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