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

Yoshihiro Nishiyama

doi:10.1140/epjb/s10051-021-00245-1

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2024 Impact factor 1.7

Condensed Matter and Complex Systems

Eur. Phys. J. B (2021) 94: 226
https://doi.org/10.1140/epjb/s10051-021-00245-1

Regular Article - Statistical and Nonlinear Physics

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

Yoshihiro Nishiyama^a

Department of Physics, Faculty of Science, Okayama University, 700-8530, Okayama, Japan

^a nisiyama@science.okayama-u.ac.jp

Received: 30 July 2021
Accepted: 9 November 2021
Published online: 19 November 2021

Abstract

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