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