Dual topology in jacutingaite ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$

Topological phases of electronic systems often coexist in a material, well-known examples being systems which are both strong and weak topological insulators. More recently, a number of materials have been found to have the topological structure of both a weak topological phase and a mirror-protected topological crystalline phase. In this work, we first focus on the naturally occurring mineral called jacutingaite, ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$, and show based on density-functional calculations that it realizes this dual topological phase and that the same conclusion holds for ${\mathrm{Pd}}_2{\mathrm{HgSe}}_3$. Second, we introduce tight-binding models that capture the essential topological properties of this dual topological phase in materials with threefold rotation symmetry and use these models to describe the main features of the surface spectral density of different materials in the class.

Figure 1

(a) Crystal structure of ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$. The three mirror planes are shown as shaded areas (top) and as green lines (bottom). (b) Bulk and surface Brillouin zones (BZ). Time-reversal invariant points are colored blue and points belonging to the surface BZ are indicated with an overline. (c) Band structure and density of states $D\left(\varepsilon \right)$ of ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$ with (black) and without (red) spin-orbit coupling.

Figure 2

Momentum-resolved surface spectral densities. (a) [100] surface. Dirac cones associated with the weak topological index are observed at $\overline{X}$ and at $\overline{R}$. The insets show a zoom in on the Dirac cones. (b) [001] surface. A pair of Dirac cones associated with the mirror Chern number are observed at $\overline{X}$.

Figure 3

(a) Triangular lattice model of $H_{\mathrm{TI}}$, Eqs. (2) and (4). Sites are shown in black, hoppings in gray, and mirror planes as shaded areas. The blue arrows represent Bravais vectors ${\mathbf{a}}_{1,2,3}$. (b) Band structure of the model defined by Eqs. (8) and (4) computed in a slab geometry. Only states on the top surface are shown. (c) Cut of the same band structure along the $k_1=k_2$ mirror plane, with $k_m$ labeling the momentum along the mirror plane. (d) Band structure for model defined by Eqs. (8) and (5), along the same $k_1=k_2$ mirror plane. All band structures are obtained in a slab geometry with hard wall boundary conditions perpendicular to ${\mathbf{a}}_3$ and a thickness of 40 unit cells. In (b) and (c), we set $\mu =3,\lambda =1,\alpha =5$, and $\varepsilon =0.4$, whereas $\varepsilon =0.1$ in (d). The color scale in (c) and (d) denotes the integrated probability density of wave functions on the top- and bottom-most eight sites of the slab, such that surface modes appear in red (dark gray) and bulk modes in light green (light gray).

Figure 4

Band structure of ${\mathrm{Pd}}_2{\mathrm{HgSe}}_3$ and of ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$. The inset presents a zoom of the area enclosed by the red dotted-line square.

Figure 5

Momentum-resolved surface spectral densities of ${\mathrm{Pd}}_2{\mathrm{HgSe}}_3$. (a) [100] surface. (b) [001] surface. A pair of Dirac cones associated with the mirror Chern number are observed at $\overline{X}$.

Figure 6

Band structure of the model obtained using Eq. (B1) in an infinite slab geometry, with thickness of 40 unit cells in the $z$ direction, using $\mu =3$ and $\lambda =1$. Only states which are localized on the top surface and have energies $\left|E\right|\le 1$ are shown. To better visualize the surface Dirac cones, we set $\alpha =\varepsilon =0$, such that each Dirac cone is doubly degenerate. The hexagonal contour marks the boundary of the surface BZ, mirror invariant lines are shown in dashed red, and black arrows indicate the reciprocal lattice vectors of the surface ${\mathbf{b}}_1=(\sqrt{3}/2,1/2)$ and ${\mathbf{b}}_2=(0,-1)$.

Figure 7

Sketch of the real-space hopping terms corresponding to the function $f^{\left(X\right)}$ of Eq. (B1). Shown is the triangular lattice describing $H_{\mathrm{TI}}$ at constant $z$ coordinate. Starting from the site denoted by a black circle, the hopping amplitudes are $t_1=-3/8$ (blue dotted lines), $t_2=3/16$ (orange dashed lines), and $t_3=3/32$ (green solid lines). The structure of the hoppings preserves the three-fold rotation symmetry of the model.