PT-symmetry berry phases,topology and PT-symmetry breaking

We study the complex Berry phases in non-Hermitian systems with parity-and time-reversal(PT) symmetry.We investigate a kind of two-level system with PT symmetry.We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or Π,which originates from the topology of the Hermitian eigenstates.We also find that if we change the relative parameters of the Hamiltonian from the unbroken-PT-symmetry phase to the broken-PT-symmetry phase,the imaginary part of the complex Berry phases are divergent at the exceptional points.We exhibit two concrete examples in this work,one is a two-level toys model,which has nontrivial Berry phases;the other is the generalized Su-Schrieffer-Heeger(SSH) model that has physical loss and gain in every sublattice.Our results explicitly demonstrate the relation between complex Berry phases,topology and PT-symmetry breaking and enrich the field of the non-Hermitian physics.