Tensor Monopoles in superconducting systems

H. Weisbrich; M. Bestler; W. Belzig

doi:10.22331/q-2021-12-07-601

Tensor Monopoles in superconducting systems

H. Weisbrich, M. Bestler, and W. Belzig

Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany

Published:	2021-12-07, volume 5, page 601
Eprint:	arXiv:2109.03135v3
Doi:	https://doi.org/10.22331/q-2021-12-07-601
Citation:	Quantum 5, 601 (2021).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Topology in general but also topological objects such as monopoles are a central concept in physics. They are prime examples for the intriguing physics of gauge theories and topological states of matter. Vector monopoles are already frequently discussed such as the well-established Dirac monopole in three dimensions. Less known are tensor monopoles giving rise to tensor gauge fields. Here we report that tensor monopoles can potentially be realized in superconducting multi-terminal systems using the phase differences between superconductors as synthetic dimensions. In a first proposal we suggest a circuit of superconducting islands featuring charge states to realize a tensor monopole. As a second example we propose a triple dot system coupled to multiple superconductors that also gives rise to such a topological structure. All proposals can be implemented with current experimental means and the monopole readily be detected by measuring the quantum geometry.

Popular summary

Many robust physical phenomena are based on topological invariants, which are intrinsic geometrical properties of quantum systems. Some topological properties are attributed to synthetic monopoles in the parameter space of the quantum system. This synthetic charge is the topological invariant and creates the synthetic field – the Berry curvature – that is a measure of the local curvature of the abstract space of quantum states. One recently proposed manifestation are tensor monopoles in four dimensions that are the source of a tensor Berry curvature with no analog in our three-dimensional world. Hence, in order to realize such systems, one has to make use of synthetic instead of the usual spatial dimensions.
In this work, we theoretically demonstrate how a tensor monopole can be constructed in superconducting multi-terminal systems using four superconducting phase differences as synthetic dimensions. Hence, we pave the way to realize this recently proposed exotic topological phenomenon of a tensor monopole in the well-established platform of superconducting circuits.

► BibTeX data

@article{Weisbrich2021tensormonopolesin,
  doi = {10.22331/q-2021-12-07-601},
  url = {https://doi.org/10.22331/q-2021-12-07-601},
  title = {Tensor {M}onopoles in superconducting systems},
  author = {Weisbrich, H. and Bestler, M. and Belzig, W.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {601},
  month = dec,
  year = {2021}
}

► References

[1] P. A. M. Dirac, Proc. R. Soc. Lond. A 133, 60–72 (1931).
https://doi.org/10.1098/rspa.1931.0130

[2] X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
https://doi.org/10.1103/RevModPhys.83.1057

[3] M. V. Berry, Proc. R. Soc. A 392, 45 (1984).
https://doi.org/10.1098/rspa.1984.0023

[4] D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
https://doi.org/10.1103/PhysRevLett.49.405

[5] M. Z. Hasan, C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
https://doi.org/10.1103/RevModPhys.82.3045

[6] M. Sato, Y. Ando, Rep. Prog. Phys. 80, 30076501 (2017).
https://doi.org/10.1088/1361-6633/aa6ac7

[7] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
https://doi.org/10.1103/PhysRevLett.45.494

[8] Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
https://doi.org/10.1103/PhysRevLett.71.3697

[9] C. N. Yang, J. Math. Phys 19, 320–328 (1978).
https://doi.org/10.1063/1.523506

[10] F. Wilczek, A. Zee, Phys. Rev. Lett. 52, 2111 (1984).
https://doi.org/10.1103/PhysRevLett.52.2111

[11] S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, I. B. Spielman, Science 360, 1429 (2018).
https://doi.org/10.1126/science.aam9031

[12] M. Nakahara, Geometry, Topology, and Physics (Institute of Physics Publishing, Bristol, 2003).
https://doi.org/10.1201/9781420056945

[13] R. I. Nepomechie, Phys. Rev. D 31, 1921 (1985).
https://doi.org/10.1103/PhysRevD.31.1921

[14] C. Teitelboim, Phys. Lett. B 167, 69–72 (1986).
https://doi.org/10.1016/0370-2693(86)90547-2

[15] P. Orland, Nucl. Phys. B 205, 107–118 (1982).
https://doi.org/10.1016/0550-3213(82)90468-0

[16] T. Banks, N. Seiberg, Phys. Rev. D 83, 084019 (2011).
https://doi.org/10.1103/PhysRevD.83.084019

[17] N. E. Mavromatos, S. Sarkar, Phys. Rev. D 95, 104025 (2017).
https://doi.org/10.1103/PhysRevD.95.104025

[18] M. Montero, A. M. Uranga, I. Valenzuela, J. High Energy Phys. 7, 1–49 (2017).
https://doi.org/10.1007/JHEP07(2017)123

[19] V. Mathai, G. C. Thiang, Comm. Math. Phys. 355, 561–602 (2017).
https://doi.org/10.1007/s00220-017-2965-z

[20] V. Mathai, G. C. Thiang, J. Lond. Math. Soc. 54, 403–416 (1996).
https://doi.org/10.1112/jlms/54.2.403

[21] G. Palumbo, N. Goldman, Phys. Rev. B 99, 045154 (2019).
https://doi.org/10.1103/PhysRevB.99.045154

[22] G. Palumbo, N. Goldman, Phys. Rev. Lett. 121, 170401 (2018).
https://doi.org/10.1103/PhysRevLett.121.170401

[23] X. Tan, D.-W. Zhang, W. Zheng, X. Yang, S. Song, Z. Han, Y. Dong, Z. Wang, D. Lan, H. Yan, S.-L. Zhu, Y. Yu, Phys. Rev. Lett. 126, 126 (2021).
https://doi.org/10.1103/PhysRevLett.126.017702

[24] M. Chen, C. Li, G. Palumbo, Y.-Q. Zhu, N. Goldman, P. Cappellaro, arXiv preprint arXiv:2008.00596 (2020).
arXiv:2008.00596

[25] R.-P. Riwar, M. Houzet, J. S. Meyer, Y. V. Nazarov, Nat. Commun. 7, 11167 (2016).
https://doi.org/10.1038/ncomms11167

[26] E. Eriksson, R.-P. Riwar, M. Houzet, J. S. Meyer, Y. V. Nazarov, Phys. Rev. B 95, 075417 (2017).
https://doi.org/10.1103/PhysRevB.95.075417

[27] H.-Y. Xie, M. G. Vavilov, A. Levchenko, Phys. Rev. B 96, 161406(R) (2017).
https://doi.org/10.1103/PhysRevB.96.161406

[28] J. S. Meyer, M. Houzet, Phys. Rev. Lett. 119, 136807 (2017).
https://doi.org/10.1103/PhysRevLett.119.136807

[29] H.-Y. Xie, M. G. Vavilov, A. Levchenko, Phys. Rev. B 97, 035443 (2018).
https://doi.org/10.1103/PhysRevB.97.035443

[30] O. Deb, K. Sengupta, D. Sen, Phys. Rev. B 97, 174518 (2018).
https://doi.org/10.1103/PhysRevB.97.174518

[31] R. L. Klees, G. Rastelli, J. C. Cuevas, W. Belzig, Phys. Rev. Lett. 124, 197002 (2020).
https://doi.org/10.1103/PhysRevLett.124.197002

[32] H.-Y. Xie, A. Levchenko, Phys. Rev. B 99, 094519 (2019).
https://doi.org/10.1103/PhysRevB.99.094519

[33] R. L. Klees, J. L. Cuevas, W. Belzig, G. Rastelli, Phys. Rev. B 103, 014516 (2021).
https://doi.org/10.1103/PhysRevB.103.014516

[34] H. Weisbrich, R. L. Klees, G. Rastelli, W. Belzig, PRX Quantum 2, 010310 (2021).
https://doi.org/10.1103/PRXQuantum.2.010310

[35] V. Fatemi, A. R. Akhmerov, L. Bretheau, Phys. Rev. Research 3, 013288 (2021).
https://doi.org/10.1103/PhysRevResearch.3.013288

[36] L. Peyruchat, J. Griesmar, J.-D. Pillet, Ç. Ö. Girit, Phys. Rev. Research 3, 013289 (2021).
https://doi.org/10.1103/PhysRevResearch.3.013289

[37] T. Herrig, R.-P. Riwar, arXiv preprint arXiv:2012.10655 (2020).
arXiv:2012.10655

[38] A. Blais, S. M. Girvin, W. D. Oliver, Nat. Phys. 16, 247 (2020).
https://doi.org/10.1038/s41567-020-0806-z

[39] J. M. Martinis, M. H. Devoret, J. Clarke, Nat. Phys. 16, 234 (2020).
https://doi.org/10.1038/s41567-020-0829-5

[40] M. Gell-Mann, Phys. Rev. 125, 1067 (1962).
https://doi.org/10.1103/PhysRev.125.1067

[41] U. Vool, and M. Devoret, Int. J. Circuit Theory Appl. 45, 897 (2017).
https://doi.org/10.1002/cta.2359

Cited by

[1] Yu Zhang, Yan-Qing Zhu, Jianwen Xu, Wen Zheng, Dong Lan, Giandomenico Palumbo, Nathan Goldman, Shi-Liang Zhu, Xinsheng Tan, Z.D. Wang, and Yang Yu, "Exploring parity magnetic effects through quantum simulation with superconducting qubits", Physical Review Applied 21 3, 034052 (2024).

[2] Lev Teshler, Hannes Weisbrich, Jonathan Sturm, Raffael L. Klees, Gianluca Rastelli, and Wolfgang Belzig, "Ground state topology of a four-terminal superconducting double quantum dot", SciPost Physics 15 5, 214 (2023).

[3] David Christian Ohnmacht, Valentin Wilhelm, Hannes Weisbrich, and Wolfgang Belzig, "Non-Hermitian Topology in Multiterminal Superconducting Junctions", Physical Review Letters 134 15, 156601 (2025).

[4] Tomáš Bzdušek, "From quantum geometry to nonlinear optics and gerbes: Recent advances in topological band theory", Physical Review B 113 9, 099601 (2026).

[5] T. Herrig, J. H. Pixley, E. J. König, and R.-P. Riwar, "Quasiperiodic circuit quantum electrodynamics", npj Quantum Information 9 1, 116 (2023).

[6] L. Peyruchat, R. H. Rodriguez, J.-L. Smirr, R. Leone, and Ç. Ö. Girit, "Spectral Signatures of Nontrivial Topology in a Superconducting Circuit", Physical Review X 14 4, 041041 (2024).

[7] Hong-Yi Xie, Jaglul Hasan, and Alex Levchenko, "Non-Abelian monopoles in the multiterminal Josephson effect", Physical Review B 105 24, L241404 (2022).

[8] T. Herrig, C. Koliofoti, J. H. Pixley, E. J. König, and R.-P. Riwar, "Emulating moiré materials with quasiperiodic circuit quantum electrodynamics", Physical Review B 111 20, L201104 (2025).

[9] Giandomenico Palumbo, "Fractional quantum Hall effect for extended objects: from skyrmionic membranes to dyonic strings", Journal of High Energy Physics 2022 5, 124 (2022).

[10] Hannes Weisbrich, Raffael L. Klees, Oded Zilberberg, and Wolfgang Belzig, "Fractional transconductance via nonadiabatic topological Cooper pair pumping", Physical Review Research 5 4, 043045 (2023).

[11] Shou-Bang Yang, Pei-Rong Han, Wen Ning, Fan Wu, Zhen-Biao Yang, and Shi-Biao Zheng, "An exceptional surface and its topology", Science China Physics, Mechanics & Astronomy 69 3, 230313 (2026).

[12] Shou-Bang Yang, Pei-Rong Han, Wen Ning, Fan Wu, Zhen-Biao Yang, and Shi-Biao Zheng, "A hypersphere-like non-Abelian Yang monopole and its topological characterization", Science China Physics, Mechanics & Astronomy 69 6, 260314 (2026).

[13] Spyros Konitopoulos and Elias Koorambas, "Tensor extension of the Abelian-Higgs model for a superconductor", arXiv:2602.17073, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 04:12:11) and SAO/NASA ADS (last updated successfully 2026-08-08 04:12:21). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.