PT-symmetric Non-Hermitian Hopf metal

Hopf insulator represents a class of three-dimensional topological insulators protected by the non-trivial Hopf bundle structure of the eigenstates. In this letter, we propose the generalization of the Hopf bundle in the non-Hermitian systems. While the Hopf invariant is not a stable topological index due to the additional non-Hermitian degree of freedom, we show that the $mathcal{PT}$-symmetry stabilizes the Hopf invariant even in the presence of the non-Hermiticity. In sharp contrast to the Hopf insulator phase in the Hermitian counterpart, we discover an interesting result that the non-Hermitian Hopf bundle exhibits the topologically protected band degeneracy, which is characterized by the surface of exceptional points. In addition to the topological degeneracy, this Hopf metal phase has non-trivial bulk-boundary correspondence, which manifests as the drumhead surface state in the open boundary condition. Finally, we show that, by breaking $mathcal{PT}$-symmetry, the nodal surface deforms into the knotted exceptional lines. Our discovery of the Hopf metal phase, for the first time, shows the existence of the non-Hermitian topological phase beyond the known topological classification methods such as $K$-theory and symmetry indicator.