Non-Hermitian systems with a real spectrum and selective skin effect

In this work we first show a simple approach to constructing non-Hermitian Hamiltonians with a real spectrum, which are \textit{not} obtained by a non-unitary transformation such as the imaginary gauge transformation. They are given, instead, by the product of a Hermitian Hamiltonian $H_0$ and a positive semi-definite matrix $A$. Depending on whether $A$ has zero eigenvalue(s), the resulting $H$ can possess an exceptional point at zero energy. When $A$ is only required to be Hermitian instead, the resulting $H$ is pseudo-Hermitian that can have real and complex conjugate energy levels. In the special case where $A$ is diagonal, we compare our approach to an imaginary gauge transformation, which reveals a selective non-Hermitian skin effect in our approach, i.e., only the zero mode is a skin mode and the non-zero modes reside in the bulk. We further show that this selective non-Hermitian skin mode has a much lower lasing threshold than its counterpart in the standard non-Hermitian skin effect with the same spatial profile, when we pump at the boundary where they are localized. The form of our construction can also be found, for example, in dynamical matrices describing coupled frictionless harmonic oscillators with different masses.