  Lattice four-fermion models containing $N$ flavors of staggered fermions,
that are invariant under $Z_2$ and U(1) chiral symmetries, are known to suffer
from sign problems when formulated using the auxiliary field approach. Although
these problems have been ignored in previous studies, they can be severe. In
this talk, we show that the sign problems disappear when the models are
formulated in the fermion bag approach, allowing us to solve them rigorously
for the first time.
