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

Abstract

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.