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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2604.12950 |
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| _version_ | 1866913058876555264 |
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| author | Zhang, Zhi-Lei Yue, Xin Qiao, Guo-Jian Sun, C. P. |
| author_facet | Zhang, Zhi-Lei Yue, Xin Qiao, Guo-Jian Sun, C. P. |
| contents | In a hybrid system where two quantum dots (QDs) are coupled to a conventional $s$-wave superconductor, Poor Man's Majorana modes (PMMs) have been proposed. Existing theories often idealize the superconductor (SC) as a bulk system or an infinitely long chain, or treat it as another quantum dot with proximity-induced superconductivity, while experiments employ superconducting segments of finite length. Here, we model the SC as a finite-length 1D chain and treat the QDs and SC on equal footing. We obtain the conditions for the existence of PMMs, valid for arbitrary SC length and applicable to arbitrary tunneling strengths and magnetic fields. We find that the number of PMMs is highly sensitive to the SC length: it oscillates between zero and two with a period set by the Fermi wavelength ($\sim1\,\textÅ$), while four PMMs appear in the long-SC limit where the effective coupling between the two QDs becomes negligible. We further demonstrate that the PMMs that are separately localized at the two ends of the hybrid system do not exist in the finite-length case. Consequently, only nearly localized PMMs can be identified when the magnetic field is strong enough. In this way, the generalized `sweet spot' of the practical system can be found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12950 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sensitive dependence of Poor Man's Majorana modes on the length of the superconductor Zhang, Zhi-Lei Yue, Xin Qiao, Guo-Jian Sun, C. P. Mesoscale and Nanoscale Physics In a hybrid system where two quantum dots (QDs) are coupled to a conventional $s$-wave superconductor, Poor Man's Majorana modes (PMMs) have been proposed. Existing theories often idealize the superconductor (SC) as a bulk system or an infinitely long chain, or treat it as another quantum dot with proximity-induced superconductivity, while experiments employ superconducting segments of finite length. Here, we model the SC as a finite-length 1D chain and treat the QDs and SC on equal footing. We obtain the conditions for the existence of PMMs, valid for arbitrary SC length and applicable to arbitrary tunneling strengths and magnetic fields. We find that the number of PMMs is highly sensitive to the SC length: it oscillates between zero and two with a period set by the Fermi wavelength ($\sim1\,\textÅ$), while four PMMs appear in the long-SC limit where the effective coupling between the two QDs becomes negligible. We further demonstrate that the PMMs that are separately localized at the two ends of the hybrid system do not exist in the finite-length case. Consequently, only nearly localized PMMs can be identified when the magnetic field is strong enough. In this way, the generalized `sweet spot' of the practical system can be found. |
| title | Sensitive dependence of Poor Man's Majorana modes on the length of the superconductor |
| topic | Mesoscale and Nanoscale Physics |
| url | https://arxiv.org/abs/2604.12950 |