Fragile Topology and Wannier Obstructions

September 21, 2018

Fig. 1(e). A symmetric Wannier function  centered at the origin.

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. In the latest issue of Physical Review Letters, Harvard Physics PhD Hoi Chun (Adrian) Po, Haruki Watanabe from University of Tokyo, and Prof. Ashvin Vishwanath address the converse question: Do obstructions to Wannier representation imply stable band topology? They 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. The authors 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 their model realize a trivial band with exponentially localized symmetric Wannier functions, whereas the second half possess fragile topology. This 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.

Hoi Chun Po, Haruki Watanabe, and Ashvin Vishwanath, "Fragile Topology and Wannier Obstructions," Phys. Rev. Lett. 121, 126402 (2018). DOI: 10.1103/PhysRevLett.121.126402.