Braid protected topological band structures with unpaired exceptional points

We explore loopholes in the non-Hermitian Fermion doubling theorem. In particular, we show how this gives rise to non-trivial exceptions in systems with three or more bands, where the topological structure is non-abelian and the winding number description does not provide the full picture. In a simple three-band model, we show the occurrence of a single, unpaired, exceptional point on a Brillouin zone. This third-order degeneracy cannot be gapped by any small perturbation due to the non-trivial boundary braid. It can only split into simpler (second order) constituent degeneracies that need to be moved around topologically non-trivial loops of the Brillouin zone before being able to fully gap out by pairwise annihilation.