The vortex phase is an essential quantum phenomenon in a type-II superconductor, and vortex physics is significant to research on unconventional superconductivity. Caroli-de Gennes-Matricon (CdGM) states were predicted in 1964 as low-energy excitations within vortex cores in type-II superconductors. For the first time, we observed clear, discrete energy levels of CdGM states in some vortex cores in FeTe_0.55Se_0.45 [1], a result attributed to the shallow band structure of this material. The energy of the first three energy levels is close to the theoretical predicted ratio of 1:3:5. We also observed discrete CdGM states in another iron-based superconductor KCa₂Fe₄As₄F₂, but the energy ratio of the first three bound states is about 1:1.6:2.3, deviating from the theoretical prediction of 1:3:5 [2,3]. Analysis and calculation revealed that the measurement satisfies the extreme quantum limit; thus, the superconducting gap exhibits a Friedel-like spatial oscillation, whereas the early 1:3:5 ratio is based on the assumption that the energy gap varies linearly with space. Based on theoretical predictions, the vortex pattern should be circular. However, we observed a completely new necklace-like vortex pattern at high-order bound state energies in KCa₂Fe₄As₄F₂, which was not predicted by any theory. Based on theoretical calculations, the necklace-shaped vortex pattern arises from selective off-shell interference between vortex bound states of opposite angular momenta, driven by rotational symmetry breaking due to disorder. This fascinating effect can be observed in a system with a small Fermi energy and wave vector, conditions fortuitously met in our samples [4]. These works challenge the traditional theoretical predictions about vortex bound states and provide important clues for a deeper understanding of superconductivity.

Reference:
1. Mingyang Chen, Xiaoyu Chen, Huan Yang, Zengyi Du, Xiyu Zhu, Enyu Wang, and Hai-Hu Wen, Nat. Commun. 9, 970 (2018).
2. Xiaoyu Chen, Wen Duan, Xinwei Fan, Wenshan Hong, Kailun Chen, Huan Yang, Shiliang Li, Huiqian Luo, and Hai-Hu Wen, Phys. Rev. Lett. 126, 257002 (2021).
3. Wen Duan, Kailun Chen, Wenshan Hong, Xiaoyu Chen, Shiliang Li, Huiqian Luo, Huan Yang, and Hai-Hu Wen, Nano Lett. 22, 9450 (2022).
4. Zhiyong Hou, Kailun Chen, Wenshan Hong, Da Wang, Wen Duan, Huan Yang, Shiliang Li, Huiqian Luo, Qiang-Hua Wang, Tao Xiang, and Hai-Hu Wen, Phys. Rev. X 15, 011027 (2025).