Quantized Axial Charge in the Hamiltonian Approach to Wilson Fermions

We investigate the Hamiltonian formulation of 1+1~D staggered fermions and reconstruct vector and axial charge operators, found by Arkya Chatterjee et al., using the Wilson fermion formalism. These operators commute with the Hamiltonian and become the generators of vector and axial $\mathrm{U}(1)$ symmetries in the continuum limit. An interesting feature of the axial charge operator is that it acts locally on operators and has quantized eigenvalues in momentum space. Therefore, the eigenstates of this operator can be interpreted as fermion states with a well-defined integer chirality, analogous to those in the continuum theory. This allows us to realize a gauge theory where the axial $\mathrm{U}(1)_A$ symmetry acts as a gauge symmetry. We construct a Hamiltonian using the eigenstates of the axial charge operator, preserving exact axial symmetry on the lattice and vector symmetry in the continuum. As applications, we examine the implementation of the Symmetric Mass Generation (SMG) mechanism in the $1^4(-1)^4$ and 3-4-5-0 models. Our formulation supports symmetry-preserving interactions with quantized chiral charges, although further numerical studies are required to verify the SMG mechanism in interacting models.