Probability-preserving evolution in a non-Hermitian two-band model

A non-Hermitian PT-symmetric system can have full real spectrum but does not ensure probability preserving time evolution, in contrast to that of a Hermitian system. We present a non-Hermitian two-band model, which is comprised of dimerized hopping terms and staggered imaginary on-site potentials, and study the dynamics in the exact PT-symmetric phase based on the exact solution. It is shown that an initial state, which does not involve two equal-momentum-vector eigenstates in different bands, obeys perfectly probability-preserving time evolution in terms of the Dirac inner product. Beyond this constriction, the quasi-Hermitian dynamical behaviors, such as non-spreading propagation and fractional revival of a Gaussian wave packet, are also observed.