Dynamical spin structure factor of one-dimensional interacting fermions

We revisit the dynamic spin susceptibility, $\chi(q,\omega)$, of one-dimensional interacting fermions. To second order in the interaction, backscattering results in a logarithmic correction to $\chi(q,\omega)$ at $q\ll k_F$, even if the single-particle spectrum is linearized near the Fermi points. Consequently, the dynamic spin structure factor, $\mathrm{Im}\chi(q,\omega)$, is non-zero at frequencies above the single-particle continuum. In the boson language, this effect results from the marginally irrelevant backscattering operator of the sine-Gordon model. Away from the threshold, the high-frequency tail of $\mathrm{Im}\chi(q,\omega)$ due to backscattering is larger than that due to finite mass by a factor of $k_F/q$. We derive the renormalization group equations for the coupling constants of the $g$-ology model at finite $\omega$ and $q$ and find the corresponding expression for $\chi(q,\omega)$, valid to all orders in the interaction but not in the immediate vicinity of the continuum boundary, where the finite-mass effects become dominant.