Lattice Chiral Fermion without Hermiticity

Our review of the lattice chiral fermion delves into some critical areas of lattice field theory. By abandoning Hermiticity, the non-Hermitian formulation circumvents the Nielsen-Ninomiya theorem while maintaining chiral symmetry, a novel approach. Comparing the Wilson and overlap fermions gives insight into how lattice formulations handle chiral symmetry. The Wilson fermion explicitly breaks chiral symmetry to eliminate doublers. In contrast, the overlap fermion restores a modified form of chiral symmetry using the Ginsparg-Wilson relation. We investigate how the (1+1)D Wilson fermion relates to the (1+1)D overlap fermion in the Hamiltonian formulation. This connection could provide a clearer physical understanding of how chiral symmetry manifests at the lattice level. Depending on Hermiticity for efficiency, Monte Carlo methods face unique challenges in a non-Hermitian setting. We investigate how to correctly apply this method to non-Hermitian lattice fermions, which is essential for practical simulations. Finally, the review of topological charge is crucial, as topological features in lattice formulations are strongly connected to chiral symmetry, anomalies, and the index theorem.