Artificial quantum-dot helium molecules: Electronic spectra, spin structures, and Heisenberg clusters

Abstract

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 four-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.