Quaternionic superconductivity with a single-field Bogoliubov-de Gennes--Ginzburg-Landau framework and charge-4e couplings

Abstract: We recast spinful superconductivity as a quaternion field theory -- where a quaternion is a four-component hypercomplex number with units $(\boldsymbol{i},\boldsymbol{j},\boldsymbol{k})$ -- that encodes the spin-singlet/triplet gap in a single field $q(\mathbf{k})$. This yields a compact Bogoliubov--de Gennes (BdG) Hamiltonian $H_{\rm BdG}=ξ{\mathbf{k}}τ_z+τ{+}q+τ{-}\,\overline{q}$ 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 $Π(0)$ (with prefactors) gives a quantitative vestigial charge-$4e$ criterion $μ{\rm eff}=μ-g^2Π(0)<0$. Numerically, we verify: (i) a two-dimensional (2D) class-DIII lattice model whose $\mathbb{Z}2$ index, computed directly from $q(\mathbf{k})$ using the matrix Pfaffian of an antisymmetric sewing matrix at time-reversal-invariant momenta (TRIM), matches helical edge spectra; (ii) a GL simulation of a pure-$Q$ vortex carrying $hc/4e$ flux within $\sim2\%$ and exhibiting $ξ_Q\propto\sqrt{η/|μ{\rm eff}|}$; 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 (ac) Josephson emission and even-only Shapiro steps. The framework provides a compact, symmetry-faithful route from microscopic pairing to device-level charge-$4e$ signatures.