Fractional excitations in the Luttinger liquid

TL;DR

This paper reinterprets the spectrum of the Luttinger liquid using fractional states, connecting bosonization with Bethe Ansatz, and explores the physical implications of fractional quasiparticles in spin chains and under magnetic fields.

Contribution

It introduces a fractional state representation for the Luttinger liquid spectrum, linking it with Bethe Ansatz predictions and providing new insights into spin-charge separation and quasiparticle states.

Findings

The $S_{z}=0$ continuum arises from fractional quasiparticle-quasihole pairs.

The spinon operator is not a semion in general.

In magnetic fields, spin-charge separation is replaced by new fractional states.

Abstract

We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows to make contact with Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wavefunction is also given for that spinon state. Our construction of the new spectroscopy based on fractional states leads to several new physical insights: in the low-energy limit, we find that the $S_{z}=0$ continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the 1D counterpart of the Laughlin FQHE quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states which we are able to describe.