Invariant magnetic helicity of...

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In this paper a theoretical studies of the effective Hamiltonian of nanohelices of graphite are presented. For this aim the solution of the three dimensional Schrödinger equations for electrons of nanohelices in graphite in magnetic field is found. The three-dimensional Schrödinger equation which is separated on one-dimensional and two-dimensional equations by separated of variables are shown to be concerned with the effective Hamiltonian of nanohelices in graphite by apply Bogoliubov transformation. The one-dimensional Schrödinger equation are shown to be concerned with the momentum anisotropy of electrons of nanohelices in graphite and the law of conservation of helicity is found. The law of conservation of helicity are shown to be related with the invariance comparatively the symmetry group of $D_{6h}$.

Landau quantization $\epsilon=\hbar\omega_{c}m\pm\frac{\delta'}{2}\sqrt{1+\frac{4\hbar^{2}\omega_{c }^{2}\beta^{2}m}{\delta'^{2}}}$ for fermions from mass chiral Dirac equation for the two-dimensional direction with spin-orbit interactions in a second quantization is found exactly. Quantum Hall effect in two-dimensional materials with spin-orbit interactions is predicted.

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