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Geometrical crossover in two-body systems in a magnetic field

An algebraic approach is formulated in the harmonic approximation to describe a dynamics of two-fermion systems, confined in three-dimensional axially symmetric parabolic potential, in an external magnetic field. The fermion interaction is considered in the form U_(M)(r)= alpha_(M)r^{-M} (alpha_(M)>0, M>0). The formalism of a semi-simple Lie group is applied to analyse symmetries of the considered system. Explicit algebraic expressions are derived in terms of system's parameters and the magnetic field strength to trace the evolution of the equilibrium shape. It is predicted that the interplay of classical and quantum correlations may lead to a quantum shape transition from a lateral to a vertical localization of fermions in the confined system. The analytical results demonstrate a good agreement with numerical results for two-electron quantum dots in the magnetic field, when classical correlations dominate in the dynamics.