Bosonization approach to charge and spin dynamics of 1D fermions with band-curvature

We consider one-dimensional (1D) spin-1/2 fermions in a clean quantum wire, with forward scattering interactions and a non-linear single-particle spectrum, ${\xi }_k=v\mathrm{\mid }k\mathrm{\mid }+k^2\mathrm{/}2m$ where $v$ is the Fermi velocity and $1\mathrm{/}m$ is the band-curvature. We calculate the dynamical structure factor (DSF) of the model at small wave-vector $q$ with the help of the bosonization technique. For spinless fermions, we show that, starting from the single-parametric spectrum: $\text{om}=u\mathrm{\mid }q\mathrm{\mid }$, bosonization emulates the 2-parametric excitation spectrum: $\om = u |q| \pm q^2/2m^$,where$m^$ decreases with increasing repulsive interactions. Moreover, away from the excitation-cone, {\it i.e.} $\om \gg u |q|$, bosonization yields the 2-pair excitation continuum of the DSF. For spinful fermions, we show that the spin-charge coupling (SCC) due to band-curvature affects charge and spin DSF in an asymmetric way. For the charge DSF, SCC manifests as a two-peak structure: a charge peak at $\text{om}=u_{\rho }\mathrm{\mid }q\mathrm{\mid }$ but also a spin peak at $\text{om}=u_{\sigma }\mathrm{\mid }q\mathrm{\mid }$, as charge fluctuations may decay via chargeless spin-singlet excitations. For the magnetic DSF, SCC manifests as a continuous transfer of magnetic spectral weight to frequencies $\om > u_\sigma |q|$, as spin fluctuations decay via pairs of chargeless spin and spinless charge-neutral excitations.