XY-VBS phase boundary for the square-lattice $$J_1$$ J 1 - $$J_2$$ J 2 XXZ model with the ring exchange

The square-lattice $$J_1$$ J 1 - $$J_2$$ 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 $$J_1/2$$ J 1 / 2 , namely, the $$J_1$$ 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, $$J_2/J_1 \rightarrow 0.5^-$$ J 2 / J 1 → 0 . 5 - . As a scaling parameter for the multi-criticality, the distance from the multi-critical point $$\delta (\ge 0)$$ δ ( ≥ 0 ) is introduced. To detect the phase transition, we employed the high-order fidelity susceptibility $$\chi ^{(3)}_F$$ χ F ( 3 ) , which is readily evaluated via the exact-diagonalization scheme. As a demonstration, for a fixed value of $$\delta $$ δ , the XY-VBS criticality was analyzed by the probe $$\chi ^{(3)}_F$$ χ F ( 3 ) . Thereby, with properly scaling $$\delta $$ δ , the $$\chi ^{(3)}_F$$ χ F ( 3 ) data were cast into the crossover-scaling formula to determine the multi-criticality. Graphic abstract