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 $\left|\mu \right|>B$, 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 $\left|\mu \right|<B$, the low-energy excitations are gapped, and we determine $S(q,\omega )$ by using an analogy to exciton physics.

Figure 1

(a) Regions in the $(q,\omega )$ plane where the noninteracting dynamical structure factor is nonzero. Yellow (red and green) regions denote contributions from the intraband (interband) transitions. The vertical dotted line at $q=0.9k_F$ indicates the position of the cut of $S(q,\omega )$ for fixed momentum. (b) The dynamical structure factor as function of $\omega$ (for fixed $q=0.9k_F$) exhibits discontinuities at the edge thresholds.

Figure 2

(a) and (b) Illustrate the transitions giving rise to singularities at the thresholds ${\omega }_{\mathrm{L}}^{\mathrm{Intra}}-,{\omega }_{\mathrm{U}}^{\mathrm{Intra}},\text{and}{\omega }_{\mathrm{L}}^{\mathrm{Inter}}+,{\omega }_{\mathrm{L}}^{\mathrm{Inter}}-$, respectively [see Fig. 1 for the edge thresholds]. In the figures the lower band ${\epsilon }_{-}\left(p\right)$ is completely filled, whereas the upper band ${\epsilon }_{+}\left(p\right)$ is filled up to the chemical potential $\mu >B$ (indicated by the top of the blue shaded area). For HgTe, $B$ and $\omega$ should be within $\lesssim$5 meV (i.e., smaller than the bulk gap), and $k_F\lesssim {10}^{-3}{\text{A}}^{-1}$ [4].

Figure 3

Ladder diagram resummation for $\left|\mu \right|<B$. Solid lines with $\pm$ denote single-particle Green's function corresponding to the upper and lower bands, respectively. The Greek letters represent spin indices (repeated indices imply summation), and wiggly lines denote interactions. Note the first term in the series is given by Eq. (3).