Fractional Quantum Hall Effect at researches of nanosize and Landau quantized quasiparticle solutions of Schrödinger boundary problem of domain wall for data storage of thin SmBaMn_{2}O_{6} nanofilms at contact with graphene channel with magnetic field

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In the article we have found the analytical chiral solutions of stationary Schrödinger boundary problem for domain wall for data storage of nanosize PmBaMn_{2}O_{6} as well as SmBaMn_{2}O_{6} ferroic in magnetic field applied in Landau symmetric gauge with incommensurately modulated sinusoidal polarization waves Q_{T}, Q_{P}, Q_{b}. The computer solution of quantized energies or wave vectors of stationary inhomogeneous Schrödinger boundary problem for Q_{T}, Q_{P}, Q_{b} with sinusoidal depolarization waves for the thin PmBaMn_{2}O_{6} as well as SmBaMn_{2}O_{6} in magnetic field applied in symmetric gauge ferroic nanofilms are specified. We have presented the computation solutions of quantized normalized charges densities of the Eq. (130) stationary Schrödinger boundary problem for different thickness h of PmBaMn_{2}O_{6} as well as SmBaMn_{2}O_{6} in magnetic field applied in symmetric gauge ferroelectric (FE) nanofilms. We have found the computation solutions of order parameters of ferroelectric phase transition and applicable that Fractional Quantum Hall Effect (FQHE) in the form of nanosize quantized as well as Landau quantized hysteresis loops that were clearly visible like Fairy-tale Phoenix. We have found the quantized out-of-plane polarizations P_{3} dispersion for LSMO/BFO interface of polar-active ferroelectric nanofilm with semiconductor with different boundary parameters. The thin Os_{x}Ra_{1-x}FeO_{3} nanofilms have quasiparticle excitations that can be described by (2+1)-dimensional Dirac theory from developed a four-sublattice model (FSM) for the analytical description of A-cation displacements in (anti)ferroelectric-antiferrodistortive perovskites of ABO_{3} type. We have found that FSM explains the coexistence of rhombohedral (R), orthorhombic (O) and spatially modulated phases observed by atomic-resolution scanning transmission electron microscopy (STEM) in Os-doped RaFeO_{3}. Using this approach we atomically resolve the theoretical model of the sublattice asymmetry inherent to the case of the A-site Os/Ra cation sublattice in Os_{x}Ra_{1-x}FeO_{3} polymorphs. We have shown that this produces an unconventional form of the quantized Hall conductivity sigma_{xy}=-(2e^{2}/h)(2n+1) with n=0,1,2,... This unconventional quantization is caused by the analytical chirality of n=0 Landau level.

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