We study a cluster of quantum dots defined within silicene that host confined electron states with spin and valley degrees of freedom. Atomistic tight-binding and continuum Dirac approximation are applied for few-electron system in quest for spontaneous valley polarization driven by inter-dot tunneling and electron-electron interaction, i.e. a valley counterpart of itinerary Nagaoka ferromagnetic ordering recently identified in GaAs square cluster of quantum dots with three excess electrons [P. Dehollain, et al., Nature 579, 528 (2020)]. We find that for Hamiltonian without intrinsic-spin orbit coupling -similar to the one of graphene with staggered potential -the valley polarization in the ground-state can be observed in a range of inter-dot spacing provided that the spin of the system is frozen by external magnetic field. The inter-valley scattering effects are negligible for cluster geometry that supports the valley polarized ground-state. In presence of a strong intrinsic spin-orbit coupling that is characteristic to graphene no external magnetic field is necessary for observation of ground-state that is polarized in both spin and valley. The effective magnetic field due to the spin-orbit interaction produces a perfect anticorrelation of the spin and valley isospin components in the degenerate ground-state.

We consider states bound at the flip of the electric field in buckled silicene. Along the electric flip lines a topological confinement is formed with the orientation of the charge current and the resulting magnetic dipole moment determined by the valley index. We compare the topological confinement to the trivial one that is due to a local reduction of the vertical electric field but without energy gap inversion. For the latter the valley does not protect the orientation of the magnetic dipole moment from inversion by external magnetic field. We demonstrate that the topologically confined states can couple and form extended bonding or antibonding orbitals with the energy splitting influenced by the geometry and the external magnetic field.