  We derive a new time-dependent Schr\"odinger equation(TDSE) for quantum
models with non-hermitian Hamiltonian. Within our theory, the TDSE is symmetric
in the two Hilbert spaces spanned by the left and the right eigenstates,
respectively. The physical quantities are also identical in these two spaces.
Based on this TDSE, we show that exchanging two quasi-particles in a
non-hermitian model can generate arbitrary geometric phase. The system can also
violate the Lieb-Robinson bound in non-relativistic quantum mechanics so that
an action in one place will immediately cause a change in the distance. We show
that the above two surprising behaviors can also appear in anyonic model, which
makes us propose that the non-hermitian single particle model may possess many
common features with anyonic model.
