Fragile Topology and Wannier Obstructions

Topological phases, such as Chern insulators, are defined in terms of additive indices that are stable against the addition of trivial degrees of freedom. Such topology presents an obstruction to any Wannier representation, namely, the representation of the electronic states in terms of symmetric, exponentially localized Wannier functions. Here, we address the converse question: Do obstructions to Wannier representation imply stable band topology? We answer this in the negative, pointing out that some bands can also display a distinct type of “fragile topology.” Bands with fragile topology do not admit any Wannier representation by themselves, but such a representation becomes possible once certain additional trivial degrees of freedom are supplied. We construct a physical model of fragile topology on the honeycomb lattice that also helps resolve a recent puzzle in band theory. This model provides a counterexample to the assumption that splitting of an “elementary band representation” introduced in [B. Bradlyn et al., Topological quantum chemistry, Nature (London) 547, 298 (2017)] leads to bands that are individually topological. Instead, half of the split bands of our model realize a trivial band with exponentially localized symmetric Wannier functions, whereas the second half possess fragile topology. Our work highlights an important and previously overlooked connection between band structure and Wannier functions, and is expected to have far-reaching consequences given the central role played by Wannier functions in the modeling of real materials.

Figure 1

Model for fragile topology. [(a) and (b)] Construction of model with fragile topology. (a) The nontrivial ${\mathbb{Z}}_2$ index of the Kane-Mele model [34, 35] can be diagnosed from the inversion eigenvalues of the bands [36], as we labeled for $\mathrm{\Gamma }$. The role of spin-orbit coupling is to open a gap. (b) A further band inversion at $\mathrm{\Gamma }$ trivializes the ${\mathbb{Z}}_2$ index. (c) Terms in the Hamiltonian ${\widehat{H}}_0$. (d) Band structure of the honeycomb Hamiltonian, with a clear band gap at a filling of two electrons per unit cell. (e) A symmetric Wannier function $w_{\uparrow }(\mathit{r})$ centered at the origin. $a$ denotes the lattice constant. The other Wannier function, also centered at the origin, can be obtained by applying time reversal. We visualize it by an arrow and a circle attached to every site at which $|w_{\uparrow }(\mathit{r}){|}^2>0.3\times {10}^{-2}$. Red (blue) filled circles indicates that, locally, the spin is tilting up (down), and their sizes represent the relative weights of the Wannier function. The arrow shows the in-plane component of the spin. (f) $|w_{\uparrow }(\mathit{r}){|}^2$ decays exponentially at large distances (red dashed line).

Figure 2

Fragile topology and decomposable elementary band representations. (a) In a $K$-theory-based classification [1, 9, 10, 11, 12, 13], it could be that two band insulators $a$ and $b$ are not smoothly connected, but the obstruction is resolved once we append to both sides an additional set of bands $c$. We say $a$ and $b$ are stably equivalent, denoted by $a{\sim }_{s}b$. (b) Using similar ideas, we introduce the notion of fragile topology. We say a set of bands possesses fragile topology if there is a topological obstruction in deforming it to any trivial (atomic) limit, but the obstruction is resolved once we allow for the introduction of additional trivial d.o.f. (c) In contrast, bands with stable topology remain nontrivial upon the stacking of any trivial bands. (d) Up to an overall sign change of the tight-binding Hamiltonian, there are generally six distinct splitting patterns for a decomposable elementary band representation. Three cases are ruled out by definitions, indicated by crosses. Our model ${\widehat{H}}_0$ shows that case (v), the only splitting pattern with trivial valence bands, is possible.