Bulk-boundary correspondence in topological systems with the momentum dependent energy shift

Bulk-boundary correspondences (BBCs) remain the central topic in modern condensed matter physics, and are gaining increasing interests with the recent discovery of non-Hermitian skin effects. However, there still exist profound features of BBCs that are beyond the existing framework. Here, we report the unexpected behavior of BBC when the Hamiltonian contains term of the form $d_0(k) I$, which serves as a momentum dependent energy shift. For Hermitian cases, momentum dependent energy shift can force the system to be semimetal, where topological phase transitions can take place with the upper and the lower bands keeping untouched. The proper modified BBC should be reconstructed from the perspective of the direct band gap. In non-Hermitian cases, skin effects are found to be capable of coexisting with the preserved BBC, of which the process can be greatly facilitated by the complex $d_0(k)I$. Remarkably, such results can be led a further step and contrary to the intuitive consideration, the modified BBC in Hermitian systems can be restored to be conventional by including extra non-Hermiticity. The physical origin for these phenomena lies in that $d_0(k)I$ can drastically change the point gap topology. Finally, the corresponding experimental simulations are proposed via the platforms of electric circuits.