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

TL;DR

This paper analyzes the dynamical structure factor of 1D spinless fermions beyond the Tomonaga approximation, revealing new diagram classes, temperature effects, and implications for optical conductivity and scattering rates.

Contribution

It introduces a diagrammatic approach to compute the structure factor beyond the Tomonaga approximation, including temperature effects and the role of different diagram classes.

Findings

Identification of three classes of diagrams affecting the structure factor.

Derivation of optical conductivity behavior at different temperatures.

Analysis of the attenuation rate crossover for coherent modes.

Abstract

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: $\sigma \sim 1/\om$ 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 $\Omega_q$ crosses over from $\gamma_q \propto \Omega_q (q/k_F)^2$, {\it e.g.} $\gamma_q \sim |q|^3$ for an acoustic mode, to $\gamma_q \propto T (q/k_F)^2$, independent of $\Omega_q$, as temperature increases. Finally, we show that the $2-$pair excitation continuum yields subleading curvature corrections to the electron-electron scattering rate: $\tau^{-1} \propto V^2 T + V^4 T^3/\eps_F^2$, where $V$ is the dimensionless strength of the interaction.