Fragile topology on solid grounds: a mathematical perspective

This paper provides a mathematical perspective on fragile topology phenomena in condensed matter physics. In dimension $d \leq 3$, vanishing Chern classes of bundles of Bloch eigenfunctions characterize operators with exponentially localized Wannier functions (these functions form convenient bases of spectrally determined subspaces of $L^2$). However, for systems with additional symmetries, such as the $C_{2}T$ (space-time reversal) or the $PT$ (parity-time) symmetry, a set of exponentially localized Wannier functions compatible with such symmetry may not exist. We show that for rank 2 Bloch bundles with such symmetry, non-trivial Euler classes are obstructions to constructing exponentially localized compatible Wannier functions. We also show that this obstruction can be lifted by adding additional Bloch bundles with the symmetry, even though the Stiefel--Whitney class of the total bundle is non-trivial. This allows a construction of exponentially localized Wannier functions compatible with the symmetry and that is referred to as topological fragility.