Exactly pairing two-dimensional charged particles using a magnetic field

It is demonstrated that a uniform magnetic field can exactly pair the two-dimensional (2D) charged particles only for some quantized magnetic intensity values. For the particle-pair consisting of two like charges the Landau level of the center-of-mass motion is multiple degenerate that implies the shell structure. However, any particle-pair has only two non-degenerate relative levels, which are associated with the diamagnetic and paramagnetic states respectively. There exist a upper critical magnetic strength and a lower critical magnetic length across which two like charges cannot be paired. The theoretical results agree with the experimental data on the sites and widths of the integral and fractional quantum Hall plateaus in a CaAs-Al$_x$Ga$_{1-x}$As heterojunction, that gives a new explanation for the quantum Hall effect.