Structure factor of interacting one-dimensional helical systems

We calculate the dynamical structure factor $S(q,\omega)$ of a weakly interacting helical edge state in the presence of a magnetic field $B$. The latter opens a gap of width $2B$ in the single-particle spectrum, which becomes strongly nonlinear near the Dirac point. For chemical potentials $|\mu|>B$, the system then behaves as a nonlinear helical Luttinger liquid, and a mobile-impurity analysis reveals power-law singularities in $S(q,\omega)$ which depend on the interaction strength as well as on the spin texture of the edge states. For $|\mu|<B$, the low-energy excitations are gapped, and we determine $S(q,\omega)$ by using an analogy to exciton physics.