  Hopf insulator is a representative class of three-dimensional topological
insulators beyond the standard topological classification methods based on
K-theory. In this letter, we discover the metallic counterpart of the Hopf
insulator 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 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 non-Hermitian degeneracy,
characterized by the two-dimensional surface of exceptional points. Despite the
non-Hermiticity, the Hopf metal has the quantized Zak phase, which results in
bulk-boundary correspondence by showing drumhead-like surface states at the
boundary. Finally, we show that, by breaking PT-symmetry, the nodal surface
deforms into the knotted exceptional lines. Our discovery of the Hopf metal
phase firstly confirms the existence of the non-Hermitian topological phase
outside the framework of the standard topological classifications.
