Intrinsic anomalous, spin and valley Hall effects in 'ex-so-tic' van-der-Waals structures

We consider the anomalous, spin, valley, and valley spin Hall effects in a pristine graphene-based van-der-Waals (vdW) heterostructure consisting of a bilayer graphene (BLG) sandwiched between a semiconducting van-der-Waals material with strong spin-orbit coupling (e.g., WS2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {WS}_2$$\end{document}) and a ferromagnetic insulating vdW material (e.g. Cr2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Cr}_2$$\end{document}Ge2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Ge}_2$$\end{document}Te6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Te}_6$$\end{document}). Due to the exchange proximity effect from one side and spin-orbit proximity effect from the other side of graphene, such a structure is referred to as graphene based 'ex-so-tic' structure. First, we derive an effective Hamiltonian describing the low-energy states of the structure. Then, using the Green's function formalism, we obtain analytical results for the Hall conductivities as a function of the Fermi energy and gate voltage. For specific values of these parameters, we find a quantized valley Hall conductivity.