We recast spinful superconductivity as a \textit{ quaternion field theory}, where a quaternion is a four-component hypercomplex number with units $(\boldsymbol{ e}_x,\boldsymbol{ e}_y,\boldsymbol{ e}_z)$, that encodes the spin-singlet/triplet gap in a single field $q(\mathbf{ k})$. This yields a compact Bogoliubov-de Gennes (BdG) Hamiltonian $H_{ \rm BdG}=\xi_{ \mathbf{ k}}\tau_z+\tau_+q+\tau_-\,q^\ddagger$ and keeps time-reversal symmetry, Altland-Zirnbauer classification, and topological diagnostics in the same variables. We introduce a quarteting field $Q\!\propto\!\mathrm{ Sc}(q^2)$ and a minimal Ginzburg-Landau (GL) functional with covariant derivatives $(\nabla-2ie\mathbf A)q$ and $(\nabla-4ie\mathbf A)Q$. Analytically, a one-loop evaluation of the fluctuation bubble $\Pi(0)$ gives a quantitative vestigial charge-$4e$ criterion $\mu_{ \rm eff}=\mu-g^2\Pi(0)<0$. Numerically, we verified: (i) a two-dimensional (2D) class-DIII lattice model whose $\mathbb{ Z}_2$ index, computed from the occupied BdG eigenvectors via the standard sewing-matrix (Pfaffian) construction at time-reversal-invariant momenta, matches helical edge spectra; (ii) a GL simulation of a pure-$Q$ vortex carrying $h/4e$ flux within $\sim2\%$ and exhibiting $\xi_Q\!\propto\!\sqrt{ \eta/|\mu_{ \rm eff}|}$; and (iii) a short-junction current-phase relation with a controlled window where the second harmonic dominates ($I_2\!\gg\!I_1$), together with doubled alternating-current Josephson emission and a Shapiro response consistent with $4e$-dominated transport. The framework provides a compact, symmetry-faithful route from microscopic pairing to device-level charge-$4e$ signatures.
Categories: cond-mat.supr-con