Spin-dependent rotating Wigner molecules in quantum dots

The spin-dependent trial wave functions with rotational symmetry are introduced to describe rotating Wigner molecular states with spin degree of freedom in four- and five-electron quantum dots under magnetic fields. The functions are constructed with unrestricted Hartree-Fock orbits and projection technique in long-range interaction limit. They highly overlap with the exact-diagonalized ones and give the accurate energies in strong fields. The zero points, i.e. vortices of the functions have straightforward relations to the angular momenta of the states. The functions with different total spins automatically satisfy the angular momentum transition rules with the increase of magnetic fields and explicitly show magnetic couplings and characteristic oscillations with respect to the angular momenta. Based on the functions, it is demonstrated that the entanglement entropies of electrons depend on the z-component of total spin and rise with the increase of angular momenta.