标题： 一维超导体边缘的量子杂质中的热力学：分裂希尔伯特空间 显示英文标题

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

摘要： 我们提出了一种对超导线边缘单个磁性杂质的热力学描述。 该杂质表现出四个相 $\unicode{x2014}$ Kondo，Yu-Shiba-Rusinov (YSR) I 和 II，以及局域磁矩 $\unicode{x2014}$ 相图比无能隙情况更丰富，这与人们预期的在有能隙基质中的杂质效应不那么重要相反。 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 $\delta \to 0$. 这些行为，包括YSR和局域磁矩相中的熵过冲，源于希尔伯特空间的分裂成不同的激发塔：在Kondo相中有一个，在YSR~I中有两个，在YSR~II和局域磁矩相中有三个。 解析这些塔结构，并由此超越传统的TBA，可以得到杂质对自由能和熵在整个相图中的闭式解析表达式。 显示英文摘要

摘要： 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.