Ginzburg-Landau approximation for the Hubbard model in the external magnetic field

The Hubbard model is studied in the external magnetic field. The analysis is carried out phenomenologically within the framework of the Ginzburg-Landau theory with the order parameter describing the opposite spin electrons. The study is performed for the nearly half-filled lower Hubbard band in the metallic state. The final equations are the Pauli-like ones for the opposite spins and nonlinear as a result of interaction between electrons with the opposite spins. The equations can analytically be solved for the spatially homogeneous distributions in a number of most interesting cases. In particular, the problem on the metal-insulator transition is analyzed for the nearly half-filled Hubbard sub-bands. The critical magnetic field at which the transition from the metallic state to the insulator one takes place is found under the paramagnetic spin effect.