Topological charge and bulk-surface correspondence for quad-helicoid surface states in topological semimetals with two glide-time-reversal symmetries

TL;DR

This work addresses quad-helicoid surface states in topological semimetals protected by two glide-time-reversal symmetries. It introduces a local $Z_2\times Z_2$ monopole charge for GT-invariant high-symmetry points and a global $Z_2$ charge on a GT-enclosing torus, then establishes a bulk-surface correspondence linking the global charge to the presence of quad-helicoid surface states on the (001) surface. The authors provide simplified charge formulas under additional symmetries, discuss filling-enforced conditions, and support the theory with a tight-binding model that exhibits QHSSs and persists under symmetry-preserving perturbations. The results extend QHSS physics beyond spinless, time-reversal–protected settings to broader GT-symmetric semimetals, with implications for material realizations and experimental detection.

Abstract

Quad-helicoid surface states (QHSSs) are unique surface states with two pairs of helicoid surface states in topological semimetals such as Dirac semimetals. So far, topologically protected QHSSs are shown to appear in spinless systems with two $\mathcal{GT}$ symmetries and $\mathcal{T}$ symmetry ($\mathcal{G}$: glide, $\mathcal{T}$: time-reversal). In this paper, we show that topologically protected QHSSs also appear in spinful/spinless systems with only two $\mathcal{GT}$ symmetries by defining new topological charges and establishing the bulk-surface correspondence. We first define a local $Z_2\times Z_2$ monopole charge for gapless nodes at $\mathcal{GT}$-invariant high-symmetry points and a global $Z_2$ charge reflecting the global topological feature of $\mathcal{GT}$-symmetric topological semimetals. Next, we show that the latter $Z_2$ classification corresponds to the presence or absence of QHSSs on the surface with two $\mathcal{GT}$ symmetries. In addition, we provide simplified formulas of the $Z_2$ charge under additional symmetries, and clarify some symmetry conditions where QHSSs are filling-enforced.

Figures (8)

• Figure 1: Quad-helicoid surface states. (a) Brillouin zone for MSG #45.236 ($Iba21'$). We assume that Dirac points exist at high-symmetry points $P$ and $P'$. (b) Quad-helicoid surface states. They can be seen as the superposition of two helicoid surface states and two anti-helicoid surface states.
• Figure 2: Topological charges for quad-helicoid surface states. (a) Brillouin zone for MSG #32.138 ($Pb'a'2$). $R$ and $S$ are high-symmetry points with $\tilde{\Theta}_x^2=\tilde{\Theta}_y^2=-1$. (b) Sphere $S^2$ used to define $Z_2\times Z_2$ charge $\mathcal{Q}_{xy}[S^2]=(\mathcal{Q}_{xy}^{(0)}[S^2], \mathcal{Q}_{xy}^{(1)}[S^2])$ for $S$. $K_0, K_2$ are $\tilde{\Theta}_x$-invariant points on $S^2$, $K_1, K_3$ are $\tilde{\Theta}_y$-invariant points on $S^2$, and $Q, Q'$ are $C_{2z}$-invariant points on $S^2$. (c) Torus $T$ used to define $Z_2$ charge $\mathcal{Q}_{xy}[T]$. $K_0, K_0'$ are $\tilde{\Theta}_x$-invariant points on $T$, and $K_1, K_1'$ are $\tilde{\Theta}_y$-invariant points on $T$.
• Figure 3: Example of gapless nodes with the nontrivial topological charge $\mathcal{Q}_{xy}[S^2]=(1,1)$. (a) Two Dirac points on the line $k_y=k_z=\pi$ related by $C_{2z}$ symmetry. (b) Two double Weyl points on the line $k_x=k_y=\pi$.
• Figure 4: Bulk-surface correspondence for quad-helicoid surface states. (a1) Brillouin zone for MSG #32.138 ($Pb'a'2$). The torus $T$ is projected onto the circle enclosing the point $(\pi,\pi)$ in the (001) surface Brillouin zone. (a2) Quad-helicoid surface states. They are obtained by considering the surface spectrum on the projection of the torus $T$ (yellow circle) with arbitrary values of the radius $r>0$. (b1) $(k,t)$-parameter space for the torus $T$. We can recognize this space as the 2D Brillouin zone for the 2D system $H(t,k)$ obtained by restricting the original 3D system to the torus. (b2) Edge spectrum of $H(t,k)$. When $\mathcal{Q}_{xy}[T]$ has a nontrivial value, then unique edge states appear, which can be considered as a section of QHSSs. (c1) 2D Brillouin zone for the 2D system $\tilde{H}(s,k)$. $\tilde{H}(s,k)$ is constructed from $H(t,k)$ such that $H(t,k)$ with $0\le t\le\pi/2$ and $\tilde{H}(s,k)$ with $0\le s\le\pi$ have the same bulk/edge spectra by identifying $s=2t$. (c2) Edge spectrum of $\tilde{H}(s,k)$. When $\mathcal{Q}_{xy}[T]$ has a nontrivial value, $\tilde{H}(s,k)$ can be seen as a $Z_2$ topological insulator, and thus helical edge states appear on the edge spectrum.
• Figure 5: Band structures of $H_{\mathrm{TBM}}(\bm{k})$ and $H'_{\mathrm{TBM}}(\bm{k})$. (a) Brillouin zone for MSG #73.551 ($Ib'c'a$). (b) Band structures of $H_{\mathrm{TBM}}(\bm{k})$. (b1) Bulk band structure. There is a Dirac point at $W:(\pi,\pi,\pi)$. Since the model has $\mathcal{P}$ symmetry, there is another Dirac point at $WA:(\pi,\pi,-\pi)$. (b2) (001) surface band structure on the blue line $(k_x,\pi)\ \ (0\le k_x\le 2\pi)$. QHSSs exist around the projection of Dirac points. (b3) (001) surface band structure on the green circle $(\pi+0.5\cos t,\pi+0.5\sin t)\ \ (0\le t\le 2\pi)$. The bands intersect the Fermi energy four times. This situation characterizes QHSSs. (c) Band structures of $H'_{\mathrm{TBM}}(\bm{k})=H_{\mathrm{TBM}}(\bm{k})+\Delta(\bm{k})$. (c1) Bulk band structure. Dirac points of $H_{\mathrm{TBM}}(\bm{k})$ split into pairs of two Weyl points. (c2) (001) surface band structure on the blue line $(k_x,\pi)\ \ (0\le k_x\le 2\pi)$. QHSSs exist around the projection of the pairs of Weyl points. (b3) (001) surface band structure on the green circle $(\pi+0.5\cos t,\pi+0.5\sin t)\ \ (0\le t\le 2\pi)$. Even after we add a perturbation, the bands intersect the Fermi energy four times.