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

TL;DR

This paper calculates the dynamic structure factor of one-dimensional spinless fermions with quadratic dispersion and long-range interactions, revealing non-Lorentzian line-shapes and different regimes based on momentum scales.

Contribution

It introduces a functional bosonization method to analyze the structure factor without mass-shell singularities, extending understanding of collective modes in long-range interacting Luttinger liquids.

Findings

Identifies a momentum scale q_c separating different regimes of the structure factor.

Finds non-Lorentzian line-shapes with a width scaling as q^3/(m q_c).

Shows a threshold singularity at the lower edge of the spectrum.

Abstract

We calculate the dynamic structure factor S (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 << 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^{2} q^2/2 + O (q^4) with finite f_0^{2} we show that the momentum scale q_c = 1/ | m f_0^{2} | separates two regimes characterized by a different q-dependence of the width gamma_q of the collective zero sound mode and other features of S (omega, q). For q_c << q << k_F we find that the line-shape in this regime is non-Lorentzian with an overall width gamma_q of order q^3/(m q_c) and a threshold singularity at the lower edge.