  We develop the $(1+1)$d lattice $U(1)$ gauge theory in order to define
$2$-flavor massless Schwinger model, and discuss its connection with Haldane
conjecture. We propose to use the central-branch Wilson fermion, which is
defined by relating the mass, $m$, and the Wilson parameter, $r$, as $m+2r=0$.
This setup gives two massless Dirac fermions in the continuum limit, and it
turns out that no fine-tuning of $m$ is required because the extra $U(1)$
symmetry at the central branch, $U(1)_{\bar{V}}$, prohibits the additive mass
renormalization. Moreover, we show that Dirac determinant is positive
semi-definite and this formulation is free from the sign problem, so the Monte
Carlo simulation of the path integral is possible. By identifying the symmetry
at low energy, we show that this lattice model has the mixed 't Hooft anomaly
between $U(1)_{\bar{V}}$, lattice translation, and lattice rotation. We discuss
its relation to the anomaly of half-integer anti-ferromagnetic spin chains, so
our lattice gauge theory is suitable for numerical simulation of Haldane
conjecture. Furthermore, it gives new and strict understanding on parity-broken
phase (Aoki phase) of $2$d Wilson fermion.
