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Interplay between photon condensation and electron-electron interactions in molecular systems

Matteo Parisi1, Elisabetta Paladino1,2, Giuseppe A. Falci1,2, Gian Marcello Andolina3, Salvatore Savasta4, Marco Polini5,6, and Francesco M. D. Pellegrino1,2

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Phys. Rev. B 114, 225416 – Published 20 October, 2026

DOI: https://doi.org/10.1103/7zlp-y14w

Abstract

We investigate a minimal molecular model consisting of square planar plaquettes hosting multiple electrons, whose dynamics is governed by a tight-binding Hamiltonian supplemented by on-site Hubbard repulsion. By coupling this system to a spatially nonuniform cavity mode, we analyze the emergence of a magnetostatic instability, namely, photon condensation, originating from the paramagnetic Van Vleck mechanism. The global behavior of the system is analyzed for different electronic filling factors, and we find that, except for the special cases of half filling and a single electron, where the transition, if it occurs, is necessarily a second-order phase transition, the global system may also undergo a first-order transition because of the action of the electron-electron interaction. The polaritonic excitation energies are analyzed, providing clear spectroscopic signatures of the magnetostatic instability and of its order.

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References (86)

  1. F. Schlawin, D. M. Kennes, and M. A. Sentef, Cavity quantum materials, Appl. Phys. Rev. 9, 011312 (2022).
  2. F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Cavity QED with quantum gases: New paradigms in many-body physics, Adv. Phys. 70, 1 (2021).
  3. J. Bloch, A. Cavalleri, V. Galitski, M. Hafezi, and A. Rubio, Strongly correlated electron–photon systems, Nature (London) 606, 41 (2022).
  4. F. J. Garcia-Vidal, C. Ciuti, and T. W. Ebbesen, Manipulating matter by strong coupling to vacuum fields, Science 373, eabd0336 (2021).
  5. A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
  6. G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, Magneto-transport controlled by Landau polariton states, Nat. Phys. 15, 186 (2019).
  7. F. Appugliese, J. Enkner, G. L. Paravicini-Bagliani, M. Beck, C. Reichl, W. Wegscheider, G. Scalari, C. Ciuti, and J. Faist, Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect, Science 375, 1030 (2022).
  8. G. Jarc, S. Y. Mathengattil, A. Montanaro, F. Giusti, E. M. Rigoni, R. Sergo, F. Fassioli, S. Winnerl, S. Dal Zilio, D. Mihailovic, P. Prelovšek, M. Eckstein, and D. Fausti, Cavity-mediated thermal control of metal-to-insulator transition in 1T−TaS2, Nature (London) 622, 487 (2023).
  9. G. Chiriacò, Thermal Purcell effect and cavity-induced renormalization of dissipations, Phys. Rev. B 110, L161107 (2024).
  10. M. A. Sentef, M. Ruggenthaler, and A. Rubio, Cavity quantum-electrodynamical polaritonically enhanced electron-phonon coupling and its influence on superconductivity, Sci. Adv. 4, eaau6969 (2018).
  11. J. B. Curtis, Z. M. Raines, A. A. Allocca, M. Hafezi, and V. M. Galitski, Cavity quantum Eliashberg enhancement of superconductivity, Phys. Rev. Lett. 122, 167002 (2019).
  12. F. Schlawin, A. Cavalleri, and D. Jaksch, Cavity-mediated electron-photon superconductivity, Phys. Rev. Lett. 122, 133602 (2019).
  13. G. M. Andolina, A. De Pasquale, F. M. D. Pellegrino, I. Torre, Frank H. L. Koppens, and M. Polini, Amperean superconductivity cannot be induced by deep subwavelength cavities in a two-dimensional material, Phys. Rev. B 109, 104513 (2024).
  14. S. Latini, D. Shin, S. A. Sato, C. Schäfer, U. De Giovannini, H. Hübener, and A. Rubio, The ferroelectric photo ground state of SrTiO3: Cavity materials engineering, Proc. Natl. Acad. Sci. USA 118, e2105618118 (2021).
  15. Y. Ashida, A. İmamoğlu, J. Faist, D. Jaksch, A. Cavalleri, and E. Demler, Quantum electrodynamic control of matter: Cavity-enhanced ferroelectric phase transition, Phys. Rev. X 10, 041027 (2020).
  16. C. Ciuti, Cavity-mediated electron hopping in disordered quantum Hall systems, Phys. Rev. B 104, 155307 (2021).
  17. A. Chiocchetta, D. Kiese, C. P. Zelle, F. Piazza, and S. Diehl, Cavity-induced quantum spin liquids, Nat. Commun. 12, 5901 (2021).
  18. O. Dmytruk and M. Schirò, Controlling topological phases of matter with quantum light, Commun. Phys. 5, 271 (2022).
  19. Z. Bacciconi, G. M. Andolina, and C. Mora, Topological protection of Majorana polaritons in a cavity, Phys. Rev. B 109, 165434 (2024).
  20. L. Winter and O. Zilberberg, Fractional quantum Hall edge polaritons, Phys. Rev. B 112, L241105 (2025).
  21. P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa, Adv. Quantum Technol. 2, 1800043 (2019).
  22. K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature (London) 464, 1301 (2010).
  23. M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Phys. Rep. 93, 301 (1982).
  24. G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Cavity quantum electrodynamics of strongly correlated electron systems: A no-go theorem for photon condensation, Phys. Rev. B 100, 121109(R) (2019).
  25. G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Theory of photon condensation in a spatially varying electromagnetic field, Phys. Rev. B 102, 125137 (2020).
  26. G. M. Andolina, F. M. D. Pellegrino, A. Mercurio, O. Di Stefano, M. Polini, and S. Savasta, A non-perturbative no-go theorem for photon condensation in approximate models, Eur. Phys. J. Plus 137, 1348 (2022).
  27. G. C. Duncan, Effect of antiresonant atom-field interactions on phase transitions in the Dicke model, Phys. Rev. A 9, 418 (1974).
  28. K. Rzażewski, K. Wódkiewicz, and W. Żakowicz, Phase transitions, two-level atoms, and the A2 term, Phys. Rev. Lett. 35, 432 (1975).
  29. I. Bialynicki-Birula and K. Rzążenwski, No-go theorem concerning the superradiant phase transition in atomic systems, Phys. Rev. A 19, 301 (1979).
  30. K. Gawędzki and K. Rząźnewski, No-go theorem for the superradiant phase transition without dipole approximation, Phys. Rev. A 23, 2134 (1981).
  31. P. Nataf, T. Champel, G. Blatter, and D. M. Basko, Rashba cavity QED: A route towards the superradiant quantum phase transition, Phys. Rev. Lett. 123, 207402 (2019).
  32. D. Guerci, P. Simon, and C. Mora, Superradiant phase transition in electronic systems and emergent topological phases, Phys. Rev. Lett. 125, 257604 (2020).
  33. D. Guerci, P. Simon, and C. Mora, Moiré lattice effects on the orbital magnetic response of twisted bilayer graphene and Condon instability, Phys. Rev. B 103, 224436 (2021).
  34. J. Román-Roche, F. Luis, and D. Zueco, Photon condensation and enhanced magnetism in cavity QED, Phys. Rev. Lett. 127, 167201 (2021).
  35. G. Manzanares, T. Champel, D. M. Basko, and P. Nataf, Superradiant quantum phase transition for Landau polaritons with Rashba and Zeeman couplings, Phys. Rev. B 105, 245304 (2022).
  36. J. H. Condon, Nonlinear de Haas-van Alphen effect and magnetic domains in beryllium, Phys. Rev. 145, 526 (1966).
  37. E. C. S., The theory of electric and magnetic susceptibilities, Nature (London) 130, 490 (1932).
  38. G. Grosso and G. P. Parravicini, Solid State Physics, 2nd ed. (Academic Press, New York, 2013).
  39. A. Mercurio, G. M. Andolina, Francesco M. D. Pellegrino, O. Di Stefano, P. Jarillo-Herrero, C. Felser, Frank H. L. Koppens, S. Savasta, and M. Polini, Photon condensation, Van Vleck paramagnetism, and chiral cavities, Phys. Rev. Res. 6, 013303 (2024).
  40. J. Li, D. Golez, G. Mazza, A. J. Millis, A. Georges, and M. Eckstein, Electromagnetic coupling in tight-binding models for strongly correlated light and matter, Phys. Rev. B 101, 205140 (2020).
  41. O. Dmytruk and M. Schiró, Gauge fixing for strongly correlated electrons coupled to quantum light, Phys. Rev. B 103, 075131 (2021).
  42. J. M. Luttinger, The effect of a magnetic field on electrons in a periodic potential, Phys. Rev. 84, 814 (1951).
  43. W. Kohn, Theory of Bloch electrons in a magnetic field: The effective Hamiltonian, Phys. Rev. 115, 1460 (1959).
  44. A. S. Alexandrov and H. Capellmann, Orbital diamagnetism of two-dimensional electrons, Phys. Rev. Lett. 66, 365 (1991).
  45. M. Graf and P. Vogl, Electromagnetic fields and dielectric response in empirical tight-binding theory, Phys. Rev. B 51, 4940 (1995).
  46. T. B. Boykin, Tight-binding-like expressions for the continuous-space electromagnetic coupling Hamiltonian, Am. J. Phys. 69, 793 (2001).
  47. J. Ibañez-Azpiroz, A. Eiguren, A. Bergara, G. Pettini, and M. Modugno, Breakdown of the Peierls substitution for the Haldane model with ultracold atoms, Phys. Rev. A 90, 033609 (2014).
  48. H. Bruus and K. Flensberg, Many–Body Quantum Theory in Condensed Matter Physics: An Introduction (Oxford University Press, Oxford, 2004).
  49. Z. Bacciconi, G. M. Andolina, T. Chanda, G. Chiriacò, M. Schirò, and M. Dalmonte, First-order photon condensation in magnetic cavities: A two-leg ladder model, SciPost Phys. 15, 113 (2023).
  50. Y. K. Wang and F. T. Hioe, Phase transition in the Dicke model of superradiance, Phys. Rev. A 7, 831 (1973).
  51. P. Nataf and C. Ciuti, No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED, Nat. Commun. 1, 72 (2010).
  52. A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, Cambridge, 2010).
  53. C. P. Koch, J. P. Palao, R. Kosloff, and F. Masnou-Seeuws, Stabilization of ultracold molecules using optimal control theory, Phys. Rev. A 70, 013402 (2004).
  54. A. Klein, Y. Shagam, W. Skomorowski, P. S. Żuchowski, M. Pawlak, L. M. C. Janssen, N. Moiseyev, S. Y. T. van de Meerakker, A. van der Avoird, C. P. Koch, and E. Narevicius, Directly probing anisotropy in atom–molecule collisions through quantum scattering resonances, Nat. Phys. 13, 35 (2017).
  55. D. N. Basov, A. Asenjo-Garcia, P. J. Schuck, X. Zhu, and A. Rubio, Polariton panorama, Nanophotonics 10, 549 (2021).
  56. J. Fregoni, F. J. Garcia-Vidal, and J. Feist, Theoretical challenges in polaritonic chemistry, ACS Photonics 9, 1096 (2022).
  57. T. W. Ebbesen, A. Rubio, and G. D. Scholes, Introduction: Polaritonic chemistry, Chem. Rev. 123, 12037 (2023).
  58. C. Gonzalez-Ballestero, J. Feist, E. Gonzalo Badía, E. Moreno, and F. J. Garcia-Vidal, Uncoupled dark states can inherit polaritonic properties, Phys. Rev. Lett. 117, 156402 (2016).
  59. A. Mandal, M. A. Taylor, B. M. Weight, E. R. Koessler, X. Li, and P. Huo, Theoretical advances in polariton chemistry and molecular cavity quantum electrodynamics, Chem. Rev. 123, 9786 (2023).
  60. K. Hirai, J. A. Hutchison, and H. Uji-i, Molecular chemistry in cavity strong coupling, Chem. Rev. 123, 8099 (2023).
  61. B. Gu, Y. Gu, V. Y. Chernyak, and S. Mukamel, Cavity control of molecular spectroscopy and photophysics, Acc. Chem. Res. 56, 2753 (2023).
  62. B. Xiang and W. Xiong, Molecular polaritons for chemistry, photonics and quantum technologies, Chem. Rev. 124, 2512 (2024).
  63. R. F. Ribeiro, L. A. Martínez-Martínez, M. Du, J. Campos-Gonzalez-Angulo, and J. Yuen-Zhou, Polariton chemistry: Controlling molecular dynamics with optical cavities, Chem. Sci. 9, 6325 (2018).
  64. T. T. Heikkilä, The Physics of Nanoelectronics: Transport and Fluctuation phenomena at Low Temperatures (Oxford University Press, Oxford, 2013).
  65. H. Lin, J. Gubernatis, H. Gould, and J. Tobochnik, Exact diagonalization methods for quantum systems, Comput. Phys. 7, 400 (1993).
  66. A. Honet, L. Henrard, and V. Meunier, Exact and many-body perturbation solutions of the Hubbard model applied to linear chains, AIP Adv. 12, 035238 (2022).
  67. M. Sharma and M. Ahsan, Organization of the Hilbert space for exact diagonalization of Hubbard model, Comput. Phys. Commun. 193, 19 (2015).
  68. D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annu. Rev. Condens. Matter Phys. 13, 239 (2022).
  69. H. Tasaki, The Hubbard model - an introduction and selected rigorous results, J. Phys.: Condens. Matter 10, 4353 (1998).
  70. Y. Nagaoka, Ferromagnetism in a narrow, almost half-filled s band, Phys. Rev. 147, 392 (1966).
  71. H. Tasaki, Extension of Nagaoka's theorem on the large- U Hubbard model, Phys. Rev. B 40, 9192 (1989).
  72. M. Kollar, R. Strack, and D. Vollhardt, Ferromagnetism in correlated electron systems: Generalization of Nagaoka's theorem, Phys. Rev. B 53, 9225 (1996).
  73. C. Emary and T. Brandes, Quantum chaos triggered by precursors of a quantum phase transition: The Dicke model, Phys. Rev. Lett. 90, 044101 (2003).
  74. C. Emary and T. Brandes, Chaos and the quantum phase transition in the Dicke model, Phys. Rev. E 67, 066203 (2003).
  75. G. Chiriacò, M. Dalmonte, and T. Chanda, Critical light-matter entanglement at cavity mediated phase transitions, Phys. Rev. B 106, 155113 (2022).
  76. G. Passetti, C. J. Eckhardt, M. A. Sentef, and D. M. Kennes, Cavity light-matter entanglement through quantum fluctuations, Phys. Rev. Lett. 131, 023601 (2023).
  77. C. J. Eckhardt, G. Passetti, M. Othman, C. Karrasch, F. Cavaliere, M. A. Sentef, and D. M. Kennes, Quantum Floquet engineering with an exactly solvable tight-binding chain in a cavity, Commun. Phys. 5, 122 (2022).
  78. I. Amelio, L. Korosec, I. Carusotto, and G. Mazza, Optical dressing of the electronic response of two-dimensional semiconductors in quantum and classical descriptions of cavity electrodynamics, Phys. Rev. B 104, 235120 (2021).
  79. E. Jeckelmann, F. Gebhard, and F. H. L. Essler, Optical conductivity of the half-filled Hubbard chain, Phys. Rev. Lett. 85, 3910 (2000).
  80. I. Gimeno, W. Kersten, M. C. Pallarés, P. Hermosilla, M. J. Martínez-Pérez, M. D. Jenkins, A. Angerer, C. Sánchez-Azqueta, D. Zueco, J. Majer, A. Lostao, and F. Luis, Enhancedmolecular spin-photon coupling at superconductingnanoconstrictions, ACS Nano 14, 8707 (2020).
  81. M. Rubín-Osanz, D. Rodriguez, I. Gimeno, W. Kersten, N. González-Prato, M. C. Pallarés, S. Roca-Jerat, M. C. de Ory, M. Mas-Torrent, J. A. de Sousa, L. Tesi, D. Granados, J. Veciana, D. Zueco, A. Lostao, J. Schmiedmayer, I. Ratera, J. van Slageren, N. Crivillers, A. Gomez, et al., Optimizing magnetic coupling in lumped element superconducting resonators for molecular spin qubits, arXiv:2511.00857.
  82. K. Hepp and E. H. Lieb, Equilibrium statistical mechanics of matter interacting with the quantized radiation field, Phys. Rev. A 8, 2517 (1973).
  83. F. M. D. Pellegrino, L. Chirolli, R. Fazio, V. Giovannetti, and M. Polini, Theory of integer quantum Hall polaritons in graphene, Phys. Rev. B 89, 165406 (2014).
  84. J. J. Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals, Phys. Rev. 112, 1555 (1958).
  85. D. Lamberto, G. Orlando, and S. Savasta, Superradiant quantum phase transition in open systems: System-bath interaction at the critical point, Quantum 10, 1970 (2026).
  86. Z. Kurucz and K. Mølmer, Multilevel Holstein-Primakoff approximation and its application to atomic spin squeezing and ensemble quantum memories, Phys. Rev. A 81, 032314 (2010).

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Phys. Rev. B 114, 225416– Published 20 October, 2026
  • Received 11 June 2026
  • Revised 5 October 2026
  • Accepted 8 October 2026
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