Giant nature optical activity coupled with incommensurate phase transition and integer quantum Hall effect in Weil 2D half layered materials: Group-VI Dichalcogenides and Rb_{2}ZnBr_{4}, Ca_{2}RuO_{4}, Cr_{2}O_{3}, MnTiO_{3}, La-doped BiFeO_{3}, PtCl_{3}

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In the article we have found that the Weyl Hall semiconductor (WHS) states allow the topological phase transition happened between two quantum anomalous Hall (QAH) insulator phase with opposite Chern numbers and we have based on phonon dispersion of Weil 2D half semiconductor monolayers group like PtCl_{3}. Landau-Ginzburg-Devonshire theory of thin ferroelectric polar-active nanofilms in incommensurate phases and semiconductor heterostructures is presented. The self-consistent solutions of the Euler-Lagrange equation for the polarization vector and the Maxwell equations for light which propagates along Oz axis in thin ferroelectric polar-active nanofilms have been found. Quantized solutions of one-dimensional Maxwell equations for thin ferroelectric films in Incommensurate phase with space dispersion have been specified. The analytical solutions of the Maxwell wave equations as well as natural optical gyrotropy effects are found in Rb_{2}ZnBr_{4} as well as K_{2}SeO_{4} Incommensurate phases crystals connected with giant light velocity as well as via interaction with coherent phonon oscillations. In the framework of the superspace symmetry group theories the Maxwell wave equations are solved which are shown to be connected with the symmetry group of D_{2h}^{16} or isomorphic groups. In the paper the non-zero gyration g_{33} and gyrotropic birefringence epsilon_{12} tensors of K_{2}SeO_{4} and Rb_{2}ZnBr_{4} materials based on D_{2h}^{16} space symmetry group were found. The values of natural optical gyrotropy as well as Rashba spin splitting are shown to be specified like (k^{(0)}\pm\tilde{k}^{(2)})^{2} as displacements of two symmetrically allocated parabolas from Brillouin zone center. Hence if sine-type edges of Group-VI Dichalcogenides and Rb_{2}ZnBr_{4}, Ca_{2}RuO_{4}, Cr_{2}O_{3}, MnTiO_{3}, La-doped BiFeO_{3}, PtCl_{3}, PdBr_{3}, RuCl_{3}, PtI_{3}, and carbon nanoribbons may be approximated armchair or zigzag edges then in the sections 5 are found that structure phase transitions to be related with giant spin-orbit interaction (SOI). In the article a creation of giant spin-orbit splitting (2.5 eV) coupled with the carbon Dirac cones (K and K') are shown to be related with structure phase transition in carbon nanoribbon with armchair or zigzag edges of honeycomb lattice of Group-VI Dichalcogenides and Rb_{2}ZnBr_{4}, Ca_{2}RuO_{4}, Cr_{2}O_{3}, MnTiO_{3}, La-doped BiFeO_{3}, PtCl_{3}, PdBr_{3}, RuCl_{3}, PtI_{3}, and carbon nanoribbons. In the article the natural optical gyrotropy effects are shown to be found with light velocity like epsilon=hbar ck/eV=14.0798 eV, epsilon=hbar ck/eV=27.6009 eV, epsilon=hbar ck/eV=7.5726 eV, with the corresponding considered in Tables 4,5,6 of gyrotropic birefringences Delta n_{11} and gyrotropies g.

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