Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery

We identify all symmetry-enforced band crossings in nonmagnetic orthorhombic crystals with and without spin-orbit coupling and discuss their topological properties. We find that orthorhombic crystals can host a large number of different band degeneracies, including movable Weyl and Dirac points with hourglass dispersions, fourfold double Weyl points, Weyl and Dirac nodal lines, almost movable nodal lines, nodal chains, and topological nodal planes. Interestingly, spin-orbit coupled materials in space groups 18, 36, 44, 45, and 46 can have band pairs with only two Weyl points in the entire Brillouin zone. This results in simpler connectivity of the Fermi arcs and more pronounced topological responses than in materials with four or more Weyl points. In addition, we show that the symmetries of space groups 56, 61, and 62 enforce nontrivial weak ${\mathbb{Z}}_2$ topology in materials with strong spin-orbit coupling, leading to helical surface states. With these classification results in hand, we perform extensive database searches for orthorhombic materials crystallizing in the relevant space groups. We find that ${\mathrm{Sr}}_2{\mathrm{Bi}}_3$ and ${\mathrm{Ir}}_2\mathrm{Si}$ have bands crossing the Fermi energy with a symmetry-enforced nontrivial ${\mathbb{Z}}_2$ invariant, CuIrB possesses nodal chains near the Fermi energy, ${\mathrm{Pd}}_7{\mathrm{Se}}_4$ and ${\mathrm{Ag}}_2\mathrm{Se}$ exhibit fourfold double Weyl points, the latter one even in the absence of spin-orbit coupling, whereas the fourfold degeneracies in AuTlSb are made up from intersecting nodal lines. For each of these examples, we compute the ab initio band structures, discuss their topologies, and for some cases also calculate the surface states.

Figure 1

Brillouin zones (BZs) for lattices with $a<b<c$ in the orthorhombic SGs. Of the several types in the face-centered case, the one for $\frac{1}{a^2}=\frac{1}{b^2}+\frac{1}{c^2}$ is shown. TRIMs are labeled in blue, rotation axes are shown in green and labeled by the points they are connecting.

Figure 2

(a) Distribution of ordered, orthorhombic phases in the Materials Project and ICSD by SG number (MP $\cap$ ICSD, $N$ = 12,292). 14 SGs of particular interest are indicated, as well as the number of phases in these SGs passing the automated screening criteria described in the text. The crystallographic point groups are indicated above the plot. (b) 2D histogram of phases passing the automated screening criteria by fraction of the projected DOS near the Fermi energy contributed by heavy atoms (see Ref. [49]) and number of atoms in the primitive unit cell ($N$ = 1259). We note that all the phases that passed the screening have an even number of atoms in the primitive cell, since all orthorhombic SGs with nonsymmorphic symmetries have even Wyckoff multiplicities. The specific example materials discussed in the text are highlighted by the stars.

Figure 3

(a) Scheme of an hourglass dispersion along a path between two degenerate points $K_1$ and $K_2$ along a twofold rotation axis or in a mirror plane. Colors indicate the sign of the symmetry eigenvalue of the twofold symmetry. (b) Band structure for ${\mathrm{Pd}}_7{\mathrm{Se}}_4$ in SG 18 with SOC showing hourglass states on $\mathrm{\Gamma }$-X (less clearly at $\mathrm{\Gamma }$-Y) colored in gray and fourfold double Weyl points at S, highlighted by the orange line.

Figure 4

Band crossings enforced by compatibility relations comprising three rotation axes. (a) Movable Weyl point in spinless band structures of SG 19 on one of the three rotation axes connecting to $\mathrm{\Gamma }$. Blue (orange) colored bands correspond to bands labeled $+$ ($-$) according to the definition in Eq. (4.3). (b) Possible arrangement of spinless bands in SG 24 with rotation eigenvalue $+1$ in blue and $-1$ in orange of the corresponding axis. The product of eigenvalues is fixed to $+1$ at $\mathrm{\Gamma }$ and $-1$ at Z, enforcing a movable Weyl point on one of the axes.

Figure 5

Dispersion of the fourfold double Weyl point in ${\mathrm{Pd}}_7{\mathrm{Se}}_4$ within the $k_z=0$ plane around $\mathrm{S}=(\pi ,\pi ,0)$. The colors indicate the two Weyl cones, related by the rotation $2_{100}$ or $2_{010}$. The twofold degenerate lines along $(k_x,\pi ,0)$ and $(\pi ,k_y,0)$ are part of nodal planes.

Figure 6

(a) Compatibility relations at R in SG 19 for spinful bands. Colors indicate the sign of the rotation eigenvalue according to Eq. (4.3). Note that on the paths shown all bands are twofold degenerate with identical rotation eigenvalues paired. The figure adapted from Ref. [21]. (b) Fourfold double Weyl point in ${\mathrm{Ag}}_2\mathrm{Se}$ without SOC at R (black). Including SOC (red) moves the fourfold double Weyl point to the axis R-T. In both cases, bands are twofold degenerate.

Figure 7

(a) Nodal chain in CuIrB formed by the 10th and 11th valence band counted from the band gap at the Fermi energy. The green line is within the $k_y=0$ plane and encircles Y, the red line is restricted to the other mirror plane $k_x=2\pi$ and winds around T. (b) Dispersion on the intersection of both planes, shown in turquoise in (a). Blue and orange indicate positive and negative signs of the mirror eigenvalues according to Eq. (5.1), the upper color belongs to $M_{100}$ and the lower to $M_{010}$.

Figure 8

Dispersion of the two different types of fourfold degenerate points in AuTlSb in the $k_y\text{−}{k}_z$-plane. Colors indicate the relative sign of $M_{100}$ eigenvalues. (a) Around S all lines are made of pairs $(+,-)$ and a pointlike degeneracy is formed. (b) At R, the nodal lines on R-$\mathrm{Z}=(\pi ,v,\pi )$ are paired $(+,+)$ and $(-,-)$, and an additional hourglass nodal line closes the gap, and thus the fourfold degenerate point is not a point crossing but part of several twofold degenerate nodal lines.

Figure 9

Possible arrangement of bands along three rotation axes in (a) SG 61 in the spinless case for the TRIMs without brackets and the spinful case for the TRIMs in brackets, (b) SG 73 in the spinless case. Blue and orange mark the sign of both mirror eigenvalues on each axis according to Eq. (5.1).

Figure 10

(a) Bulk bands of ${\mathrm{Sr}}_2{\mathrm{Bi}}_3$ in the $k_z=\pi$ plane with the sign of the product of inversion eigenvalues for each set of band at all TRIMs in the plane. (b) Surface density of states on the (100) surface along a high-symmetry path in the surface BZ. The nontrivial plane maps on the line Z-T, see inset in (a), with symmetry enforced surface states crossing the projected bulk gap.

Figure 11

(a) Bulk bands of ${\mathrm{Ir}}_2\mathrm{Si}$ in the $k_z=\pi$ plane with the sign of the product of of inversion eigenvalues for each set of band at all TRIMs in the plane. (b) Surface density of states on the (100)-surface, where the nontrivial plane maps on the line Z-T, see inset in (a). A pair of time-reversal symmetric surface states connect the bands below the projected bulk gap to the upper ones.

Figure 12

DFT band structures including SOC along the high-symmetry paths for all examples discussed in the main text. The positions of symmetry-enforced fourfold degenerate points are highlighted in orange and correspond to fourfold double Weyl points, fourfold degenerate points with $\mathcal{C}=0$ and Dirac points in the first, second and third row, respectively. The movable fourfold double Weyl points in (b) are highlighted with arrows. Gray areas highlight connected bands in path segments with an hourglass dispersion. Green areas indicate the set of connected bands in a plane with nontrivial ${\mathbb{Z}}_2$-invariant.

Figure 13

High-symmetry path for CuIrB in SG 43 used in Fig. 12.

Figure 14

Elementary bands representations in SG 19 without SOC (blue) and with strong SOC (orange). The upper bands are a schematic band structure with minimal number of crossings, based on compatibility relations alone. The lower set of bands are from the minimal tight-binding model defined in Eq. (C1) with hopping parameters $t_0=-t_5=0.25$, $t_1=-t_4=0.55$, $t_2=-0.6$, $t_3=-0.95$, and $l_i=0$ for vanishing SOC. For the case with SOC $l_0=l_1=l_7=0$, $l_2=-l_5=0$, $l_3=0.7$, $l_4=-l_8=0.2$, $l_5=-0.3$, $l_9=l_{10}=-l_{11}=0.1$. Parameters were chosen for maximal visibility of the features mentioned in the text.