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

Symmetry protected topological phases of 1D interacting fermions with spin-charge separation