$\mathcal{PT}$-Symmetric Topological Edge-Gain Effect

We demonstrate a non-Hermitian topological effect that is characterized by having complex eigenvalues only in the edge states of a topological material, despite the fact that the material is completely uniform. Such an effect can be constructed in any topological structure formed by two gapped sub-systems, e.g., a quantum spin-Hall system, with a suitable non-Hermitian coupling between the spins. The resulting complex-eigenvalued edge state is robust against defects due to the topological protection. In photonics, such an effect can be used for the implementation of topological lasers, in which a uniform pumping provides gain only in the edge lasing state. Furthermore, such a topological lasing model is reciprocal and is thus compatible with standard photonic platforms.