Multicellularity of Delicate Topological Insulators

Being Wannierizable is not the end of the story for topological insulators. We introduce a family of topological insulators that would be considered trivial in the paradigm set by the tenfold way, topological quantum chemistry, and the method of symmetry-based indicators. Despite having a symmetric, exponentially localized Wannier representation, each Wannier function cannot be completely localized to a single primitive unit cell in the bulk. Such multicellular topology is shown to be neither stable nor fragile, but delicate; i.e., the topology can be nullified by adding trivial bands to either valence or conduction band.

Figure 1

Topological stability of (a) fragile vs (b) delicate topology. The blocks [colored according to legend in (c)] below (above) the Fermi energy (dashed blue lines) are valence (conduction) bands, with addition indicated by stacking and topological equivalence by “$\sim$.” In panel (a) the complement to fragile topological bands could be fragile topological, obstructed atomic limit, or fully trivial.

Figure 2

Characterization of (a)–(d) bulk and (e),(f) surface Wannier functions for the delicate topological insulator modeled by Eq. (1) with $m=-6,\chi =-1$. (a) illustrates the exponential decay parallel (orthogonal) to the rotation axis, as indicated by $R_z$ ($R_x$). (b),(e),(f) Rotation-symmetric probability distribution—projected onto the rotation-invariant plane, utilizing the color bar in (g). (c),(d) The total polarization in $z$ direction is nonanalytic at topological-phase transitions indicated by dashed lines.

Figure 3

(a) The polarization and Hopf invariants as a function of $m\in \mathbb{R}$ [cf. Eq. (1)]; colors distinguish distinct phases. (b) For each phase, the polarization along $M\mathrm{\Gamma }K$ is plotted for a representative value of $m$. (c) Rotation-invariant points in reduced Brillouin zone.

Figure 4

Spectrum of (a) Hopf-insulating and (b) RTP-insulating slab with one, respectively, two detached surface bands. Red, blue, green coloring represents contribution to the bands from basis orbitals with $\ell =0$, 1, 2. (c) For various bands discussed in the main text, ${\ell }_{\mathrm{\Gamma }},{\ell }_K,{\ell }_{K^{\prime }}$ denote the mod-three angular momenta at $C_3$-invariant wave vectors.