The Yu-Shiba-Rusinov (YSR) state appears as a bound state of a quasiparticle at a magnetic atom embedded in a superconductor. We discuss why the YSR state has energy below the superconducting gap and why the pair potential changes the sign at the magnetic atom. Although a magnetic atom in a superconductor has been considered as a pair breaker since the 1960s, we propose an alternative physical picture to explain these reasons. The analytical expression of the Green's function indicates that a magnetic atom converts a spin-singlet $s$-wave Cooper pair into odd-frequency Cooper pairs rather than breaking it and that the odd-frequency pairing correlations coexist with the YSR states below the gap. The relationships among the free-energy density, the amplitudes of pairing correlation functions, and the sign change of the pair potential at a magnetic impurity are discussed utilizing the self-consistent solution of the Eilenberger equation. We conclude that the sign change of the pair potential happens only when the amplitudes of odd-frequency pairing correlations are dominant at the magnetic impurity. In the presence of the local $\pi$-phase shift in the pair potential, odd-frequency pairs can decrease the free-energy density there because their response to a magnetic field is paramagnetic.

中文翻译：

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磁性杂质周围的奇频库珀对

Yu-Shiba-Rusinov (YSR) 状态表现为准粒子在嵌入超导体中的磁性原子处的束缚状态。我们讨论了为什么 YSR 状态的能量低于超导间隙以及为什么对电位会改变磁性原子的符号。尽管自 1960 年代以来，超导体中的磁性原子一直被认为是成对断路器，但我们提出了另一种物理图片来解释这些原因。格林函数的解析表达式表明一个磁性原子转换一个自旋单线态$s$波库珀对变成奇频率库珀对而不是打破它，并且奇频率配对相关性与间隙下方的 YSR 状态共存。利用Eilenberger方程的自洽解，讨论了磁性杂质的自由能密度、配对相关函数的幅值和配对电位的符号变化之间的关系。我们得出结论，只有当奇频率配对相关的幅度在磁性杂质中占主导地位时，才会发生配对电位的符号变化。在当地人面前$\pi$- 对电位的相移，奇数频率对可以降低那里的自由能密度，因为它们对磁场的响应是顺磁性的。