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 σ of the long-range interaction is related to the effective dimensionality D(σ) , 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(η) exhibits a reentrant behavior within 2 < D< 3.065 … , as the XY-anisotropy η 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(η) - H_c(0) ∼ | η| and 0, respectively, and the transient behavior around D≈ 2.5 remains unclear. As a preliminary survey, setting (σ, η) = (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 σ= 4 / 3 (D= 2.5), we cast the fidelity data into the crossover-scaling formula with the properly scaled η, aiming to determine the multi-criticality around η= 0. Our result indicates that the multi-criticality is identical to that of the D= (2 + 1) magnet, and H_c(η) ’s linearity might be retained down to D> 2.