  The influence of strong electron-electron interactions and Wigner-molecule
(WM) formation on the spectra of $2e$ singlet-triplet double-dot Si qubits is
presented based on a full configuration interaction (FCI) approach that
incorporates the valley degree of freedom (VDOF) in the context of the
continuous (effective mass) description of semiconductor materials. Our FCI
treats the VDOF as an isospin in addition to the regular spin. Our treatment is
able to assign to each energy curve in the qubit's spectrum a complete set of
good quantum numbers for both the spin and the valley isospin. This reveals an
underlying SU(4) $\supset$ SU(2) $\times$ SU(2) group-chain organization in the
Si double-dot spectra. With parameters in the range of actual experimental
situations, we demonstrate in a double-dot qubit that, in the (2,0) charge
configuration and compared to the expected large, and dot-size determined,
single-particle (orbital) energy gap, the strong $e-e$ interactions drastically
quench the spin-singlet$-$spin-triplet energy gap, $E_{\rm ST}$, within the
same valley, making it competitive to the small energy gap, $E_V$, between the
two valleys. We present results for both the $E_{\rm ST} < E_V$ and $E_{\rm ST}
> E_V$ cases. We investigate the spectra as a function of detuning and
demonstrate the strengthening of the avoided crossings due to a lowering of the
interdot barrier and/or the influence of valley-orbit coupling. We further
demonstrate, as a function of an applied magnetic field, the emergence of
avoided crossings in the (1,1) charge configuration due to the spin-valley
coupling. The valleytronic FCI formulated here, and implementeded for two
electrons confined in a tunable double quantum dot, offers also a most
effective tool for analyzing the spectra of Si qubits with more than two wells
and/or more than two electrons.

Mesoscale and Nanoscale Physics;Nuclear Theory;Quantum Physics