Fourier transform of the $2k_F$ Luttinger liquid density correlation function with different spin and charge velocities

TL;DR

This paper derives a closed-form expression for the Fourier transform of the $2k_F$ density correlation in a Luttinger liquid with different spin and charge velocities, analyzing singularities and temperature effects.

Contribution

It provides an exact analytical formula for the Fourier transform of the density correlation function considering different spin and charge velocities in a Luttinger liquid.

Findings

Power law singularities lead to divergences or cusps.

Temperature smooths out the singularities.

Exact integral and numerical results for finite temperature.

Abstract

We obtain a closed-form analytical expression for the zero temperature Fourier transform of the $2k_F$ component of the density-density correlation function in a Luttinger liquid with different spin and charge velocities. For frequencies near the spin and charge singularities approximate analytical forms are given and compared with the exact result. We find power law like singularities leading to either divergence or cusps, depending on the values of the Luttinger parameters and compute the corresponding exponents. Exact integral expressions and numerical results are given for the finite temperature case as well. We show in particular how the temperature rounds the singularities in the correlation function.