XY-VBS phase boundary for the square-lattice J1-J2XXZ model with the ring exchange

The square-lattice J 1 - J 2 XXZ model with the ring-exchange interaction K was investigated numerically. As for the hard-core-boson model with the nearest-neighbor hopping J 1 / 2 , namely, the J 1 - K XY model, it has been reported that the ring exchange leads to a variety of exotic phases such as the valence-bond-solid (VBS) phase. In this paper, we extend the parameter space to investigate the phase boundary between the XY (superfluid) and VBS phases. A notable feature is that the phase boundary terminates at the fully frustrated point, J 2 / J 1 → 0 . 5 - . As a scaling parameter for the multi-criticality, the distance from the multi-critical point δ ( ≥ 0 ) is introduced. To detect the phase transition, we employed the high-order fidelity susceptibility χ F ( 3 ) , which is readily evaluated via the exact-diagonalization scheme. As a demonstration, for a fixed value of δ , the XY -VBS criticality was analyzed by the probe χ F ( 3 ) . Thereby, with properly scaling δ , the χ F ( 3 ) data were cast into the crossover-scaling formula to determine the multi-criticality. Graphic abstract