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  • Letter
  • Open Access

Many-electron characterizations of higher-charge superconductors

Zi-Hao Dong and Yi Zhang*

  • International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China

  • *Contact author: frankzhangyi@pku.edu.cn
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APS Open Sci. 1, L000076 – Published 31 July, 2026

DOI: https://doi.org/10.1103/xxvz-8rj9

Abstract

The theoretical understanding of conventional superconductivity as the phonon-assisted formation and condensation of two-electron Cooper pairs is a significant triumph in condensed matter physics. Here, we propose many-electron characterizations of higher-charge superconductivity with Cooper pairs consisting of more than two electrons, by implementing translation symmetrization on parent pair-density-wave-ordered states. In particular, we demonstrate many-electron constructions with vanishing charge-2e sectors, but with sharp signatures in charge-4e or charge-6e expectation values instead. Such characterizations are consistent with previous phenomenology of vestigial order and Ginzburg-Landau theory. Furthermore, we demonstrate that at the microscopic level, momentum conservation alone may be vital and sufficient for the robust emergence of higher-charge superconductivity. Our study thus offers a general and microscopic route to understand and characterize higher-charge superconductivity, including nontrivial experimental signatures such as fractional magnetic flux and period in interferometry, as well as localized Cooper pairs at lattice topological defects.

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References (94)

  1. K. Onnes, The resistance of pure mercury at helium temperatures, Commun. Phys. Lab. Univ. Leiden, B 120, 1274 (1911).
  2. W. Meissner and R. Ochsenfeld, Ein neuer effekt bei eintritt der supraleitfähigkeit, Naturwissenschaften 21, 787 (1933).
  3. D. Van Delft and P. Kes, The discovery of superconductivity, Phys. Today 63(9), 38 (2010).
  4. V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, in On Superconductivity and Superfluidity: A Scientific Autobiography, by V. L. Ginzburg (Springer-Verlag, Berlin, Heidelberg, 2009), pp. 113–137.
  5. L. N. Cooper, Bound electron pairs in a degenerate Fermi gas, Phys. Rev. 104, 1189 (1956).
  6. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957).
  7. P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev. 135, A550 (1964).
  8. A. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Sov. Phys. JETP 20, 762 (1965).
  9. D. Agterberg and H. Tsunetsugu, Dislocations and vortices in pair-density-wave superconductors, Nat. Phys. 4, 639 (2008).
  10. E. Berg, E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Striped superconductors: How spin, charge and superconducting orders intertwine in the cuprates, New J. Phys. 11, 115004 (2009).
  11. P. A. Lee, Amperean pairing and the pseudogap phase of cuprate superconductors, Phys. Rev. X 4, 031017 (2014).
  12. E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
  13. D. F. Agterberg, J. S. Davis, S. D. Edkins, E. Fradkin, D. J. Van Harlingen, S. A. Kivelson, P. A. Lee, L. Radzihovsky, J. M. Tranquada, and Y. Wang, The physics of pair-density waves: Cuprate superconductors and beyond, Annu. Rev. Condens. Matter Phys. 11, 231 (2020).
  14. Y.-M. Wu, P. A. Nosov, A. A. Patel, and S. Raghu, Pair density wave order from electron repulsion, Phys. Rev. Lett. 130, 026001 (2023).
  15. Y. Yerin, S.-L. Drechsler, M. Cuoco, and C. Petrillo, Multiple-q current states in a multicomponent superconducting channel, J. Phys.: Condens. Matter 35, 505601 (2023).
  16. N. S. Ticea, S. Raghu, and Y.-M. Wu, Pair density wave order in multiband systems, Phys. Rev. B 110, 094515 (2024).
  17. M. Hamidian, S. D. Edkins, S. H. Joo, A. Kostin, H. Eisaki, S. Uchida, M. Lawler, E.-A. Kim, A. P. Mackenzie, K. Fujita, et al., Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x, Nature (London) 532, 343 (2016).
  18. X. Liu, Y. X. Chong, R. Sharma, and J. S. Davis, Discovery of a Cooper-pair density wave state in a transition-metal dichalcogenide, Science 372, 1447 (2021).
  19. H. Chen, H. Yang, B. Hu, Z. Zhao, J. Yuan, Y. Xing, G. Qian, Z. Huang, G. Li, Y. Ye, et al., Roton pair density wave in a strong-coupling kagome superconductor, Nature (London) 599, 222 (2021).
  20. H. Zhao, H. Li, B. R. Ortiz, S. M. Teicher, T. Park, M. Ye, Z. Wang, L. Balents, S. D. Wilson, and I. Zeljkovic, Cascade of correlated electron states in the kagome superconductor CsV3Sb5, Nature (London) 599, 216 (2021).
  21. Q. Gu, J. P. Carroll, S. Wang, S. Ran, C. Broyles, H. Siddiquee, N. P. Butch, S. R. Saha, J. Paglione, J. S. Davis, et al., Detection of a pair density wave state in UTe2, Nature (London) 618, 921 (2023).
  22. E. Berg, E. Fradkin, and S. A. Kivelson, Charge-4e superconductivity from pair-density-wave order in certain high-temperature superconductors, Nat. Phys. 5, 830 (2009).
  23. D. Agterberg, M. Geracie, and H. Tsunetsugu, Conventional and charge-six superfluids from melting hexagonal Fulde-Ferrell-Larkin-Ovchinnikov phases in two dimensions, Phys. Rev. B 84, 014513 (2011).
  24. S.-K. Jian, Y. Huang, and H. Yao, Charge-4e superconductivity from nematic superconductors in two and three dimensions, Phys. Rev. Lett. 127, 227001 (2021).
  25. L. Nie, G. Tarjus, and S. A. Kivelson, Quenched disorder and vestigial nematicity in the pseudogap regime of the cuprates, Proc. Natl. Acad. Sci. USA 111, 7980 (2014).
  26. Y.-B. Liu, J. Zhou, and F. Yang, Nematic superconductivity and its critical vestigial phases in the quasicrystal, Phys. Rev. Lett. 133, 136002 (2024).
  27. E. V. Herland, E. Babaev, and A. Sudbø, Phase transitions in a three dimensional U(1)×U(1) lattice London superconductor: Metallic superfluid and charge-4e superconducting states, Phys. Rev. B 82, 134511 (2010).
  28. R. M. Fernandes and L. Fu, Charge-4e superconductivity from multicomponent nematic pairing: Application to twisted bilayer graphene, Phys. Rev. Lett. 127, 047001 (2021).
  29. J. H. Han and P. A. Lee, Understanding resistance oscillation in the CsV3Sb5 superconductor, Phys. Rev. B 106, 184515 (2022).
  30. C. M. Varma and Z. Wang, Extended superconducting fluctuation region and 6e and 4e flux quantization in a kagome compound with a normal state of 3Q order, Phys. Rev. B 108, 214516 (2023).
  31. L.-F. Zhang, Z. Wang, and X. Hu, Higgs-Leggett mechanism for the elusive ϕ0/3=hc/6e oscillation in Little-Parks setup of kagome superconductor CsV3Sb5, Commun. Phys. 7, 210 (2024).
  32. T.-Y. Lin, F.-F. Song, and G.-M. Zhang, Theory of the charge-6e condensed phase in kagome-lattice superconductors, Phys. Rev. B 111, 054508 (2025).
  33. V. B. Geshkenbein, A. I. Larkin, and A. Barone, Vortices with half magnetic flux quanta in ‘‘heavy-fermion” superconductors, Phys. Rev. B 36, 235 (1987).
  34. M. Sigrist, T. Rice, and K. Ueda, Low-field magnetic response of complex superconductors, Phys. Rev. Lett. 63, 1727 (1989).
  35. H.-Y. Kee, Y. B. Kim, and K. Maki, Half-quantum vortex and d-soliton in Sr2RuO4, Phys. Rev. B 62, R9275 (2000).
  36. M. A. Rampp and J. Schmalian, Integer and fractionalized vortex lattices and off-diagonal long-range order, J. Phys. Commun. 6, 055013 (2022).
  37. J. Ge, P. Wang, Y. Xing, Q. Yin, A. Wang, J. Shen, H. Lei, Z. Wang, and J. Wang, Charge-4e and charge-6e flux quantization and higher charge superconductivity in kagome superconductor ring devices, Phys. Rev. X 14, 021025 (2024).
  38. Y.-F. Jiang, Z.-X. Li, S. A. Kivelson, and H. Yao, Charge-4e superconductors: A Majorana quantum Monte Carlo study, Phys. Rev. B 95, 241103(R) (2017).
  39. N. V. Gnezdilov and Y. Wang, Solvable model for a charge-4e superconductor, Phys. Rev. B 106, 094508 (2022).
  40. M. O. Soldini, M. H. Fischer, and T. Neupert, Charge-4e superconductivity in a Hubbard model, Phys. Rev. B 109, 214509 (2024).
  41. P. Li, K. Jiang, and J. Hu, Charge 4e superconductor: A wavefunction approach, Sci. Bull. 69, 2328 (2024).
  42. Y.-B. Liu, J. Zhou, C. Wu, and F. Yang, Charge-4e superconductivity and chiral metal in 45-twisted bilayer cuprates and related bilayers, Nat. Commun. 14, 7926 (2023).
  43. G. E. Volovik, Fermionic quartet and vestigial gravity, JETP Lett. 119, 330 (2024).
  44. Y.-M. Wu and Y. Wang, D-wave charge-4e superconductivity from fluctuating pair density waves, npj Quantum Mater. 9, 66 (2024).
  45. F.-F. Song and G.-M. Zhang, Phase coherence of charge-6e superconductors with a frustrated kagome XY antiferromagnet, Chin. Phys. Lett. 42, 037401 (2025).
  46. S. Zhou and Z. Wang, Chern Fermi pocket, topological pair density wave, and charge-4e and charge-6e superconductivity in kagomé superconductors, Nat. Commun. 13, 7288 (2022).
  47. Z. Pan, C. Lu, F. Yang, and C. Wu, Frustrated superconductivity and sextetting order, Sci. China Phys. Mech. Astron. 67, 287412 (2024).
  48. M. Zeng, L.-H. Hu, H.-Y. Hu, Y.-Z. You, and C. Wu, High-order time-reversal symmetry breaking normal state, Sci. China Phys. Mech. Astron. 67, 237411 (2024).
  49. R. B. Laughlin, Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50, 1395 (1983).
  50. D. Arovas, J. R. Schrieffer, and F. Wilczek, Fractional statistics and the quantum Hall effect, Phys. Rev. Lett. 53, 722 (1984).
  51. J. K. Jain, Composite-fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63, 199 (1989).
  52. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
  53. L. Saminadayar, D. Glattli, Y. Jin, and B. Etienne, Observation of the e/3 fractionally charged Laughlin quasiparticle, Phys. Rev. Lett. 79, 2526 (1997).
  54. M. C. Gutzwiller, Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett. 10, 159 (1963).
  55. X.-G. Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B 44, 2664 (1991).
  56. P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006).
  57. Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  58. S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima, H. Nojiri, Y. Shimizu, K. Miyagawa, and K. Kanoda, Thermodynamic properties of a spin-1/2 spin-liquid state in a κ-type organic salt, Nat. Phys. 4, 459 (2008).
  59. M. Yamashita, N. Nakata, Y. Senshu, M. Nagata, H. M. Yamamoto, R. Kato, T. Shibauchi, and Y. Matsuda, Highly mobile gapless excitations in a two-dimensional candidate quantum spin liquid, Science 328, 1246 (2010).
  60. H. Yokoyama and H. Shiba, Variational Monte-Carlo studies of Hubbard model. I, J. Phys. Soc. Jpn. 56, 1490 (1987).
  61. H. Yokoyama and H. Shiba, Variational Monte-Carlo studies of Hubbard model. II, J. Phys. Soc. Jpn. 56, 3582 (1987).
  62. C. Gros, Physics of projected wavefunctions, Ann. Phys. 189, 53 (1989).
  63. A. Paramekanti, M. Randeria, and N. Trivedi, Projected wave functions and high temperature superconductivity, Phys. Rev. Lett. 87, 217002 (2001).
  64. A. Himeda, T. Kato, and M. Ogata, Stripe states with spatially oscillating d-wave superconductivity in the two-dimensional t−t′−J model, Phys. Rev. Lett. 88, 117001 (2002).
  65. O. I. Motrunich, Variational study of triangular lattice spin-1/2 model with ring exchanges and spin liquid state in κ-(ET)2Cu2(CN)3, Phys. Rev. B 72, 045105 (2005).
  66. B. Edegger, V. N. Muthukumar, and C. Gros, Gutzwiller–RVB theory of high-temperature superconductivity: Results from renormalized mean-field theory and variational Monte Carlo calculations, Adv. Phys. 56, 927 (2007).
  67. Y. Iqbal, F. Becca, and D. Poilblanc, Projected wave function study of Z2 spin liquids on the kagome lattice for the spin-1/2 quantum Heisenberg antiferromagnet, Phys. Rev. B 84, 020407 (2011).
  68. Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Gapless spin-liquid phase in the kagome spin-1/2 Heisenberg antiferromagnet, Phys. Rev. B 87, 060405 (2013).
  69. C. A. Jiménez-Hoyos, T. M. Henderson, T. Tsuchimochi, and G. E. Scuseria, Projected Hartree–Fock theory, J. Chem. Phys. 136, 164109 (2012).
  70. T. Nikšić, D. Vretenar, and P. Ring, Beyond the relativistic mean-field approximation. II. Configuration mixing of mean-field wave functions projected on angular momentum and particle number, Phys. Rev. C 74, 064309 (2006).
  71. T. Sogo, R. Lazauskas, G. Röpke, and P. Schuck, Critical temperature for α-particle condensation within a momentum-projected mean-field approach, Phys. Rev. C 79, 051301 (2009).
  72. T. Mizusaki and M. Imada, Quantum-number projection in the path-integral renormalization group method, Phys. Rev. B 69, 125110 (2004).
  73. T. Heitmann and J. Schnack, Combined use of translational and spin-rotational invariance for spin systems, Phys. Rev. B 99, 134405 (2019).
  74. N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591 (1979).
  75. T. Rice and M. Sigrist, Sr2RuO4: An electronic analogue of 3He?, J. Phys.: Condens. Matter 7, L643 (1995).
  76. N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, 10267 (2000).
  77. A. P. Mackenzie and Y. Maeno, The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing, Rev. Mod. Phys. 75, 657 (2003).
  78. G. M. Luke, Y. Fudamoto, K. Kojima, M. Larkin, J. Merrin, B. Nachumi, Y. Uemura, Y. Maeno, Z. Mao, Y. Mori, et al., Time-reversal symmetry-breaking superconductivity in Sr2RuO4, Nature (London) 394, 558 (1998).
  79. K. Nelson, Z. Mao, Y. Maeno, and Y. Liu, Odd-parity superconductivity in Sr2RuO4, Science 306, 1151 (2004).
  80. C. Kallin and J. Berlinsky, Chiral superconductors, Rep. Prog. Phys. 79, 054502 (2016).
  81. See Supplemental Material at http://link.aps.org/supplemental/10.1103/xxvz-8rj9 for further details on additional models, settings, properties, and results for higher-charge superconductors, as well as our computational methods for their expectation values.
  82. We note that the PDW has a finite charge-2e Cooper-pair density in most places, yet the average of states is different from the average of expectation values.
  83. Y. Zhang, T. Grover, and A. Vishwanath, Entanglement entropy of critical spin liquids, Phys. Rev. Lett. 107, 067202 (2011).
  84. Y. Zhang, T. Grover, A. Turner, M. Oshikawa, and A. Vishwanath, Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B 85, 235151 (2012).
  85. T. Grover, Y. Zhang, and A. Vishwanath, Entanglement entropy as a portal to the physics of quantum spin liquids, New J. Phys. 15, 025002 (2013).
  86. Y. Zhang and A. Vishwanath, Establishing non-Abelian topological order in Gutzwiller-projected Chern insulators via entanglement entropy and modular S-matrix, Phys. Rev. B 87, 161113 (2013).
  87. Z.-Q. Gao, Y.-Q. Wang, H. Yang, and C. Wu, Topological charge-2ne superconductors, arXiv:2512.21325.
  88. Z. D. Shi and T. Senthil, Non-Abelian topological superconductivity from melting Abelian fractional Chern insulators, arXiv:2512.17996.
  89. J. S. Davis and D.-H. Lee, Concepts relating magnetic interactions, intertwined electronic orders, and strongly correlated superconductivity, Proc. Natl. Acad. Sci. USA 110, 17623 (2013).
  90. Y. Wang and A. Chubukov, Charge-density-wave order with momentum (2Q, 0) and (0, 2Q) within the spin-fermion model: Continuous and discrete symmetry breaking, preemptive composite order, and relation to pseudogap in hole-doped cuprates, Phys. Rev. B 90, 035149 (2014).
  91. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature (London) 518, 179 (2015).
  92. R. M. Fernandes, P. P. Orth, and J. Schmalian, Intertwined vestigial order in quantum materials: Nematicity and beyond, Annu. Rev. Condens. Matter Phys. 10, 133 (2019).
  93. G. Grüner, The dynamics of charge-density waves, Rev. Mod. Phys. 60, 1129 (1988).
  94. G. Grüner, The dynamics of spin-density waves, Rev. Mod. Phys. 66, 1 (1994).

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APS Open Sci. 1, L000076– Published 31 July, 2026
Vol. 1
  • Received 10 January 2026
  • Revised 3 May 2026
  • Accepted 21 July 2026
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