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

TL;DR

We develop a quaternionic formalism for spinful superconductivity in which the spin-singlet/triplet gap is encoded by a single field $q(\boldsymbol{k})$, yielding a compact BdG form $H_{\rm BdG}=\xi_{\boldsymbol{k}}\tau_z+\tau_+ q+\tau_- \overline{q}$ that preserves TRS and AZ class structure. A minimal GL theory is built with a quaternion order parameter $q$ and a vestigial quartet field $Q\propto \mathrm{Sc}(q^2)$, with covariant gauges $D$ for $q$ and $D^{(4e)}$ for $Q$, enabling phase locking $\varphi_Q\approx 2\varphi_q$ and a computable vestigial mass shift $\mu_{\rm eff}=\mu-g^2\Pi(0)$. Topology is formulated in the quaternion frame via quaternionic Berry data, with the Fu–Kane Pfaffian and $\mathbb{Z}_2$/$\mathbb{Z}$ invariants computed directly from $q(\boldsymbol{k})$, and the 3D winding $\nu_3$ characterized by the normalized off-diagonal $\widehat{q}$. Charge-$4e$ order arises from $Q\propto \mathrm{Sc}(q^2)$, leading to halved flux quanta $\Phi=h/(4e)$, a dominant second Josephson harmonic, doubled ac Josephson frequency, and half-period SQUID signatures; these are demonstrated numerically in 2D DIII models, GL simulations of $hc/4e$ vortices, and microscopic lattice systems. Overall, the quaternion framework provides a symmetry-faithful, compact route from microscopic pairing to device-level charge-$4e$ signatures, with clear paths to extensions to multiband/SOC and non-centrosymmetric platforms.

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.

Figures (6)

• Figure 1: Minimum positive Bogoliubov–de Gennes quasiparticle energy $E_{\mathrm{gap}}(\mathbf k)$ for the 2D class-DIII lattice model used in Sec. \ref{['sec:numerics']}. Parameters as in the text. This bulk map is used in Sec. \ref{['sec:topology']} to cross-check the $\mathbb Z_2$ index (bulk–edge correspondence).
• Figure 2: Quaternion GL simulation of a pure quarteting ($4e$) vortex: (a) amplitude and (b) phase (single winding $\simeq 2\pi$). The colorbars quantify the amplitude (arbitrary units) and phase (radians). The halved flux quantum follows from the $4e$ gauge coupling, Eq. \ref{['eq:gauge-4e']}, and flux quantization, Eq. \ref{['eq:flux-quantization-4e']}.
• Figure 3: Josephson signatures of quartet transport. (a) Quartet dominance $I_2\!\gg\! I_1$ produces a strong second harmonic in the CPR [Eqs. \ref{['eq:josephson-energy']}–\ref{['eq:CPR']}]. (b) The AC Josephson frequency doubles to $2f_J$ when the $4e$ channel dominates.
• Figure 4: SQUID critical current modulation $I_c(\Phi)$ for conventional $2e$, quartet-dominant $4e$, and mixed CPRs. Period halving to $\Phi_0/2$ in the $4e$ regime follows from the charge-$4e$ coupling and phase locking, see Secs. \ref{['sec:GL']} and \ref{['sec:4e']}.
• Figure 5: Shapiro step positions for the same drive frequency $f$: the $2e$ regime shows steps at $V_n=n\,hf/2e$, while the $4e$ regime shows steps at $V_n=n\,hf/4e$, i.e. a doubled step count over the same voltage window (Sec. \ref{['sec:4e']}).