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

Yichen Hu1,2 and Biao Lian1

1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
2The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK

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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.

Integrability of many-body quantum systems crucially affects their quantum dynamics and nonequilibrium behaviors. One of the simplest integrable systems is the (1+1)d chiral Luttinger liquid, which are interacting systems with emergent free boson descriptions. An interesting and practical question is whether realistic interacting chiral edge theories can still be integrable without being free fermions/free bosons. In this work, we make the first attempt to investigate the integrability of the interacting chiral edge states of filling 4/3 fractional quantum Hall (FQH) state, a prototypical experimentally accessible quantum system. We find strong numerical evidences supporting the possibility that this system is quantum integrable.
Our 4/3 FQH edge system consists of two interacting chiral modes with inter-mode electron tunnelling, and we show it generically maps to an interacting chiral fermion model. At two special points, the model is integrable as free fermions or free bosons, respectively. Deviating from these points, we perform a detailed exact diagonalization (ED) study and detect the level spacing statistics (LSS) of the energy spectrum in a conserved charge sector. Generically, we find Poisson LSS, which indicates the existence of hidden conserved quantities and the possibility that the generic 4/3 FQH edge theory is integrable. We further identified a series of bilayer FQH states which are expected to have similar edge state integrability behaviors. Our results revealed a novel class of potentially robust integrable chiral systems beyond free (Luttinger) theories, and provide key insights for understanding the edge state thermalization and interferometric experiments in FQH systems.

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