Dual topological insulator and insulator-semimetal transition in mirror-symmetric honeycomb materials

While the topological insulator and topological crystalline insulator phases rely on time-reversal symmetry and crystalline mirror symmetry, respectively, here, we predict that two-dimensional ${\mathrm{Na}}_2\mathrm{MgPb}$ and ${\mathrm{Na}}_2\mathrm{CdSn}$ can possess the properties of both of them, realizing dual topological insulators with nonzero ${\mathbb{Z}}_2$ invariant ${\mathbb{Z}}_2=1$ and mirror Chern number ${\mathcal{C}}_{\mathcal{M}}=-1$. Remarkably, the large band gap, as much as 0.58 eV for ${\mathrm{Na}}_2\mathrm{MgPb}$, and stable crystal structure indicate a high possibility for an experimental observation at room temperature. Moreover, we find that dual topological insulators can be turned into topological nodal-line semimetals by mimicking a magnetic field that keeps the crystalline mirror symmetry. In addition, we test and explore the emergence of such topological nodal-line semimetals on a model Hamiltonian and a realistic two-dimensional ferromagnet.

Figure 1

(a) Top and (b) side view of ${\mathrm{Na}}_{2}XY$ ($X=\text{Mg}$, Cd; $Y=\text{Pb}$, Sn) trilayers, where the unit cell is indicated by black dashed lines. (c) Main symmetry operations: threefold rotation ($R_3$), twofold rotation ($R_2$), mirror reflection ($M_z$). Phonon dispersion for (d) ${\mathrm{Na}}_2\mathrm{MgPb}$ and (e) ${\mathrm{Na}}_2\mathrm{CdSn}$, indicating that the structures are dynamically stable.

Figure 2

Orbitally resolved band structures for (a), (b) ${\mathrm{Na}}_2\mathrm{MgPb}$ and (c), (d) ${\mathrm{Na}}_2\mathrm{CdSn}$ trilayers (a), (c) without and (b), (d) with SOC, weighted with the $p_{x/y}$ and $p_z$ states of Pb/Sn. The Fermi level is indicated by the dashed line.

Figure 3

(a) Evolution of Wannier charge centers (WCCs) for ${\mathrm{Na}}_2\mathrm{MgPb}$ trilayer along $k_x$, confirming ${\mathbb{Z}}_2=1$. (b) WCCs for ${\mathrm{Na}}_2\mathrm{MgPb}$ trilayer in the mirror plane. All Bloch states can be distinguished by their eigenstates with respect to the mirror eigenvalues $+i$ and $-i$. The WCCs (open black circles) and their sum (solid red circles) for the $+i$ and $-i$ eigenstates are shown in (c) and (d), respectively, indicating the mirror Chern number ${\mathcal{C}}_M=-1$.

Figure 4

The Berry curvature distributions of all occupied bands over the 2D Brillouin zone associated with mirror eigenstates (a) $-i$ and (b) $+i$ for the ${\mathrm{Na}}_2\mathrm{MgPb}$ trilayer. The opposite sign of the Berry curvature for opposite $i$ results in a nonzero mirror Chern number. Energy dispersion of semi-infinite ribbons of (c) ${\mathrm{Na}}_2\mathrm{MgPb}$ and (d) ${\mathrm{Na}}_2\mathrm{CdSn}$, where the gapless edge states appear in the bulk band gap.

Figure 5

(a) Band evolution around the $\mathrm{\Gamma }$ point and (b) corresponding crossing points in the mirror plane for ${\mathrm{Na}}_2\mathrm{CdSn}$ under exchange fields of 0.00, 0.28, 0.33, and 0.40 eV, which are arranged from bottom to top accordingly. The band crossings can be effectively tuned by the exchange field and three configurations can be formed, i.e., two nodal lines, the coexistence of a nodal line and a nodal point, one nodal line. By using the model Hamiltonian, evolutions of band structures and crossing points can be repeated accurately as displayed in (c) and (d).

Figure 6

(a) Phonon dispersion of ${\mathrm{Na}}_2\mathrm{TiPb}$ trilayers. Side view of the corresponding unit cell is shown as an inset. (b) Spin-resolved band structures of ${\mathrm{Na}}_2\mathrm{TiPb}$ trilayers without SOC. Orange and violet lines represent the spin-up and spin-down bands, respectively. (c) Band structures with SOC. Band crossings form a closed nodal line around the $\mathrm{\Gamma }$ point. (d) Energy dispersion of a semi-infinite ribbon of ${\mathrm{Na}}_2\mathrm{TiPb}$.