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(ω,q), beyond the Tomonaga approximation focusing on it's tails, |ω| ≫vq, 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 ω→±vq, 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 $T\not=0$ case and show that the 2-pair excitation continuum dominates the single-pair one in the range: |q|T/kF ≪ω±vq ≪T (substantial for q ≪kF). As applications we first derive the small-momentum optical conductivity due to forward scattering: σ∼1/ω for T ≪ω and σ∼T/ω2 for T ≫ω. Next, within the 2-pair excitation continuum, we show that the attenuation rate of a coherent mode of dispersion Ωq crosses over from $\gamma_q \propto \Omega_q~(q/k_F)^2$, e.g. γq ∼|q|3 for an acoustic mode, to $\gamma_q \propto T~(q/k_F)^2$, independent of Ωq, as temperature increases. Finally, we show that the 2-pair excitation continuum yields subleading curvature corrections to the electron-electron scattering rate: $\tau^{-1} \propto V^2 T + V^4~T^3/\epsilon_F^2$, where V is the dimensionless strength of the interaction