Integrability of the $\nu=4/3$ fractional quantum Hall edge states

TL;DR

This paper explores the integrability of the edge states in the $ u=4/3$ fractional quantum Hall effect, identifying solvable cases and analyzing spectral statistics to suggest the model's potential integrability.

Contribution

It identifies exactly solvable cases of the $ u=4/3$ edge theory and provides evidence for integrability through spectral analysis and symmetry considerations.

Findings

Two solvable cases with free modes identified

Poisson level spacing suggests hidden conserved quantities

Irrelevant nonlinear terms induce transition to Wigner-Dyson statistics

Abstract

We investigate the homogeneous chiral edge theory of the filling $\nu=4/3$ fractional quantum Hall state, which is parameterized by a Luttinger liquid velocity matrix and an electron tunneling amplitude (ignoring irrelevant terms). We identify two solvable cases: one case where the theory gives two free chiral boson modes, and the other case where the theory yields one free charge $\frac{2e}{\sqrt{3}}$ chiral fermion and two free chiral Bogoliubov (Majorana) fermions. For generic parameters, the energy spectrum from our exact diagonalization shows Poisson level spacing statistics (LSS) in each conserved charge and momentum sector, indicating the existence of hidden conserved quantities and the possibility that the generic edge theory of the $\nu=4/3$ fractional quantum Hall state is integrable. We further show that a global symmetry preserving irrelevant nonlinear kinetic term will lead to the transition of LSS from Poisson to Wigner-Dyson at high energies. This further supports the possibility that the model without irrelevant terms is integrable.