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

We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal $\left(PT\right)$ 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. (paper)