Numerical study of a quantum spin in an s-wave superconductor using the natural orbitals renormalization group method

doi:10.1088/1674-1056/adf4af

中国物理B ›› 2026, Vol. 35 ›› Issue (2): 27104-027104.doi: 10.1088/1674-1056/adf4af

Numerical study of a quantum spin in an s-wave superconductor using the natural orbitals renormalization group method

Wen-Jing Zhang(张文静)¹, Ru Zheng(郑汝)^1,†, Rong-Qiang He(贺荣强)^2,3, and Zhong-Yi Lu(卢仲毅)^2,3,4,‡

1 School of Physical Science and Technology, Ningbo University, Ningbo 315211, China;
2 School of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of China (RUC), Beijing 100872, China;
3 Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, China;
4 Hefei National Laboratory, Hefei 230088, China

收稿日期:2025-06-06 修回日期:2025-07-25 接受日期:2025-07-28 发布日期:2026-02-05
通讯作者: Ru Zheng, Zhong-Yi Lu E-mail:zhengru@nbu.edu.cn;zlu@ruc.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 12104247 and 11934020).

Numerical study of a quantum spin in an s-wave superconductor using the natural orbitals renormalization group method

Wen-Jing Zhang(张文静)¹, Ru Zheng(郑汝)^1,†, Rong-Qiang He(贺荣强)^2,3, and Zhong-Yi Lu(卢仲毅)^2,3,4,‡

1 School of Physical Science and Technology, Ningbo University, Ningbo 315211, China;
2 School of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of China (RUC), Beijing 100872, China;
3 Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin University of China, Beijing 100872, China;
4 Hefei National Laboratory, Hefei 230088, China

Received:2025-06-06 Revised:2025-07-25 Accepted:2025-07-28 Published:2026-02-05
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 12104247 and 11934020).

摘要/Abstract

摘要： In a superconductor embedded with a quantum magnetic impurity, the Kondo effect is involved, leading to the competition between the Kondo singlet phase and the superconductivity phase. By means of the natural orbitals renormalization group (NORG) method, we revisit the problem of a quantum magnetic impurity coupled with a conventional s-wave superconductor. Here we present a detailed study focusing on the impurity spin polarization and susceptibility, the Kondo screening cloud, as well as the number and structures of the active natural orbitals (ANOs). In the superconducting phase, the impurity spin is partially polarized, indicating that the impurity remains partially screened by the quantum fluctuations. Furthermore, the impurity spin susceptibility becomes divergent, resulting from the presence of residual local moment formed at the impurity site. Correspondingly, a non-integral (incomplete) Kondo cloud is formed, although the ground state is a spin doublet in this phase. In comparison, the Kondo cloud is complete in the Kondo singlet phase as expected. We also quantify the critical point, where the quantum phase transition from a Kondo singlet phase to a superconducting phase occurs, which is consistent with that in previous works. On the other hand, it is illustrated that only one ANO emerges in both quantum phases. The structures of the ANO, projected into both the real space and momentum space, are distinct in the Kondo singlet phase from that in the superconducting phase. More specifically, in the Kondo singlet phase, the ANO keeps fully active with half-occupied, and the superconducting gap has negligible influence on its structure. On the contrary, in the superconducting phase, the ANO tends to be inactive and its structure changes significantly as the superconducting gap increases. Additionally, our investigation demonstrates that the NORG method is reliable and convenient to solve the quantum impurity problems in superconductors as well, which will promote further theoretical studies on the Kondo problems in such systems using numerical methods.

关键词: Kondo effect, superconductor, natural orbitals renormalization group, active natural orbitals

Abstract: In a superconductor embedded with a quantum magnetic impurity, the Kondo effect is involved, leading to the competition between the Kondo singlet phase and the superconductivity phase. By means of the natural orbitals renormalization group (NORG) method, we revisit the problem of a quantum magnetic impurity coupled with a conventional s-wave superconductor. Here we present a detailed study focusing on the impurity spin polarization and susceptibility, the Kondo screening cloud, as well as the number and structures of the active natural orbitals (ANOs). In the superconducting phase, the impurity spin is partially polarized, indicating that the impurity remains partially screened by the quantum fluctuations. Furthermore, the impurity spin susceptibility becomes divergent, resulting from the presence of residual local moment formed at the impurity site. Correspondingly, a non-integral (incomplete) Kondo cloud is formed, although the ground state is a spin doublet in this phase. In comparison, the Kondo cloud is complete in the Kondo singlet phase as expected. We also quantify the critical point, where the quantum phase transition from a Kondo singlet phase to a superconducting phase occurs, which is consistent with that in previous works. On the other hand, it is illustrated that only one ANO emerges in both quantum phases. The structures of the ANO, projected into both the real space and momentum space, are distinct in the Kondo singlet phase from that in the superconducting phase. More specifically, in the Kondo singlet phase, the ANO keeps fully active with half-occupied, and the superconducting gap has negligible influence on its structure. On the contrary, in the superconducting phase, the ANO tends to be inactive and its structure changes significantly as the superconducting gap increases. Additionally, our investigation demonstrates that the NORG method is reliable and convenient to solve the quantum impurity problems in superconductors as well, which will promote further theoretical studies on the Kondo problems in such systems using numerical methods.

Key words: Kondo effect, superconductor, natural orbitals renormalization group, active natural orbitals

中图分类号: (Theories and models of many-electron systems)

71.10.-w

71.10.Fd (Lattice fermion models (Hubbard model, etc.)) 71.27.+a (Strongly correlated electron systems; heavy fermions)

Wen-Jing Zhang(张文静), Ru Zheng(郑汝), Rong-Qiang He(贺荣强), and Zhong-Yi Lu(卢仲毅). Numerical study of a quantum spin in an s-wave superconductor using the natural orbitals renormalization group method[J]. 中国物理B, 2026, 35(2): 27104-027104.

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Numerical study of a quantum spin in an s-wave superconductor using the natural orbitals renormalization group method

Numerical study of a quantum spin in an s-wave superconductor using the natural orbitals renormalization group method

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