A 1d realization of lattice chiral fermions with a non-Hermitian Hamiltonian

It is generally believed that a 1+1D chiral fermion state can not exist by itself on lattice. The obstruction to such a lattice realization is originated from the quantum anomalies of chiral fermion in quantum field theory. It has been recognized that such an anomalous state have a topologically nontrivial bulk. For example, a chiral fermion can be realized as the boundary state of a 2+1D Quantum Hall state. In this talk, we demonstrate that such a 1+1D chiral fermion state can be realized, without its corresponding bulk, by a one-spatial-dimensional non-Hermitian lattice Hamiltonian. The model possesses correct chiral anomaly and gravitational anomaly. Furthermore, the low energy effective theory of the model is a field theory of unitary chiral fermion.