  Energy spectra and spin configurations of a system of N=4 electrons in
lateral double quantum dots (quantum dot Helium molecules) are investigated
using exact diagonalization (EXD), as a function of interdot separation,
applied magnetic field (B), and strength of interelectron repulsion. As a
function of the magnetic field, the energy spectra exhibit a low-energy band
consisting of a group of six states, with the number six being a consequence of
the conservation of the total spin and the ensuing spin degeneracies for four
electrons. The energies of the six states appear to cross at a single value of
the magnetic field, and with increasing Coulomb repulsion they tend to become
degenerate, with a well defined energy gap separating them from the
higher-in-energy excited states. The appearance of the low-energy band is a
consequence of the formation of a Wigner supermolecule, with the four electrons
(two in each dot) being localized at the vertices of a rectangle. Using
spin-resolved pair-correlation distributions, a method for mapping the
complicated EXD many-body wave functions onto simpler spin functions associated
with a system of four localized spins is introduced. Detailed interpretation of
the EXD spin functions and EXD spectra associated with the low-energy band via
a 4-site Heisenberg cluster (with B-dependent exchange integrals) is
demonstrated. Aspects of spin entanglement, referring to the well known N-qubit
Dicke states, are also discussed.
