Complex Berry phase and imperfect non-Hermitian phase transitions

In many classical and quantum systems described by an effective non-Hermitian Hamiltonian, spectral phase transitions, from entirely real-energy spectrum to a complex spectrum, can be observed as parameter in the system is increased above critical value. A paradigmatic example provided possessing parity-time ($\mathcal{PT}$) symmetry, where energy remains real unbroken $\mathcal{PT}$ while transition energies broken phase. Such transitions are universally sharp. However, when slowly periodically cycled, become smooth, i.e., imperfect, owing Berry associated cyclic adiabatic evolution of system. This remarkable phenomenon illustrated considering Wannier-Stark ladders $\mathcal{PT}$-symmetric class two-band lattices subjected external dc field, revealing that nonvanishing imaginary part Zak phase---the picked up Bloch eigenstate evolving across entire Brillouin zone---is responsible for imperfect transitions.