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Vertically coupled double quantum rings submitted to a perpendicular magneticfield $B$ are addressed within the local spin-density functional theory. Wedescribe the structure of quantum ring molecules containing up to 40 electronsconsidering different inter-ring distances and intensities of the appliedmagnetic field. When the rings are quantum mechanically strongly coupled, onlybonding states are occupied and the addition spectrum of the artificialmolecules resembles that of a single quantum ring, with some small differencesappearing as an effect of the magnetic field. Despite the latter has thetendency to flatten the spectra, in the strong coupling limit some clear peaksare still found even when $B\neq 0$ that can be interpretated from thesingle-particle energy levels analogously as at zero applied field, namely interms of closed-shell and Hund's-rule configurations. Increasing the inter-ringdistance, the occupation of the first antibonding orbitals washes out suchstructures and the addition spectra become flatter and irregular. In the weakcoupling regime, numerous isospin oscillations are found as a function of $B$.

Isospin phases of vertically coupled double quantum rings under the influence of perpendicular magnetic fields