Fidelity-mediated analysis of the transverse-field XY chain with the long-range interactions: anisotropy-driven multi-criticality

The transverse-field $XY$ chain with the long-range interactions was investigated by means of exact-diagonalization method. algebraic decay rate $\sigma$ interaction is related to effective dimensionality $D(\sigma)$, which governs criticality transverse-field-driven phase transition at $H=H_c$. According large-$N$ analysis, boundary $H_c(\eta)$ exhibits a reentrant behavior within $2<D < 3.065\dots $, as $XY$-anisotropy $\eta$ changes. On one hand, for $D=(2+1)$ and $(1+1)$ short-range magnets, singularities have been determined $H_c(\eta) -H_c(0) \sim |\eta|$ $ 0 respectively, transient around $D \approx 2.5$ remains unclear. As preliminary survey, setting $(\sigma,\eta)=(1 ,0.5)$, we investigate agency fidelity, seems detect singularity $H=H_c$ rather sensitively. Thereby, under \sigma=4/3$ ($D=2.5$), cast fidelity data into crossover-scaling formula properly scaled $\eta$, aiming determine multi-criticality $\eta=0$. Our result indicates that identical magnet, $H_c(\eta)$'s linearity might be retained down $D>2$.