  We present a rigorous but elementary index theory for a class of
one-dimensional systems of interacting (and possibly disordered) fermions with
$\Uone\rtimes\bbZ_2$ symmetry defined on the infinite chain. The class includes
the Su-Schrieffer-Heeger (SSH) model as a special case. For any locally-unique
gapped (fixed-charge) ground state of a model in the class, we define a
$\bbZ_2$ index in terms of the sign of the expectation value of the local twist
operator. We prove that the index is topological in the sense that it is
invariant under continuous modification of models in the class with a
locally-unique (fixed-charge) gapped ground state. This establishes that any
path of models in the class that connects the two extreme cases of the SSH
model must go through a phase transition. Our rigorous $\bbZ_2$ classification
is believed to be optimal for the class of models considered here. We also show
an interesting duality of the index, and prove that any topologically
nontrivial model in the class has a gapless edge excitation above the ground
state when defined on the half-infinite chain. The results extend to other
classes of models, including the extended Hubbard model. Our strategy to focus
on the expectation value of local unitary operators makes the theory intuitive
and conceptually simple. The paper also contains a careful discussion about the
notion of unique gapped ground states of a particle system on the infinite
chain.
