The Fractional Hall hierarchy from duality

TL;DR

This paper develops a unified theoretical framework for fractional quantum Hall states using dualities and composite fermion theories, revealing a landscape of gapped and gapless phases at various filling fractions.

Contribution

It introduces a novel application of 3D dualities to composite fermion Landau levels, unifying descriptions of different fractional quantum Hall states within a single effective field theory.

Findings

Unified description of gapped and gapless states

Identification of critical points at even-denominator fillings

Matching hierarchy states with abelian Chern-Simons theory

Abstract

We show that a modified version of Son's Dirac composite fermion theory proposed by Seiberg et al gives a candidate unified description of the gapped and gapless fractional quantum Hall states within a single Landau level. Our main tool is the successive application of three-dimensional dualities to partially filled Landau levels of composite fermions, which imply that this theory has a complicated landscape of gapped vacua and critical points. This construction is the Lagrangian, or effective field theory, analogue of the flux attachment procedure. The critical points exist at even denominator filling and are well-described by a Fermi surface for a weakly coupled composite fermion coupled to an abelian Chern-Simons theory. The gapped states include odd-denominator filling fraction states with an abelian Chern-Simons description which we show matches the one expected for hierarchy states, as well as non-abelian states at even-denominator filling that arise from pair instabilities of the composite fermion's Fermi surface.