We present a full thermodynamic description of a magnetic impurity at the edge of a superconducting wire. We find that, depending on the relative strengths of the impurity and the superconducting bulk couplings, the impurity exhibits four phases: Kondo, Yu-Shiba-Rusinov (YSR) I and II, and local moment. This rich phase diagram is due to the competition between the Kondo effect and superconductivity, and contrary to the expectation that the effects of impurities in gapped hosts are inconsequential, this phase diagram is richer than in the gapless case. We derive the impurity contribution to free energy $F_{\mathrm{imp}}\left(T\right)$ and entropy in each phase: in Kondo phase, the entropy flows monotonically from $ln2$ (ultraviolet) to 0 (infrared) with critical exponents same as that of the conventional Kondo model; in YSR phases, thermal activation of a midgap bound state produces entropy overshoots above $ln2$, saturating to $ln2$ at high $T$, and approaching either 0 or $ln2$ at low $T$ depending on whether impurity is screened or not; in the local-moment phase the impurity remains effectively decoupled, with entropy near $ln2$, with some intermediate-temperature features that progressively fade as the bulk interaction increases. These behaviors, including the entropy overshoots in the YSR and local-moment phases, stem from a splitting of the Hilbert space into distinct excitation towers: one in the Kondo phase, two in YSR I, and three in YSR II and the local-moment phase. Resolving these tower structures and thereby going beyond conventional thermodynamic Bethe ansatz yields closed-form analytic expressions for the impurity contribution to the free energy and entropy across the entire phase diagram.

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• Impurities in superconductors
• Kondo effect
• Quantum field theory
• Quantum thermodynamics
• Bethe ansatz