Mirror symmetry and the half-filled Landau level

TL;DR

This paper uses mirror symmetry to analyze the half-filled Landau level of Dirac fermions, deriving a dual description involving neutral composite fermions and emergent gauge fields, revealing a controlled model for strong interactions.

Contribution

It provides a supersymmetric duality-based derivation of flux attachment and the emergent Fermi liquid state for Dirac fermions in a magnetic field.

Findings

Coulomb interactions induce a dynamical exponent z=2 for the gauge field.

The dual theory maps the problem to a perturbatively controlled model.

Analysis of BCS interactions in the emergent composite Fermi liquid.

Abstract

We study the dynamics of the half-filled zeroth Landau level of Dirac fermions using mirror symmetry, a supersymmetric duality between certain pairs of $2+1$-dimensional theories. We show that the half-filled zeroth Landau level of a pair of Dirac fermions is dual to a pair of Fermi surfaces of electrically-neutral composite fermions, coupled to an emergent gauge field. Thus, we use supersymmetry to provide a derivation of flux attachment and the emergent Fermi liquid-like state for the lowest Landau level of Dirac fermions. We find that in the dual theory the Coulomb interaction induces a dynamical exponent $z=2$ for the emergent gauge field, making the interactions classically marginal. This enables us to map the problem of $2+1$-dimensional Dirac fermions in a finite transverse magnetic field, interacting via a strong Coulomb interaction, into a perturbatively controlled model. We analyze the resulting low-energy theory using the renormalization group and determine the nature of the BCS interaction in the emergent composite Fermi liquid.