Abstract
We present a thermodynamic description of a single magnetic impurity at the edge of a superconducting wire. The impurity exhibits four phases x2014 Kondo, Yu-Shiba-Rusinov (YSR) I and II, and local moment x2014 a phase diagram richer than in the gapless case, contrary to the expectation that the effects of impurities in gapped hosts are less consequential. We derive the impurity contribution to free energy F_ imp(T) and entropy in each phase: in Kondo phase, the entropy flows monotonically from ln 2 (UV) to 0 (IR) 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 ln 2, saturating to ln 2 at high T and approaching either 0 or ln 2 at low T depending on whether impurity is screened or not; in the local-moment phase the impurity remains effectively decoupled, with entropy near ln 2, with some intermediate-temperature features that progressively fade as δ → 0. 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 TBA yields closed-form analytic expressions for the impurity contribution to the free energy and entropy across the entire phase diagram.