  The low energy behavior of a huge variety of one-dimensional interacting
spinful fermionic systems exhibits spin-charge separation, described in the
continuum limit by two sine-Gordon models decoupled in the charge and spin
channels. Interaction is known to induce, besides the gapless Luttinger liquid
phase, eight possible gapped phases, among which are the Mott, Haldane,
charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery
liquid. Here we prove that some of these physically distinct phases have
nontrivial topological properties, notably the presence of degenerate protected
edge modes with fractionalized charge/spin. Moreover, we show that the eight
gapped phases are in one-to-one correspondence with the symmetry-protected
topological (SPT) phases classified by group cohomology theory in the presence
of particle-hole symmetry P. The latter result is also exploited to
characterize SPT phases by measurable nonlocal order parameters which follow
the system evolution to the quantum phase transition. The implications on the
appearance of exotic orders in the class of microscopic Hubbard Hamiltonians,
possibly without P symmetry at higher energies, are discussed.
