Microscopic determination of intervalley-Umklapp phase pinning

Determine the microscopic intervalley Umklapp pinning potential and the resulting depinning threshold for the intervalley-coherent phase in the superconducting $S|\mathrm{IVC}|S$ junction, by treating intervalley Umklapp processes explicitly.

Background

The paper models the domain-wall region between time-reversal-related chiral superconductors as an intervalley-coherent state with phase $\theta$. Josephson coupling depends on the gauge-invariant combination $\Phi=\varphi-2\theta$, while a three-electron Umklapp process generates a pinning term proportional to $\cos(3\theta)$. Consequently, the junction has both a conventional Josephson critical current and an intervalley-coherent depinning current.

The relative magnitudes of these currents determine whether the junction remains dissipationless or develops a voltage through dynamical precession of the intervalley-coherent phase. The paper does not microscopically calculate the Umklapp-induced pinning energy $E_U$ or the associated depinning threshold, leaving their determination as an explicit future problem.