  In the Hamiltonian picture, free spin-$1/2$ Dirac fermions on a bipartite
lattice have an $O(4)$ (spin-charge) symmetry. Here we construct an interacting
lattice model with an interaction $V$, which is similar to the Hubbard
interaction but preserves the spin-charge flip symmetry. By tuning the coupling
$V$, we show that we can study the phase transition between the massless
fermion phase at small-$V$ and a massive fermion phase at large-$V$. We
construct a fermion bag algorithm to study this phase transition and find
evidence for it to be second order. Numerical study shows that the universality
class of the transition is different from the one studied earlier involving the
Hubbard coupling $U$. Here we obtain some critical exponents using lattices up
to $L=48$.
