Transmission representation of the Josephson free energy beyond the short-junction limit
Abstract
Beenakker's formula, U(φ)−U(0)=Δ∑n(1−√(1−τnsin2(φ/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≤αr≤7/80 at finite temperature. We also construct a four-site junction with αr>0, which disproves the conjectured upper bound αr≤0. 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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BibTeX
@misc{akhmerov2026transmission,
title = {{Transmission representation of the Josephson free energy beyond the short-junction limit}},
author = {Anton R. Akhmerov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18980},
eprint = {2610.18980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18980},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Transmission representation of the Josephson free energy beyond the short-junction limit
AU - Anton R. Akhmerov
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18980
UR - https://arxiv.org/abs/2610.18980
AB - Beenakker's formula, U(φ)−U(0)=Δ∑_n(1−√(1−τ_nsin^2(φ/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≤α_r≤7/80 at finite temperature. We also construct a four-site junction with α_r>0, which disproves the conjectured upper bound α_r≤0. So that these results can be trusted without repeating the derivations, every numbered equation is formally verified in the Lean proof assistant.
ER -