Phase theory and critical exponents for the Tomonaga-Luttinger model with harmonic confinement

TL;DR

This paper develops a phase operator approach for a 1D fermion model in a harmonic trap, calculating critical exponents and correlation functions, revealing similarities and differences with open boundary conditions in Luttinger liquids.

Contribution

It introduces a novel phase operator formulation for the harmonic trap model, extending the Luttinger model approach to new geometries and calculating critical exponents.

Findings

Critical exponents match open boundary conditions except for boundary scaling.

Derived correlation functions for static and dynamic properties.

Analyzed local spectral density in the harmonic trap context.

Abstract

A phase operator formulation for a recent model of interacting one-dimensional fermions in a harmonic trap is developed. The resulting theory is similar to the corresponding approach for the Luttinger model with open boundary conditions (OBC). However, in place of the spatial coordinate $z$, a dimensionless variable $u$ defined on the unit circle appears as argument of the phase fields and $u$ is non-linearly related to $z$. Furthermore, form factors appear which reflect the harmonic trap geometry. The theory is applied to calculate one-particle correlation functions. In a properly defined thermodynamic limit, bulk and boundary critical exponents are calculated for the static two-point correlation function and the dynamic local correlation function. The local spectral density is also considered. The critical exponents found are in agreement with those known for OBC with the exception of the boundary scaling exponent $\Delta_\perp$.