Novel topological end modes and breaking of bulk boundary correspondence in skin-effect absent superconductive chain

Topological nontrivial systems feature isolated gapless edge modes, and play a key role in advancing our understanding of quantum matter. A most profound way to characterize edge modes above is through bulk topological invariants, which is known as bulk boundary correspondence. Recent studies on non-Hermitian physics have pronounced the broken bulk-boundary correspondence with the presence of skin effect. Here, we propose a new type of fermionic topological edge modes η, satisfying η+= iη,η2=-i. Remarkably, we demonstrate that for both two cases: superconductive chain with purely η modes and quantum chain with η, Majorana modes γ on different ends, fermion parity can be well defined. Interestingly, for the latter case, broken bulk boundary correspondence is observed despite the absence of skin effects . The phenomenon above is unique to open quantum systems. For the junction with both η,γ modes, the current will not remain sinusoid form but decay exponentially. The exchange of η modes obeys the rules of non-abelian statistics, and can find its applications in topological quantum computing.