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 <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$T\not=0$</EquationSource> </InlineEquation> case and show that the 2-pair excitation continuum dominates the single-pair one in the range: |q|T/k<Subscript>F</Subscript> ≪ω±vq ≪T (substantial for q ≪k<Subscript>F</Subscript>). As applications we first derive the small-momentum optical conductivity due to forward scattering: σ∼1/ω for T ≪ω and σ∼T/ω<Superscript>2</Superscript> for T ≫ω. Next, within the 2-pair excitation continuum, we show that the attenuation rate of a coherent mode of dispersion Ω<Subscript>q</Subscript> crosses over from <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\gamma_q \propto \Omega_q~(q/k_F)^2$</EquationSource> </InlineEquation>, e.g. γ<Subscript>q</Subscript> ∼|q|<Superscript>3</Superscript> for an acoustic mode, to <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\gamma_q \propto T~(q/k_F)^2$</EquationSource> </InlineEquation>, independent of Ω<Subscript>q</Subscript>, as temperature increases. Finally, we show that the 2-pair excitation continuum yields subleading curvature corrections to the electron-electron scattering rate: <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\tau^{-1} \propto V^2 T + V^4~T^3/\epsilon_F^2$</EquationSource> </InlineEquation>, where V is the dimensionless strength of the interaction. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006