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

The two-dimensional quantum-spin $$S=1$$ S = 1 square-lattice $$J_1$$ J 1 – $$J_2$$ J 2 XY model with the single-ion anisotropy D was investigated numerically, placing an emphasis on the end-point singularity of the phase boundary separating the XY and paramagnetic phases in proximity to the fully frustrated point, $$J_2/J_1 \rightarrow 0.5^{-}$$ J 2 / J 1 → 0 . 5 - . We employed the exact diagonalization method to circumvent the negative sign problem of the quantum Monte Carlo method, and evaluated the fidelity susceptibility $$\chi _F$$ χ F as a probe to detect the phase transition. As a preliminary survey, for an intermediate value of $$J_2/J_1$$ J 2 / J 1 , the D-driven XY-paramagnetic phase transition was investigated via the probe $$\chi _F$$ χ F . It turned out that the criticality belongs to the 3D-XY universality class. Thereby, the $$\chi _F$$ χ F data were cast into the crossover-scaling formula with the properly scaled distance from the multi-critical point, $$0.5-J_2/J_1$$ 0.5 - J 2 / J 1 . The set of multi-critical indices was obtained, and compared to those of the quantum Lifshitz criticality. Graphic Abstract