  It is generally believed that the 1+1D model for a single chiral fermion does
not exist by itself alone on lattice. The obstruction to such a lattice
realization is the failure to reproduce the quantum anomalies of a chiral
fermion in continuum. The conventional way to escape is to associate the
anomalous 1d system with a 2d bulk, which is in a topologically non-trivial
state, as the boundary of the latter. In this paper, we propose a 1+1D chiral
fermion model on 1d spatial lattice, {standing alone} -- without being
associated with a 2d bulk -- with a simple {non-Hermitian} hopping Hamiltonian.
We demonstrate, using various methods, that the model possesses the same chiral
anomaly and gravitational anomaly as in continuum theory. Furthermore, with
appropriate parameters, the low energy effective theory of the model remains a
field theory for unitary chiral fermions. The essential reason for the success
is that the usual "doubled" fermion mode with opposite chirality is rapidly
damped out because of non-Hermicity of the Hamiltonian.
