Tails of the dynamical structure factor of 1D spinless fermions beyond the Tomonaga approximation

We consider one-dimensional (1D) interacting spinless fermions with a non-linear spectrum in a clean quantum wire (non-linear bosonization). We compute diagrammatically the 1D dynamical structure factor, $S(\text{om},q)$, beyond the Tomonaga approximation focusing on it’s tails, $\mathrm{\mid }\text{om}\mathrm{\mid }\gg vq$, {\it i.e.} the 2-pair excitation continuum due to forward scattering. Our methodology reveals three classes of diagrams: two “chiral” classes which bring divergent contributions in the limits $\text{om}\to \pm vq$, {\it i.e.} near the single-pair excitation continuum, and a “mixed” class (so-called Aslamasov-Larkin or Altshuler-Shklovskii type diagrams) which is crucial for the f-sum rule to be satisfied. We relate our approach to the T=0 ones present in the literature. We also consider the $T≠0$ case and show that the 2-pair excitation continuum dominates the single-pair one in the range: $\mathrm{\mid }q\mathrm{\mid }T\mathrm{/}{k}_F\ll \text{om}\mp vq\ll T$ (substantial for $q\ll k_F$). As applications we first derive the small-momentum optical conductivity due to forward scattering: $\sigma \sim 1/\om$for$T \ll \om$and$\sigma \sim T/\om^2$for$T \gg \om$. Next, within the $2-$pair excitation continuum, we show that the attenuation rate of a coherent mode of dispersion ${\mathrm{\Omega }}_q$ crosses over from $\gamma_q \propto \Omega_q (q/k_F)^2$,e.g.$\gamma_q \sim |q|^3$foranacousticmode,to$\gamma_q \propto T (q/k_F)^2$,independentof$\Omega_q$, as temperature increases. Finally, we show that the $2-$pair excitation continuum yields subleading curvature corrections to the electron-electron scattering rate: ${\tau }^{-1}\propto V^{2}T+V^4{T}^3\mathrm{/}{\text{eps}}_F^2$, where $V$ is the dimensionless strength of the interaction.