lines. (b) Surface spectrum on the (100) face as a function of both $k_y$ and $k_z$ for the same parameters as in panel (a). Nondegenerate zero-energy flat bands protected by the winding number $n_{\mathbb{Z}}=1$ [see Eq. (62)] appear within the region $1.3<cos{k}_y+cos{k}_z<1.7$ of the surface BZ (green area). Doubly degenerate flat bands protected by $n_{\mathbb{Z}}=2$ exist within the region $cos{k}_y+cos{k}_z>1.7$ (brown area). (c) Surface spectrum in the presence of a staggered chemical potential (55) with ${\mu }_s=0.05$. Linearly dispersing surface states (red traces) connect the projected Fermi rings in the surface BZ.

Figure 6

Surface band structure of the reflection-symmetric nodal superconductor (70) (class DIII with $R_{--})$ for the (001) face as a function of (a) surface momentum $k_y$ with $k_x=0$ and (b) surface momentum $k_x$ with $k_y=0$. A zero-energy arc surface state (red trace) connects the projected point nodes in the surface BZ. (c) Surface spectrum on the (001) face as a function of both $k_x$ and $k_y$. The surface states and bulk states are indicated in green and gray, respectively.