Point-gap bound states in non-Hermitian systems

In non-Hermitian band systems, point gaps are frequently present, and the coalescence of the point gaps with nonzero spectral winding leads to the non-Hermitian skin effect, which can be regarded as a mechanism that competes with the impurity for bound states. Here, we systematically investigate impurity-induced bound states in one-dimensional non-Hermitian systems. By establishing the relationship between bound-state energy and the requisite impurity potential, we conveniently construct an impurity potential diagram corresponding to point gaps. This diagram indicates both the minimal impurity potential required to generate bound states within each point gap and the distribution of bound states across these point gaps for a given impurity potential. From this, we reveal that a finite impurity potential is required to generate bound states in the absence of Bloch saddle points; otherwise, even a negligible impurity potential can yield bound states. Additionally, we show that bound states in point gaps with nonzero spectral winding numbers are sensitive to boundary conditions and abruptly shift to the edges upon opening the boundaries, signifying the bulk-boundary correspondence in point-gap topology.