As well known, the integer and fractional quantum Hall effect...

The spin polarization of a fractional quantum Hall state shows very interesting properties. The curve of polarization versus magnetic field has wide plateaus. The fractional quantum Hall effect is caused by the Coulomb interaction because the 2D electron system without the Coulomb interac-tion yields no energy gap at the fractional filling factor. Therefore, the wide plateau in the polari-zation curve is also caused by the Coulomb interaction. When the magnetic field is weak, some electrons have up-spins and the others down-spins. Therein the spin-exchange transition occurs between two electrons with up and down spins via the Coulomb interaction. Then the charge dis-tribution before the transition is the same as one after the transition. So these two states have the same classical Coulomb energy. Accordingly, the partial Hamiltonian composed of the spin ex-change interaction should be treated exactly. We have succeeded in diagonalizing the spin ex-change interaction for the first and second nearest electron pairs. The theoretical results repro-duce the wide plateaus very well. If the interval modulations between Landau orbitals are taken into the Hamiltonian, the total energy has the Peierls instability. We can diagonalize the Hamilto-nian with the interval modulation. The results reproduce wide plateaus and small shoulder