Non-Abelian descendant of Abelian duality in a two-dimensional frustrated quantum magnet

Several recent works on quantum criticality beyond the Landau-Ginzburg-Wilson paradigm have led to a number of field theories, potentially important for certain two-dimensional magnetic insulating systems, where criticality is not very well understood. This situation highlights the need for nonperturbative information about criticality in two spatial dimensions (three space-time dimensions), which is a longstanding challenge. As a step toward addressing these issues, we present evidence that the O(4) vector model is dual to a theory of Dirac fermions coupled to both SU(2) and U(1) gauge fields. Both field theories arise as low-energy long-wavelength descriptions of a frustrated $XY$ model on the triangular lattice. Abelian boson-vortex duality of the lattice model, together with the emergence of larger non-Abelian symmetry at low energies, leads to this rare example of duality in two spatial dimensions involving non-Abelian global symmetry and fermions but without supersymmetry. The duality can also be viewed as a bosonization of the Dirac fermion gauge theory.

Figure 1

(a) Depiction of the coplanar $120°$ ordered state, and (b) one of the two collinear states. The filled circles represent sites with zero ordered moment. The other collinear state is depicted in Ref. 8.

Figure 2

Relation between the dual honeycomb lattice (dashed lines) and the triangular lattice of the original $XY$ model (solid lines). Honeycomb lattice sites correspond to triangular lattice plaquettes.

Figure 3

Illustration of the $2\pi$-monopole state whose corresponding operator is the QCED3 partner of the O(4) field $Z$. The spectrum of each two-component fermion ${\eta }_{a\alpha }$ is shown. The fermions ${\eta }_{11}$ and ${\eta }_{21}$ both feel a net $2\pi$ flux through the sphere. The energy-level spectrum for each of these fermions is symmetric about zero energy, where for each fermion there is a single zero-energy state. The ${\eta }_{12}$ and ${\eta }_{22}$ fermions feel no flux, and also have a spectrum symmetric about zero energy but with no zero-energy state. All negative-energy levels are filled, and one of the two zero mode states is filled to obtain a state with zero total charge under both the $a_{\mu }$ and ${\alpha }_{\mu }^3$ gauge fields.