Particle-vortex duality of 2d Dirac fermion from electric-magnetic duality of 3d topological insulators

TL;DR

This paper introduces a duality for 2+1D Dirac fermions based on 3D topological insulator electric-magnetic duality, leading to a new surface theory and insights into topological states.

Contribution

It establishes a particle-vortex duality for Dirac fermions using 3D topological insulator dualities, deriving a dual QED3 theory and connecting to surface topological order.

Findings

Derivation of dual QED3 as a surface theory for Dirac fermions

Explicit construction of the T-Pfaffian state from dual fermions

Connection of duality to particle-hole symmetric quantum Hall states

Abstract

Particle-vortex duality is a powerful theoretical tool that has been used to study bosonic systems. Here we propose an analogous duality for Dirac fermions in 2+1 dimensions. The physics of a single Dirac cone is proposed to be described by a dual theory, QED3 with a dual Dirac fermion coupled to a gauge field. This duality is established by considering two alternate descriptions of the 3d topological insulator (TI) surface. The first description is the usual Dirac cone surface state. The second description is accessed via an electric-magnetic duality of the bulk TI coupled to a gauge field, which maps it to a gauged topological superconductor. This alternate description ultimately leads to a new surface theory - dual QED3. The dual theory provides an explicit derivation of the T-Pfaffian state, a proposed surface topological order of the TI, which is simply the paired superfluid state of the dual fermions. The roles of time reversal and particle-hole symmetry are exchanged by the duality, which connects some of our results to a recent conjecture by Son on particle-hole symmetric quantum Hall states.