Luttinger liquid with complex forward scattering: robustness and Berry phase

Luttinger liquids (LLs) are one dimensional systems with well-understood instabilities due to umklapp or backscattering. We study a generalization of the Luttinger model, which incorporates a time reversal symmetry breaking interaction producing a complex forward scattering amplitude ($g_2$ process). The resulting low energy state is still a LL, and belongs to the family of interacting Schulz-Shastry models. Remarkably, it becomes increasingly robust against additional perturbations -- for purely imaginary $g_2$, both umklapp and local backscattering are always irrelevant. Changing the phase of the interaction generates a non-trivial Berry phase, with a universal geometric phase difference between ground and a one boson excited state depending only on the LL parameter.