Topological semimetals protected by off-centered symmetries in nonsymmorphic crystals

Topological semimetals have energy bands near the Fermi energy sticking together at isolated points/lines/planes in the momentum space, which are often accompanied by stable surface states and intriguing bulk topological responses. Although it has been known that certain crystalline symmetries play an important role in protecting band degeneracy, a general recipe for stabilizing the degeneracy, especially in the presence of spin-orbit coupling, is still lacking. Here we show that a class of novel topological semimetals with point/line nodes can emerge in the presence of an off-centered rotation/mirror symmetry whose symmetry line/plane is displaced from the center of other symmorphic symmetries in nonsymmorphic crystals. Due to the partial translation perpendicular to the rotation axis/mirror plane, an off-centered rotation/mirror symmetry always forces two energy bands to stick together and form a doublet pair in the relevant invariant line/plane in momentum space. Such a doublet pair provides a basic building block for emerging topological semimetals with point/line nodes in systems with strong spin-orbit coupling.

Figure 1

3D Dirac points protected by an off-centered twofold rotation ${\tilde{C}}_{2z}^{\perp }$. (a) A schematic figure describing the distribution of 3D Dirac points in momentum space. The location of four 3D Dirac points is marked in red dots. The bold blue arrow indicates the axis for ${\tilde{C}}_{2z}^{\perp }$ symmetry. The pink square indicates the plane of the glide mirror symmetry ${\tilde{M}}_z^{\parallel }$, which is dual to ${\tilde{C}}_{2z}^{\perp }$. (b) The band structure along the $U-X-U$ line on which $\{P,{\tilde{C}}_{2z}^{\perp }\}=0$. A pair of degenerate bands (a doublet pair) form a 3D Dirac point at each time-reversal invariant momentum (TRIM). (c) The band structure when two doublet pairs cross on the $U-X-U$ line. There are in total $4n$ ($n$ is an integer) 3D Dirac points, which are all away from TRIMs. In (b) and (c), the integers in the bottom indicate the number $N\left(k_z\right)$ defined in Eq. (B2) from which the topological charge $Q$ can be found by using Eq. (B1).

Figure 2

3D Dirac lines protected by an off-centered mirror symmetry ${\tilde{M}}_x^{\perp }$. (a) A schematic figure describing the distribution of nodal lines in momentum space. The location of two line nodes in the $k_x=\pi$ plane is marked in red color. The blue square indicates the plane of the off-centered mirror symmetry ${\tilde{M}}_x^{\perp }$. The bold pink arrow indicates the axis for the screw rotation ${\tilde{C}}_{2x}^{\parallel }$ symmetry, which is dual to ${\tilde{M}}_x^{\perp }$ symmetry. (b) Distribution of the integer $N(\pi ,k_y,k_z)$ defined in Eq. (B9) in the $k_x=\pi$ plane from which the topological charge $Q^{\prime }$ of the line node can be computed by using Eq. (B8). The corresponding band structure along the $S-X-S$ line is shown in the bottom. The doublet pair are degenerate at each TRIM, which is a part of line nodes in the $k_x=\pi$ plane. (c) Distribution of the integer $N(\pi ,k_y,k_z)$, when two doublet pairs cross in the $k_x=\pi$ plane. There are in total $4n$ ($n$ is an integer) nodal lines, which are away from TRIM. The corresponding band structure along the $S-X-S$ line is shown in the bottom.

Figure 3

Construction of 3D lattice models by stacking 2D layers. (a) A schematic figure describing a 3D lattice model obtained by vertical stacking of 2D square lattices. (b) Structure of a 2D layer where the B site in a unit cell is shifted along the $z$ direction, thus the whole system has nonsymmorphic symmetries. (c) An additional shifting of B sites along the $y$ direction, which breaks the symmetry ${\tilde{C}}_{2z}^{\perp }$ (or equivalently, ${\tilde{M}}_z^{\parallel }$). (d) An additional distortion of a unit cell, which breaks the symmetry ${\tilde{M}}_x^{\perp }$ and ${\tilde{M}}_y^{\perp }$ (or equivalently, ${\tilde{C}}_{2x}^{\parallel }$ and ${\tilde{C}}_{2y}^{\parallel }$).

Figure 4

Band structure of a semimetal with Dirac line nodes protected by ${\tilde{M}}_x^{\perp }=\left\{M_x\right|\frac{1}{2}\widehat{x}\}$ symmetry. (a) From four-band lattice models with $P, T$, and ${\tilde{M}}_x^{\perp }$ symmetries. Here eigenstates are doubly degenerate at each momentum. Doublet pairs on the $k_x=\pi$ plane form two Dirac line nodes, each passes two TRIMs. The bold red lines in the left panel mark the points in the BZ where two bands stick together. (b) Similar plots from an eight-band lattice model. Crossing points between two doublet pairs form four Dirac line nodes on the $k_x=\pi$ plane, which correspond to those marked by red circles in Fig. 2. In the figures, $U^{\prime }$ and $S^{\prime }$ indicate the momenta equivalent to $U$ and $S$, respectively.

Figure 5

Band structure of a semimetal with Dirac point nodes protected by ${\tilde{C}}_{2z}^{\perp }=\left\{C_{2z}\right|\frac{1}{2}\widehat{x}+\frac{1}{2}\widehat{y}\}$ symmetry. (a) From four-band lattice models with $P, T$, and ${\tilde{C}}_{2z}^{\perp }$ symmetries. Doublet pairs on the $\mathit{k}=(\pi ,0,k_z)$ and $\mathit{k}=(0,\pi ,k_z)$ lines with $k_z\in [-\pi ,\pi ]$, form Dirac point nodes at every TRIM. A bold red dot on the left marks the points where two bands stick together. (b) Similar plots from an eight-band lattice model. Crossing points between two doublet pairs form four Dirac points away from TRIM in both $\mathit{k}=(\pi ,0,k_z)$ and $\mathit{k}=(0,\pi ,k_z)$ lines with $k_z\in [-\pi ,\pi ]$, which correspond to those marked by red circles in Fig. 1. In the figures, $U^{\prime }$ and $S^{\prime }$ indicate the momenta equivalent to $U$ and $S$, respectively.

Figure 6

Magnetic field induced transition of Dirac points into Weyl points. The bulk band structure of the four-band lattice model with $P, T$, and ${\tilde{C}}_{2z}^{\perp }$ symmetries is shown in the lower panels after projection onto the surface BZ of a slab structure with a finite length $L_x$ along the $x$ direction. The states localized on the $x=0$ ($x=L_x$) surface are indicated by red (green) lines, respectively. (a) In the absence of magnetic field. There are four Dirac points at TRIMs ($X, U, Y$, and $T$) at $\mathit{k}=(\pi ,0,0), (\pi ,0,\pi ), (0,\pi ,0), (0,\pi ,\pi )$, respectively. There are nontopological surface states on both surfaces which can be merged into bulk states through smooth deformation. (b) In the presence of magnetic field along the $z$ direction ($h_z=0.2$). A Dirac point with fourfold degeneracy splits into two Weyl points, each with twofold degeneracy. Splitting of a Dirac point along the $k_z$ direction accompanies emergent surface states (Fermi arcs) connecting two split Weyl points, which are marked with red dotted circles. In the figures, $U^{\prime }$ and $S^{\prime }$ indicate the momenta equivalent to $U$ and $S$, respectively.

Figure 7

Magnetic field induced transition of Dirac line nodes into Weyl line nodes and three-dimensional quantum Hall insulators. The bulk band structure of the four-band lattice model with $P, T$, and ${\tilde{M}}_x^{\perp }$ symmetries is shown in the right panels after projection onto the surface Brillouin zone of a slab structure with a finite length $L_x$ along the $x$ direction. The states localized on the $x=0$ ($x=L_x$) surface are indicated by red (green) lines, respectively. (a) In the absence of magnetic field. There are two Dirac line nodes on the $k_x=\pi$ plane. There are some nontopological surface states which can be merged with bulk states through smooth deformation. (b) In the presence of magnetic field along the $x$ direction ($h_x=0.3$). A Dirac line node with fourfold degeneracy splits into two Weyl line nodes, each with twofold degeneracy. Splitting of a Dirac line node accompanies emergent surface states connecting two split Weyl line nodes, which are marked with red dotted circles. (c) In the presence of magnetic field along the $z$ direction ($h_z=0.7$). A Dirac line node is fully gapped, and two-dimensional chiral surface states emerge, which are marked with red dotted circles. In the figures, $U^{\prime }$ and $S^{\prime }$ indicate the momenta equivalent to $U$ and $S$, respectively.

Figure 8

Magnetic field induced transition of Dirac line nodes into Weyl line nodes in systems with both ${\tilde{M}}_x^{\perp }$ and ${\tilde{C}}_{2z}^{\perp }$. (a) In the absence of magnetic field. There are three open straight Dirac line nodes along $\mathit{k}=(\pi ,0,k_z), (0,\pi ,k_z)$ and $(\pi ,\pi ,k_z)$ lines with $k_z\in [-\pi ,\pi ]$ due to the simultaneous presence of ${\tilde{M}}_x^{\perp }$ and ${\tilde{M}}_y^{\perp }$. (b) The bulk band structure of the four-band lattice model with $P, T, {\tilde{C}}_{2z}^{\perp }$, and ${\tilde{M}}_x^{\perp }$ symmetries. (c) The band structure of the slab structure with a finite length along the $x$ direction are shown. The states localized on the $x=0$ ($x=L_x$) surface are indicated by red (green) lines, respectively. (d)–(f) Similar plots in the presence of magnetic field along the $x$ direction ($h_x=0.3$). A Dirac line node with fourfold degeneracy splits into two Weyl line nodes, each with twofold degeneracy. Splitting of a Dirac line node accompanies emergent surface states connecting two split Weyl line nodes, which are marked with dotted circles. In the figures, $U^{\prime }$ and $S^{\prime }$ indicate the momenta equivalent to $U$ and $S$, respectively.

Figure 9

Dirac points/lines protected by off-centered screw/glide symmetries. (a)–(c) Dirac points protected by off-centered twofold screw rotation ${\tilde{C}}_{2z}^{\parallel ,\perp }=\left\{C_{2z}\right|\frac{1}{2}\widehat{x}+\frac{1}{2}\widehat{z}\}$. (a) A schematic figure describing the distribution of 3D Dirac points in momentum space. The location of two 3D Dirac points is marked in red dots. Here high-symmetry momenta are labeled as in Ref. [33]. (b) The band structure along the $D-B-D$ line on which $\{P,{\tilde{C}}_{2z}^{\parallel ,\perp }\}=0$. A pair of degenerate bands (a doublet pair) form a single 3D Dirac point at the time-reversal invariant momentum (TRIM) with $k_z=0$. (c) The band structure when two doublet pairs cross on the $D-B-D$ line. There are in total $2n$ ($n$ is an integer) 3D Dirac points, which are all away from TRIMs. (d)–(f) Dirac lines protected by off-centered glide mirror ${\tilde{M}}_z^{\parallel ,\perp }=\left\{M_z\right|\frac{1}{2}\widehat{x}+\frac{1}{2}\widehat{z}\}$. (d) A schematic figure describing the location of the nodal line in momentum space. The location of the line node in the $k_z=\pi$ plane is marked in red color. (e) Shape of a nodal line in the $k_z=\pi$ plane in which $\{P,{\tilde{M}}_z^{\parallel ,\perp }\}=0$ is satisfied. The corresponding band structure along the $D-Z-D$ line is shown in the bottom. The doublet pair are degenerate at two TRIMs ($C$ and $Z$ points), which are a part of line nodes in the $k_z=\pi$ plane. (f) Shape of nodal lines when two doublet pairs cross in the $k_z=\pi$ plane. The corresponding band structure along the $D-Z-D$ line is shown in the bottom.