Broken bulk-boundary correspondence in the non-Hermitian superconductive chain with the identity determinant of transfer matrix

We have constructed a new type of topological end mode, named η, satisfying η†=iη,η2=−i, and demonstrate the topological characteristics of a quantum chain with isolated η modes on separated ends. Remarkably, we present that for a non-Hermitian superconductive chain with combined η mode and Majorana mode γ on different ends, the bulk-boundary correspondence is broken, even though the determinant of the transfer matrix is identity. When the tunneling parameters are tuned to make the system Hermitian, the bulk-boundary correspondence gets recovered. We demonstrate that such broken bulk-boundary correspondence has unique physical origins. Meanwhile, it is observed that the fractional Josephson effect does not exist in the junction with combined η−γ modes and the AC will not remain the sinusoidal form, despite of the well-defined fermion parity. Such effects can be utilized to detect the dissipation rate of the system. Experimentally, we propose to simulate chains with η modes, as well as combined η−γ modes, via electrical circuits.