Transmission representation of the Josephson free energy beyond the short-junction limit

Beenakker’s formula, $U(\phi )-U(0)=\mathrm{\Delta }{\sum }_n(1-\sqrt{1-{\tau }_n{\sin }^2(\phi \mathrm{/}2)})$, expresses the zero-temperature energy-phase relation of a short Josephson junction through its normal-state transmission eigenvalues. We generalize this transmission representation beyond the short-junction limit and express the phase-dependent free energy of a time-reversal-symmetric $s$-wave junction as a Matsubara sum over channels with frequency-dependent transmissions. The representation constrains the small-phase expansion of the free energy. In particular, we prove that the normalized ratio of its quartic and quadratic coefficients satisfies $-1\le {\alpha }_r\le 7\mathrm{/}80$ at finite temperature. We also construct a four-site junction with ${\alpha }_r>0$, which disproves the conjectured upper bound ${\alpha }_r\le 0$. So that these results can be trusted without repeating the derivations, every numbered equation is formally verified in the Lean proof assistant.