Deconfined quantum critical point on the triangular lattice

In this work we propose a theory for the deconfined quantum critical point (DQCP) for spin-1/2 systems on a triangular lattice, which is a direct unfine-tuned quantum phase transition between the standard “$\sqrt{3}\times \sqrt{3}$” noncollinear antiferromagnetic order (or the so-called ${120}^{\circ }$ state) and the “$\sqrt{12}\times \sqrt{12}$” valence solid bond (VBS) order, both of which are very standard ordered phases often observed in numerical simulations. This transition is beyond the standard Landau-Ginzburg paradigm and is also fundamentally different from the original DQCP theory on the square lattice due to the very different structures of both the magnetic and VBS order on frustrated lattices. We first propose a topological term in the effective-field theory that captures the “intertwinement” between the $\sqrt{3}\times \sqrt{3}$ antiferromagnetic order and the $\sqrt{12}\times \sqrt{12}$ VBS order. Then using a controlled renormalization-group calculation, we demonstrate that an unfine-tuned direct continuous DQCP exists between the two ordered phases mentioned above. This DQCP is described by the $N_f=4$ quantum electrodynamics (QED) with an emergent PSU(4)=SU(4)/$Z_4$ symmetry only at the critical point. The aforementioned topological term is also naturally derived from the $N_f=4$ QED. We also point out that physics around this DQCP is analogous to the boundary of a $3d$ bosonic symmetry- protected topological state with only on-site symmetries.

Figure 1

The global phase diagram of spin-1/2 systems on the triangular lattice. The intertwinement between the order parameters is captured by the WZW term (3). Our RG analysis concludes that there is a direct unfine-tuned SO(3)-to-SO(3) transition, which is a direct unfine-tuned transition between the noncollinear magnetic order and the VBS order. The detailed structure of the shaded areas demands further studies.