Two-Dimensional Electron Gas in a Periodic Potential and External Magnetic Field: States of Pairs and Three-Particle Systems

The group-theoretical classification of states of identical particle pairs is presented. Then obtained states are coupled with those of an antiparticle to construct states of a three-particle system. Investigations are performed using products of irreducible projective representations of the 2D translation group. For a given Born-von Kármán period N degeneracy of pair states is N, whereas three-particle states are N$\text{}^{2}$-fold degenerated. It has to be underlined that the case of even N is more complicated since pair states are labelled by four inequivalent irreducible projective representations. The problem of symmetry properties with respect to particles transposition is briefly discussed.