  Gapless Luttinger liquid is conventionally viewed as topologically trivial,
unless it hosts degenerate ground states and or entanglement spectrum, which
necessitates partial bulk degree of freedom to be gapped out. Here we predict
an emergent gapless topological Luttinger liquid which is beyond the
conventional scenarios and is characterized by the nontrivial many-body bulk
spin texture, and propose feasible scheme for experimental observation. We
consider a one-dimensional spin-orbit coupled Fermi-Hubbard model with
fractional filling, whose low-energy physics is effectively described by a
spinless Luttinger liquid and is trivial in the conventional characterization.
We show that, as being tuned by the filling factor and interaction strength,
the many-body ground state may exhibit nontrivial winding in its bulk spin
texture in the projected momentum space, manifesting an emergent topological
phase. A topological transition occurs when the projected spin-state at a high
symmetry momentum becomes fully mixed one, resulting from competing processes
of particle scattering to the lower and higher subbands, for which the spin
texture at such momentum point is ill-defined, but the Luttinger liquid keeps
gapless through the transition. Surprisingly, at relatively small filling the
Luttinger liquid remains topologically nontrivial even at infinitely strong
interaction. The results can be generalized to finite temperature which
facilitates the real experimental detection. This work shows a novel gapless
topological Luttinger liquid whose characterization is beyond the low-energy
effective theory, and can be verified based on current experiments.
