Bosonizing one-dimensional cold atomic gases

TL;DR

This paper uses bosonization to analyze correlation functions in one-dimensional cold atomic gases, providing detailed formulas applicable to various boundary conditions, temperatures, and interaction regimes, with validation against numerical data.

Contribution

It offers a comprehensive bosonization framework for 1D cold atomic gases, including boundary effects and finite temperature, validated by Bethe-ansatz solutions and Monte Carlo comparisons.

Findings

Correlation functions derived for different boundary conditions

Accurate formula for one-body density matrix of bosons

Validation against Monte Carlo results within 10% accuracy

Abstract

We present results for the long-distance asymptotics of correlation functions of mesoscopic one-dimensional systems with periodic and open (Dirichlet) boundary conditions, as well as at finite temperature in the thermodynamic limit. The results are obtained using Haldane's harmonic-fluid approach (also known as ``bosonization''), and are valid for both bosons and fermions, in weakly and strongly interacting regimes. The harmonic-fluid approach and the method to compute the correlation functions using conformal transformations are explained in great detail. As an application relevant to one-dimensional systems of cold atomic gases, we consider the model of bosons interacting with a zero-range potential. The Luttinger-liquid parameters are obtained from the exact solution by solving the Bethe-ansatz equations in finite-size systems. The range of applicability of the approach is discussed, and the prefactor of the one-body density matrix of bosons is fixed by finding an appropriate parametrization of the weak-coupling result. The formula thus obtained is shown to be accurate, when compared with recent diffusion Montecarlo calculations, within less than 10%. The experimental implications of these results for Bragg scattering experiments at low and high momenta are also discussed.