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arXiv:2508.19330 · Physical review. B./Physical review. B

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

Pradip Kattel, Abay Zhakenov, Natan Andrei

Published May 11, 2026• 1 citation
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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 $δ\to 0$. 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.

DOI
10.1103/qw2d-k8wv

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We present a thermodynamic description of a single magnetic impurity at the edge of a superconducting wire. The impurity exhibits four phases \unicodex2014\unicode{x2014}\unicodex2014 Kondo, Yu-Shiba-Rusinov (YSR) I and II, and local moment \unicodex2014\unicode{x2014}\unicodex2014 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 Fimp(T)F_{\rm imp}(T)Fimp​(T) and entropy in each phase: in Kondo phase, the entropy flows monotonically from ln⁡2\ln 2ln2 (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\ln 2ln2, saturating to ln⁡2\ln 2ln2 at high TTT and approaching either 0 or ln⁡2\ln 2ln2 at low TTT depending on whether impurity is screened or not; in the local-moment phase the impurity remains effectively decoupled, with entropy near ln⁡2\ln 2ln2, with some intermediate-temperature features that progressively fade as δ→0δ\to 0δ→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 YSRI, and three in YSRII 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.
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Boundary phases and thermodynamics of the Kondo spin-sss chain: from overscreened Kondo to boundary-bound states
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Entanglement entropy (EE) is commonly studied using real-space bipartitions. We show that, in quantum impurity models, an energy-space bipartition, equivalent to the momentum-space bipartition of the bath, can display universal behavior. Motivated by poor man's scaling, we logarithmically discretize the bath and partition it into high- and low-energy sectors. For models with Fermi-liquid fixed points, including the Anderson model and fully screened or underscreened Kondo models, the low-energy EE flows to constants independent of model parameters. These constants are integer multiples of ln⁡2\ln 2ln2 plus corrections that depend only on the logarithmic discretization parameter ΛΛΛ. We show that scale invariance of the fixed-point wavefunction in energy space maps to effective translation invariance along a one-dimensional chain, allowing the fixed points to be classified by one-dimensional topological band theory. With low-energy chiral symmetry, each ln⁡2\ln 2ln2 contribution originates from a topological edge mode. We also study transitions between distinct Fermi-liquid fixed points using the local-singlet--Kondo-singlet transition in a two-orbital Anderson model driven by an inter-orbital antiferromagnetic coupling. The local-singlet phase has an effectively decoupled impurity and nearly vanishing EE, whereas the Kondo-singlet phase has finite EE larger than ln⁡2\ln 2ln2 per spin and orbital. When chiral symmetry holds at low energies, this distinction corresponds to a topological transition of the effective bath chain. At the non-Fermi-liquid critical point, the EE develops an unstable plateau. Its ΛΛΛ dependence resembles that of the overscreened two-channel Kondo model, supporting universality within the same non-Fermi-liquid universality class.
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