Kavli Affiliate: Anton Akhmerov
| Summary:
Beenakker’s formula, $U(varphi)-U(0)=Δsum_nbigl(1-sqrt1-τ_nsin^2(varphi/2)bigr)$, 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 $-1leα_rle7/80$ at finite temperature. We also construct a four-site junction with $α_r>0$, which disproves the conjectured upper bound $α_rle0$. So that these results can be trusted without repeating the derivations, every numbered equation is formally verified in the Lean proof assistant.
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