Dual topological nodal line and nonsymmorphic Dirac semimetal in three dimensions

There are two types of previously known three-dimensional Dirac semimetals (DSs)—topological DSs and nonsymmorphic DSs. Here, we present a three-dimensional DS that exhibits features of both topological and nonsymmorphic DSs. We introduce a minimal tight-binding model for the space group 100 that describes a layered crystal made of two-dimensional planes in the $p4g$ wallpaper group. Using this model, we demonstrate that double glide mirrors allow a noncentrosymmetric three-dimensional DS that hosts both symmetry-enforced Dirac points at time-reversal invariant momenta and twofold-degenerate Weyl nodal lines on a glide-mirror-invariant plane in momentum space. The proposed DS allows for rich topological physics manifested in both topological surface states and topological phase diagrams, which we discuss in detail. We also perform first-principles calculations to predict that the proposed DS is realized in a set of existing materials $\mathrm{BaLa}X\mathrm{B}{Y}_5$, where $X=\text{Cu}$ or Au, and $Y=\text{O}$, S, or Se.

Figure 1

(a) Schematic illustration of a two-dimensional (2D) layer in the $p4g$ wallpaper group (left panel) and an infinite stack of the $p4g$ layers (right panel). The red (green) lines represent mirror (glide-mirror-) invariant lines. The center of the $C_{4z}$ rotational axis is designated by the $\otimes$ symbol. (b) Corresponding BZ in two (left) and three (right) dimensions. The locations of the twofold-degenerate WPs and fourfold-degenerate DPs are indicated by the Weyl and the double Weyl (Dirac) cones, respectively. The location of WNLs in the 3D BZ is indicated by green lines.

Figure 2

(a) Model lattice for SG 100. The glide planes are represented by dashed boxes. (b) Bulk tetragonal and surface rectangular BZs. (c) Electronic energy bands for SG 100, calculated from (1) with the parameter set $\{t_1,t_2,t_3,v_0,v_1,v_2,v_3,v_4\}=\{0.35,0,0,0,0.5,0.6,0.45\}$. The bands in the + and $-$ eigensectors of $g_y$ ($g_x$) are colored red and blue, respectively. The DP is indicated by a dashed (green) circle. (d) Bulk (gray) and slab (red) energy bands. Topological surface states emerge in the interior region of the projected nodal lines, where $\overline{X}$ and $\overline{R}$ are contained. (e) Schematic illustration of band inversion at $X$ ($R$). (f) Topological characterization of twofold-degenerate nodal lines in SG 100.

Figure 3

Topological phase diagram and topological phase transitions induced by symmetry-lowering perturbations from the DS in SG 100. (a) Topological phase diagram in the $(v_{-},m_s,m_{A_{2u}})$-parameter space in the unit of $t_1$. The DS in SG 100 resides along the $|v_{-}|$ axis (colored red), which is connected with a centrosymmetric DS at the origin, where $(v_{-},m_s,m_{A_{2u}})=(0,0,0)$ (indicated by a yellow circle). A WNL semimetal appears along the $|m_{A_{2u}}|$ axis (green line). The WS phases are present in the gray-colored regions on the $m_s-\left|m_{A_{2u}}\right|$ and $|v_{-}|-|m_{A_{2u}}|$ planes. The WS phase carrying both double ($\left|\mathcal{C}\right|=2$) and single ($\left|\mathcal{C}\right|=1$) WPs is distinguished from the WS phase carrying only double ($\left|\mathcal{C}\right|=2$) WPs with a thicker gray color on the $|v_{-}|-|m_{A_{2u}}|$ plane. A WTI phase is colored blue on the $m_s-\left|m_{A_{2u}}\right|$ plane. The remaining white area represents the trivial insulator phase. (b) Evolution of the WPs as a function of $m_{A_{2u}}$ during the topological phase transition occurring along the vertical yellow line in (a). The numbers near a trajectory indicate the corresponding Chern number of the WPs. A color scheme is used to indicate the magnitude of $m_{A_{2u}}$ at $m_{\mathrm{s}}=0$ and $v_{-}=1.0$. The WPs are annihilated at the red crosses. (c) Evolution of the WPs as a function of $m_s$ during the topological phase transition occurring along the horizontal yellow line in (a). A square (circle) represents a $\left|\mathcal{C}\right|=2$ ($\left|\mathcal{C}\right|=1$) WP.

Figure 4

Atomic and electronic structures of ${\mathrm{BaLaCuBO}}_5$. (a) Top view (top panel) and side view (bottom panel) of the atomic structure. (b) Electronic energy band structure. The unit cell is indicated by a solid box. The DFT (Wannier) energy bands are colored gray (blue). The Wannierzation exactly reproduces the four DFT bands near the Fermi level. (c) Magnified views of the red rectangles in (b) at $X$ (left panel) and at $M$ (right panel). The WNL and DP are indicated by a red circle.