Perturbative Energy Shifts of the Luttinger Liquid Due to the Presence of Additional Dimensions

TL;DR

This paper calculates how additional dimensions perturb the energy levels of a Luttinger liquid in a cylindrical trap, using bosonization and a simplified model to determine the bandwidth.

Contribution

It introduces a perturbative approach to quantify energy shifts in a Luttinger liquid due to extra dimensions, linking two models for a unified result.

Findings

Calculated zero-temperature energy shifts of Luttinger bosonic modes.

Derived the Luttinger liquid bandwidth for the system.

Validated results through two different modeling approaches.

Abstract

We consider a system of spinless fermions in a nearly one-dimensional cylindrical trap. We introduce four-fermion interaction terms through which particles in low-lying modes of the trap may interact with particles in radially or angularly excited states. Treating these interactions as perturbations about a purely one-dimensional Luttinger liquid, we calculate the zero-temperature energy shifts of the bosonic quanta of the Luttinger model. We perform this calculation is two ways. First we use the bosonized forms of the fermion operators. Then we introduce a simplified, physically-motivated model; in this model, the methods used immediately generalize, so that we may also calculate the energy shifts of the Tomonaga-Luttinger model. By equating the results from the two disparate models, we obtain the numerical value of the Luttinger liquid bandwidth for this system.