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The $SU(4)-SU(2)$ crossover, driven by an external magnetic field $h$, isanalyzed in a capacitively-coupled double-quantum-dot device connected toindependent leads. As one continuously charges the dots from empty toquarter-filled, by varying the gate potential $V_g$, the crossover starts whenthe magnitude of the spin polarization of the double quantum dot, as measuredby $\langle n_{\uparrow}\rangle -\langle n_{\downarrow}\rangle$, becomesfinite. Although the external magnetic field breaks the $SU(4)$ symmetry of theHamiltonian, the ground state preserves it in a region of $V_g$, where $\langlen_{\uparrow}\rangle -\langle n_{\downarrow}\rangle =0$. Once the spinpolarization becomes finite, it initially increases slowly until a suddenchange occurs, in which $\langle n_{\downarrow}\rangle$ (polarization directionopposite to the magnetic field) reaches a maximum and then decreases tonegligible values abruptly, at which point an orbital $SU(2)$ ground state isfully established. This crossover from one Kondo state, with emergent $SU(4)$symmetry, where spin and orbital degrees of freedom all play a role, toanother, with $SU(2)$ symmetry, where only orbital degrees of freedomparticipate, is triggered by a competition between $g\mu_Bh$, the energy gainby the Zeeman-split polarized state and the Kondo temperature $T_K^{SU(4)}$,the gain provided by the $SU(4)$ unpolarized Kondo-singlet state.

The $SU(4)-SU(2)$ crossover and spin filter properties of a double quantum dot nanosystem