  By considering energy flow, we construct the one-dimensional (1d) model
consisting of the quasiparticles caused by asymmetric hopping (in carrier
position space) or the complex bosonic potential whose varying gradience with a
chiral ordering plays the role of ingredience of quasiparticles. A bosonic
potential can be generated and the chaotic dynamics of chiral excitations after
disorder average can be investigated in the presence of gauge invariance. This
feature is also shared by the well-known non-Hermitian systems.
