Interacting weak topological insulators and their transition to Dirac semimetal phases

Topological insulators in the presence of a strong Coulomb interaction constitute novel phases of matter. Transitions between these phases can be driven by single-particle or many-body effects. On the basis of ab initio calculations, we identify a concrete material, i.e., ${\mathrm{Ca}}_2{\mathrm{PtO}}_4$, that turns out to be a hole-doped weak topological insulator. Interestingly, the Pt $d$ orbitals in this material are relevant for the band inversion that gives rise to the topological phase. Therefore, Coulomb interactions should be of importance in ${\mathrm{Ca}}_2{\mathrm{PtO}}_4$. To study the influence of interactions on the weak topological insulating phase, we look at a toy model corresponding to a layer-stacked three-dimensional version of the Bernevig-Hughes-Zhang model with local interactions. For a low to intermediate interaction strength, we discover novel interaction-driven topological phase transitions between the weak topological insulator and two Dirac semimetal phases. The latter correspond to gapless topological phases. For strong interactions, the system eventually becomes a Mott insulator.

Figure 1

Electronic structure of ${\mathrm{Ca}}_2{\mathrm{PtO}}_4$ along the high-symmetry direction $\mathrm{N}(\frac{1}{2}$,0,0)-$\mathrm{\Gamma }$(0,0,0)-N(0,$\frac{1}{2}$,0)-$\mathrm{X}(\frac{1}{2},\frac{1}{2}$,0)-$\mathrm{Z}(\frac{1}{2},\frac{1}{2},\frac{1}{2}$)-$\mathrm{N}(\frac{1}{2}$,0,$\frac{1}{2}$)-X(0,0,$\frac{1}{2}$)-N(0,$\frac{1}{2},\frac{1}{2}$). The shaded (light-yellow) region shows the topological band gap, which contains two inversions at $\mathrm{\Gamma }$ and $Z$ between the bands with mainly O-$p_z$ (red circles) and those with mainly Pt-$d_{xy}$ (green circles) characters.

Figure 2

(a) The presence of spin-orbital coupling induces a band inversion between O-$p_z$ (red line) and Pt-$d_{xy}$ (green line) bands, which characterizes the nontrivial topology of this system. (b) A topological invariant can be defined for the topological gap shown in the shaded (light-yellow) region in Fig. 1. It is found that this gap possesses a weak topological nature. The $\pm$ sign associated with each high-symmetry point is the parity product for the bands below the topological gap at this TRIM. In the coordinate system defined in Fig. 3, the topological invariant is given as $0;\left(111\right)$.

Figure 3

(a) The topological surface states show two Dirac cones at the (001) surface and no Dirac cone at the ($11\overline{1}$) surface, characterizing the weak topological nature of ${\mathrm{Ca}}_2{\mathrm{PtO}}_4$. (b) The number of Dirac cones for a given surface can be determined from the relative sign of the projected pairs of the bulk TRIMs.

Figure 4

(a) Bulk electronic structure of the weak TI model defined in Eq. (1) along a high-symmetry path of a cubic lattice with the BZ plotted in (b). Two band inversions (presented as the interchange of color), at $\mathrm{\Gamma }$ and $Z$, give rise to topologically distinct surfaces; i.e., at the (100) surface (c), there are two Dirac cones, at surface BZ momentum points $\overline{\mathrm{\Gamma }}$ and ${\overline{X}}^{\prime }$, and at the (001) surface (d) there is no Dirac cone.

Figure 5

Depending on the strength of $M$, the 3D topological model in Eq. (1) displays four topologically distinct phases, characterized by a different number of band inversions as well as the presence (WTI and band insulator) or the absence (DSM) of a bulk gap.

Figure 6

Bulk band dispersion and TSS on the (100) surface for the two DSM phases. (a, b) When $t_z<M<(2-t_z)$, there is a bulk-band linear crossing between $\mathrm{\Gamma }$ and $Z$. As a result, there is a flat band at the Fermi level in this plot between $\overline{\mathrm{\Gamma }}$ and ${\overline{X}}^{\prime }$ connecting the Dirac cones from the surface and bulk, respectively. (c, d) When $M<t_z$, the linear crossing is between $A$ and $X$. The flat band connects the bulk Dirac point at $A$ and the topological surface Dirac cone.

Figure 7

The occupancy of orbitals changes from $\langle n\rangle =2$ (fully occupied)/$\langle n\rangle =0$ (empty) to $\langle n\rangle =1$ (equally occupied) with the increase in interaction, which indicates the band-to-Mott-insulator transition.

Figure 8

Interacting phase diagram of the 3D topological model based on Eqs. (1) and (7). Five phases are identified. At sufficiently large $M$ and $U$, the conventional band insulator and the Mott insulator phases dominate. In between, the competition between Coulomb interaction and topological order gives rise to interesting phases such as a region of a weak TI phase and two regions of DSM phases. The topological surface states with parameters given by the corresponding colored symbols are shown in Fig. 9. As we carefully explain there, the phase boundaries are subject to an approximation for large values of $U$.

Figure 9

Left: Topological states at the (100) surface with (a) $U=4.0,M=3.0$; (b) $U=7.0,M=3.0$; (c) $U=8.0,M=3.0$; and (d) $U=4.0,M=1.0$. (a–c) The topological phase transitions are driven by the increase in $U$. Right: Same as the left plot, but for the (001) surface. At the two DSM phases, the bulk gaps are closed, thus the topological surface states at these two phases are embedded in the bulk bands.