Fate of Quantum Anomalies for 1d lattice chiral fermion with a simple non-Hermitian Hamiltonian

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.

Automated summary

It is generally believed that a 1+1D model for a single chiral fermion cannot exist alone on a lattice. The main obstacle is the inability to reproduce the quantum anomalies of a chiral fermion in the continuum. This paper proposes a 1+1D chiral fermion model on a 1D spatial lattice, which stands independently without being associated with a 2D bulk, by using a simple non-Hermitian hopping Hamiltonian. It is demonstrated that the model exhibits the same chiral anomaly and gravitational anomaly as the continuum theory.