End-point singularity of the XY-paramagnetic phase boundary for the $$(2+1)$$D $$S=1$$ square-lattice $$J_1$$–$$J_2$$ XY model with the single-ion anisotropy

The two-dimensional quantum-spin $$S=1$$ square-lattice $$J_1$$ – $$J_2$$ XY model with the single-ion anisotropy D was investigated numerically, placing an emphasis on end-point singularity of phase boundary separating and paramagnetic phases in proximity to fully frustrated point, $$J_2/J_1 \rightarrow 0.5^{-}$$ . We employed exact diagonalization method circumvent negative sign problem quantum Monte Carlo method, evaluated fidelity susceptibility $$\chi _F$$ as a probe detect transition. As preliminary survey, for intermediate value $$J_2/J_1$$ , D-driven XY-paramagnetic transition via It turned out that criticality belongs 3D-XY universality class. Thereby, data were cast into crossover-scaling formula properly scaled distance from multi-critical $$0.5-J_2/J_1$$ set indices obtained, compared those Lifshitz criticality.