Dynamic structure factor of Luttinger liquids with quadratic energy dispersion and long-range interactions

Abstract

We calculate the dynamic structure factor $S(\ensuremath{\omega},q)$ of spinless fermions in one dimension with quadratic energy dispersion ${k}^{2}/2m$ and long-range density-density interaction whose Fourier transform ${f}_{q}$ is dominated by small momentum transfers $q\ensuremath{\lesssim}{q}_{0}⪡{k}_{F}$. Here ${q}_{0}$ is a momentum-transfer cutoff and ${k}_{F}$ is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter ${q}_{0}/{k}_{F}$. In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as ${f}_{q}={f}_{0}+{f}_{0}^{\ensuremath{''}}{q}^{2}/2+O({q}^{4})$ with ${f}_{0}^{\ensuremath{''}}\ensuremath{\ne}0$, we show that the momentum scale ${q}_{c}=1/|m{f}_{0}^{\ensuremath{''}}|$ separates two regimes characterized by a different $q$ dependence of the width ${\ensuremath{\gamma}}_{q}$ of the collective zero sound mode and other features of $S(\ensuremath{\omega},q)$. For ${q}_{c}⪡q⪡{k}_{F}$ we find that the line shape is non-Lorentzian with an overall width ${\ensuremath{\gamma}}_{q}\ensuremath{\propto}{q}^{3}/(m{q}_{c})$ and a threshold singularity ${[(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}){\text{ln}}^{2}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}})]}^{\ensuremath{-}1}$ at the lower edge $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}=vq\ensuremath{-}{\ensuremath{\gamma}}_{q}$, where $v$ is the velocity of the zero sound mode. Assuming that higher orders in perturbation theory transform the logarithmic singularity into an algebraic one, we find for the corresponding threshold exponent ${\ensuremath{\mu}}_{q}=1\ensuremath{-}2{\ensuremath{\eta}}_{q}$ with ${\ensuremath{\eta}}_{q}\ensuremath{\propto}{q}_{c}^{2}/{q}^{2}$. Although for $q\ensuremath{\lesssim}{q}_{c}$ we have not succeeded to explicitly evaluate our functional bosonization result for $S(\ensuremath{\omega},q)$, we argue that for any one-dimensional model belonging to the Luttinger liquid universality class, the width of the zero sound mode scales as ${q}^{2}/m$ for $q\ensuremath{\rightarrow}0$.