PT -symmetric non-Hermitian Hopf metal

The Hopf insulator is a representative class of three-dimensional topological insulators beyond the standard topological classification methods based on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>K</a:mi></a:math> theory. In this Letter, we describe the metallic counterpart of the Hopf insulator in 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi mathvariant="script">PT</b:mi></b:math> 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 describe an interesting result that the non-Hermitian Hopf bundle exhibits 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 drumheadlike surface states at the boundary. Finally, we show that, by breaking <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi mathvariant="script">PT</d:mi></d:math> symmetry, the nodal surface deforms into knotted exceptional lines. Our description of the Hopf metal phase confirms the existence of the non-Hermitian topological phase outside the framework of the standard topological classifications.