Thermodynamics in a split Hilbert space: Quantum impurity at the edge of a one-dimensional superconductor

TL;DR

This paper provides a comprehensive thermodynamic analysis of a magnetic impurity at the edge of a one-dimensional superconductor, revealing complex phase behavior and entropy dynamics across multiple impurity phases.

Contribution

It introduces a novel thermodynamic framework that captures the impurity's phase diagram and entropy behavior, including entropy overshoots, by resolving Hilbert space structures beyond traditional methods.

Findings

Identifies four impurity phases with distinct thermodynamic signatures.

Derives closed-form expressions for free energy and entropy in all phases.

Reveals entropy overshoots due to excitation tower splitting.

Abstract

We present a thermodynamic description of a single magnetic impurity at the edge of a superconducting wire. The impurity exhibits four phases $\unicode{x2014}$ Kondo, Yu-Shiba-Rusinov (YSR) I and II, and local moment $\unicode{x2014}$ 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 $F_{\rm imp}(T)$ 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 $\delta \to 0$. 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.