In this work, we focus on the fact that the Hamiltonian of the $1+1$D staggered fermion system can be smoothly deformed into that of Wilson fermions. We reinterpret the structure of the axial charge operator proposed above using Wilson fermions. The eigenstates of the axial charge are expressed as linear combinations of positive-energy creation and negative-energy annihilation operators. Consequently, the corresponding Hamiltonian includes terms that violate particle number conservation. Interestingly, by applying the insights of Chatterjee et al., it can be shown that even such Hamiltonians admit a conserved charge operator associated with the vector $\mathrm{U}(1)$ symmetry in the continuum limit. Since this operator does not commute with the axial charge, the construction is consistent with the Nielsen–Ninomiya theorem.

The resulting $1+1$D Hamiltonian formulation is expected to be useful in constructing chiral gauge theories based on the symmetric mass generation (SMG) mechanism. SMG refers to a mechanism by which gapless systems can be gapped without fermion bilinears, purely through appropriate interactions, while preserving symmetries. To demonstrate this, we examine the feasibility of realizing SMG while maintaining the $\mathrm{U}(1)_A$ gauge symmetry generated by the axial charge operator $Q_A$, using the $1^4(-1)^4$ and 3-4-5-0 models as examples.