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

Max A. Metlitski, Ashvin Vishwanath

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.

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

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.

Paper Structure

This paper contains 22 sections, 31 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. A parton construction of a topological insulator
  3. The $u(1)$ spin-liquid of bosons SL$_{\times}$
  4. Confinement to a topological insulator
  5. The surface theory - QED$_3$
  6. Derivation of surface theory from parton construction
  7. Heuristic derivation of dual surface theory
  8. Dynamics of QED$_3$
  9. Dual Descriptions of the TI Surface phases
  10. Surface phases preserving all symmetries
  11. Fermi liquid and HLR state of dual fermions
  12. Surface topological order and superconductivity of dual fermions
  13. Breaking symmetries
  14. Surface superfluid and dual surface topological order
  15. Breaking time-reversal symmetry
  16. ...and 7 more sections

Figures (1)

  • Figure 1: (a) Dual derivation of topological insulator using fermionic partons in a topological superconductor band structure ($\nu=1$ of class AIII), where time reversal flips the sign of the gauge electric charge. The bulk topological insulator phase is obtained by condensing a pair of monopoles (0,2) bound to an electron. The surface state consists of the parton Dirac cone coupled to photons that only propagate on the surface, i.e. QED$_3$(b) The 3D TI derived more directly from the partons in the topological insulator band structure, which is Higgsed by condensing the unit electric charge (1,0) (bound to an electron). The surface is the regular single Dirac cone. The gauged versions of the topological superconductor (a) and topological insulator (b) are related by electric magnetic (E-M) duality - as seen from the lattice of electric and magnetic charges by identifying the basis vectors shown. The two basis vectors are exchanged by time-reversal symmetry in both (a) and (b). The E-M duality relates the double monopole condensate and the Higgs condensate, consistent with obtaining a TI from both descriptions.