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

Bulk-boundary correspondence (BBC) remains the central topic in modern condensed matter physics and has received a boost of interest with the recent discovery of non-Hermitian skin effects. However, there still exist profound features of BBC that are beyond the existing framework. Here, we report the unexpected behavior of BBC when the Hamiltonian contains terms of the form d0(k)I , which serves as a momentum dependent energy shift. For Hermitian cases, the momentum dependent energy shift can force the system to be metallic, where topological phase transitions can take place with the upper and the lower bands kept untouched. The modified BBC should be reconstructed from the perspective of the indirect 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 d0(k)I . Remarkably, such results can be led to a further step, and contrary to the intuition, 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 d0(k)I can drastically change the point gap topology. Finally, the corresponding experimental simulations are proposed via the platforms of electric circuits.