Topological phases in a Kitaev chain with imbalanced pairing

We systematically study a Kitaev chain with imbalanced pair creation and annihilation, which is introduced by non-Hermitian pairing terms. An exact phase diagram shows that the topological phase is still robust under the influence of the conditional imbalance. The gapped phases are characterized by a topological invariant, the extended Zak phase, which is defined by the biorthonormal inner product. Such phases are destroyed at the points where the coalescence of ground states occurs, associated with the time-reversal symmetry breaking. We find that the Majorana edge modes also exist in an open chain in the time-reversal symmetry-unbroken region, demonstrating the bulk-edge correspondence in such a non-Hermitian system.