Quantum Entanglement in Heisenberg Model with Dzyaloshinskii-Moriya Interactions

Using the two-site cluster mean-field method and the concept of negativity, the magnetization and entanglement of spin-1 quantum ferromagnetic Heisenberg model with Dzyaloshinskii-Moriya (DM) interactions on d-dimensional (d=1,2,3,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1,2,3,4$$\end{document}) lattices are studied. The phase transitions and the variations of the negativity with temperature, anisotropy and DM interaction parameters are obtained. It is found that in the systems there are both second-order, first-order phase transitions and tricritical points. For the one-dimensional system, there is a maximum value of the negativity at a certain temperature which corresponds to the phase transition point for the case of first-order phase transition, and the maximum value increases with the increase of the DM interaction intensity. We also find that for two-dimensional square lattice with different temperature values, negativity increases with increasing DM interaction, and finally approaches to the same value. There is a lower limit of the DM interaction intensity (or temperature) above which negativity exists. In addition, we discuss the effect of the dimension on the magnetization, negativity and tricritical point. The results show that the tricritical temperature is independent of the exchange anisotropy parameter Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta$$\end{document}, and the lower the dimension, the more obvious the quantum effect.