Explicit Derivation of Duality between a Free Dirac Cone and Quantum Electrodynamics in ($2+1$) Dimensions

We explicitly derive the duality between a free electronic Dirac cone and quantum electrodynamics in ($2+1$) dimensions (${\mathrm{QED}}_3$) with $N=1$ fermion flavors. The duality proceeds via an exact, nonlocal mapping from electrons to dual fermions with long-range interactions encoded by an emergent gauge field. This mapping allows us to construct parent Hamiltonians for exotic topological-insulator surface phases, derive the particle-hole-symmetric field theory of a half-filled Landau level, and nontrivially constrain ${\mathrm{QED}}_3$ scaling dimensions. We similarly establish duality between bosonic topological insulator surfaces and $N=2$ ${\mathrm{QED}}_3$.

Figure 1

An array of chiral wires (left) exhibits a gauge anomaly when coupled to the electromagnetic vector potential $A_{\mu }$. The same anomaly but with opposite sign arises in a two-dimensional Chern-Simons theory with alternating coefficient (middle). The anomalies exactly cancel in the full two-dimensional theory (right) containing both the chiral wires and the bulk Chern-Simons theory.