Unveiling π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi }$$\end{document}-tangle and quantum phase transition in the one-dimensional anisotropic XY model

In this paper, the relationship between π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi }$$\end{document}-tangle and quantum phase transition (QPT) is investigated by employing the quantum renormalization-group method in the one-dimensional anisotropic XY model. The results show that all the 1-tangles increase firstly and then decrease with the anisotropy parameter γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} increasing, and the Coffman–Kundu–Wootters monogamy inequality is always tenable for this system. The entanglement’s status of subsystems depends on its site position, and this proposition can be generalized to a multipartite system. Meanwhile, with the increasing of the size of the system, the π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi }$$\end{document}-tangle decreases slowly and tends to a fixed value finally. Additionally, it exhibits a QPT and a maximum value for the next-nearest-neighbor entanglement at the critical point in our model, which is different from the case of two-body system. After several iterations of the renormalization, the quantum entanglement measure can develop two saturated values, which are associated with two different phases: spin-fluid phase and the Néel phase. To gain further insight, the nonanalytic and scaling behaviors of π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi }$$\end{document}-tangle have also been analyzed in detail.