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

Teber, Sofian

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

Authors

Description

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, $S(\om,q)$, beyond the Tomonaga approximation focusing on it's tails, $|\om| \gg vq$, {\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 $\om \to \pm vq$, {\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 $T\not=0$ case and show that the 2-pair excitation continuum dominates the single-pair one in the range: $|q|T/k_F \ll \om \mp vq \ll T$ (substantial for $q \ll k_F$). As applications we first derive the small-momentum optical conductivity due to forward scattering: $σ\sim 1/\om$ for $T \ll \om$ and $σ\sim T/\om^2$ for $T \gg \om$. Next, within the $2-$pair excitation continuum, we show that the attenuation rate of a coherent mode of dispersion $Ω_q$ crosses over from $γ_q \propto Ω_q (q/k_F)^2$, {\it e.g.} $γ_q \sim |q|^3$ for an acoustic mode, to $γ_q \propto T (q/k_F)^2$, independent of $Ω_q$, as temperature increases. Finally, we show that the $2-$pair excitation continuum yields subleading curvature corrections to the electron-electron scattering rate: $τ^{-1} \propto V^2 T + V^4 T^3/\eps_F^2$, where $V$ is the dimensionless strength of the interaction.
(v4) Published version. Details of calculations given (/ referee's comments). No change in previous results. 13 pages, 4 figures. (v3) Extended version (/ referee's comments and recent literature). No change in previous results. 8 pages, 4 figures. (v2) 4 pages, 4 figures. Submitted version (rapid note in EPJB). Kinetic arguments reduced to a footnote. 2 references added