Effect of three-spin interaction on thermal entanglement in Heisenberg XXZ model

The effect of three-spin interaction k on thermal entanglement between alternate qubits is studied using pairwise concurrence C and energy-level diagram. It is found that k breaks the symmetry about the effect of magnetic field h on C. It shifts a dip structure and gradually effaces a boot structure when k<J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| k \right| <\left| J \right| $$\end{document} (J is spin exchange coupling). A peak with C maintains 1 appears and expands, elbowing the dip backwards when k>J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| k \right| >\left| J \right| $$\end{document}. A sudden change in the concurrence occurs around k=J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| k \right| =\left| J \right| $$\end{document}, h=-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=-k$$\end{document}. Similar conclusions about nearest-neighbor qubits are directly given.