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=Hc. According to the large-N analysis, the phase boundary Hc(eta) exhibits a reentrant behavior within 2<D<3.065 horizontal ellipsis , 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 Hc(eta)-Hc(0)similar to vertical bar eta vertical bar and 0, respectively, and the transient behavior around D approximate to 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.