  The $SU(4)-SU(2)$ crossover, driven by an external magnetic field $h$, is
analyzed in a capacitively-coupled double-quantum-dot device connected to
independent leads. As one continuously charges the dots from empty to
quarter-filled, by varying the gate potential $V_g$, the crossover starts when
the magnitude of the spin polarization of the double quantum dot, as measured
by $\langle n_{\uparrow}\rangle -\langle n_{\downarrow}\rangle$, becomes
finite. Although the external magnetic field breaks the $SU(4)$ symmetry of the
Hamiltonian, the ground state preserves it in a region of $V_g$, where $\langle
n_{\uparrow}\rangle -\langle n_{\downarrow}\rangle =0$. Once the spin
polarization becomes finite, it initially increases slowly until a sudden
change occurs, in which $\langle n_{\downarrow}\rangle$ (polarization direction
opposite to the magnetic field) reaches a maximum and then decreases to
negligible values abruptly, at which point an orbital $SU(2)$ ground state is
fully established. This crossover from one Kondo state, with emergent $SU(4)$
symmetry, where spin and orbital degrees of freedom all play a role, to
another, with $SU(2)$ symmetry, where only orbital degrees of freedom
participate, is triggered by a competition between $g\mu_Bh$, the energy gain
by the Zeeman-split polarized state and the Kondo temperature $T_K^{SU(4)}$,
the gain provided by the $SU(4)$ unpolarized Kondo-singlet state.
