Theory of competing Chern-Simons orders and emergent phase transitions

Based on the Chern-Simons fermionization of spin-1/2 operators, we propose a systematic framework to investigate the competition between emergent phases in frustrated two-dimensional XY quantum magnets. Application of the method to the antiferromagnetic honeycomb spin-1/2 $J_1$-$J_2$ XY model reveals an unconventional phase transition between two Chern-Simons orders: the Chern-Simons superconductor and the exciton insulator of Chern-Simons fermions. We show that in the spin language, this transition translates to the transition from the planar N\'{e}el state to the non-uniform chiral spin-liquid that was proposed earlier in the literature. Namely, the Chern-Simons superconductor describes the planar N\'{e}el state, while the Chern-Simons exciton insulator corresponds to the non-uniform chiral spin-liquid. These results are further confirmed by our high-precision tensor network calculation, which provides the first numerical evidence for the transition from N\'{e}el order to a non-uniform chiral spin-liquid. We argue that the developed method can be applied to other frustrated quantum magnets of XXZ type and can detect unconventional phase transitions.