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Exactly solvable phase transition in a cavity-coupled one-dimensional Ising chain

Shuntaro Otake1,2,* and Motoaki Bamba2,3,4

  • *Contact author: shuntarootake@g.ecc.u-tokyo.ac.jp
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Phys. Rev. Research 8, 023150 – Published 11 May, 2026

DOI: https://doi.org/10.1103/fxml-nv46

Abstract

Although one-dimensional classical spin chains do not exhibit phase transitions, we found that a phase transition does occur when they are coupled to a cavity photon mode. This provides the simplest exactly solvable examples demonstrating that finite-temperature superradiant phase transitions can emerge from long-range fully connected interactions mediated by photons and interactions within the material.

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References (39)

  1. E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Phys. 31, 253 (1925).
  2. P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light–matter interaction, Rev. Mod. Phys. 91, 025005 (2019).
  3. A. F. Kockum, A. Miranowicz, S. D. Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
  4. W. L. Faust and C. H. Henry, Mixing of visible and near-resonance infrared light in GaP, Phys. Rev. Lett. 17, 1265 (1966).
  5. Z. Zhang, H. Hirori, F. Sekiguchi, A. Shimazaki, Y. Iwasaki, T. Nakamura, A. Wakamiya, and Y. Kanemitsu, Ultrastrong coupling between THz phonons and photons caused by an enhanced vacuum electric field, Phys. Rev. Res. 3, L032021 (2021).
  6. M. Barra-Burillo, U. Muniain, S. Catalano, M. Autore, F. Casanova, L. E. Hueso, J. Aizpurua, R. Esteban, and R. Hillenbrand, Microcavity phonon polaritons from the weak to the ultrastrong phonon–photon coupling regime, Nat. Commun. 12, 6206 (2021).
  7. G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, Sub-cycle switch-on of ultrastrong light-matter interaction, Nature (London) 458, 178 (2009).
  8. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nat. Phys. 6, 772 (2010) .
  9. F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling regime, Nat. Phys. 13, 44 (2017).
  10. T. Schwartz, J. A. Hutchison, C. Genet, and T. W. Ebbesen, Reversible switching of ultrastrong light-molecule coupling, Phys. Rev. Lett. 106, 196405 (2011).
  11. G. Scalari, C. Maissen, D. Turcinková, D. Hagenmüller, S. De Liberato, C. Ciuti, C. Reichl, D. Schuh, W. Wegscheider, M. Beck, and J. Faist, Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial, Science 335, 1323 (2012).
  12. A. Bayer, M. Pozimski, S. Schambeck, D. Schuh, R. Huber, D. Bougeard, and C. Lange, Terahertz light–matter interaction beyond unity coupling strength, Nano Lett. 17, 6340 (2017).
  13. J. George, T. Chervy, A. Shalabney, E. Devaux, H. Hiura, C. Genet, and T. W. Ebbesen, Multiple Rabi splittings under ultrastrong vibrational coupling, Phys. Rev. Lett. 117, 153601 (2016).
  14. N. S. Mueller, Y. Okamura, B. G. M. Vieira, S. Juergensen, H. Lange, E. B. Barros, F. Schulz, and S. Reich, Deep strong light–matter coupling in plasmonic nanoparticle crystals, Nature (London) 583, 780 (2020).
  15. I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, A. A. Golubov, M. Y. Kupriyanov, V. V. Ryazanov, and A. V. Ustinov, Approaching deep-strong on-chip photon-to-magnon coupling, Phys. Rev. Appl. 16, 034029 (2021).
  16. K. Nagarajan, A. Thomas, and T. W. Ebbesen, Chemistry under vibrational strong coupling, J. Am. Chem. Soc. 143, 16877 (2021).
  17. K. Hepp and E. H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: The Dicke maser model, Ann. Phys. (NY) 76, 360 (1973).
  18. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  19. J. Rohn, M. Hörmann, C. Genes, and K. P. Schmidt, Ising model in a light-induced quantized transverse field, Phys. Rev. Res. 2, 023131 (2020).
  20. T. Nakamoto, K. Takasan, and N. Tsuji, One-dimensional extended Hubbard model coupled with an optical cavity, Phys. Rev. B 112, 155150 (2025).
  21. A. Langheld, M. Hörmann, and K. P. Schmidt, Quantum phase diagrams of Dicke-Ising models by a wormhole algorithm, Phys. Rev. B 112, L161123 (2025).
  22. D. S. Shapiro, Y. Weber, T. Bode, F. K. Wilhelm, and D. Bagrets, Digital-analog simulations of Schrödinger cat states in the Dicke-Ising model, Phys. Rev. A 112, 042412 (2025).
  23. J. A. Koziol, A. Langheld, and K. P. Schmidt, Melting of devil's staircases in the long-range Dicke-Ising model, Phys. Rev. B 111, 224427 (2025).
  24. E. Cortese, L. Garziano, and S. De Liberato, Polariton spectrum of the Dicke-Ising model, Phys. Rev. A 96, 053861 (2017).
  25. C. F. Lee and N. F. Johnson, First-order superradiant phase transitions in a multiqubit cavity system, Phys. Rev. Lett. 93, 083001 (2004).
  26. A. Schellenberger and K. P. Schmidt, (Almost) everything is a Dicke model—mapping non-superradiant correlated light-matter systems to the exactly solvable Dicke model, SciPost Phys. Core 7, 038 (2024).
  27. J. Román-Roche, A. Gómez-León, F. Luis, and D. Zueco, Bound polariton states in the Dicke-Ising model, Nanophotonics 14, 2053 (2025).
  28. J. P. Mendonça, K. Jachymski, and Y. Wang, Role of matter interactions in superradiant phenomena, Phys. Rev. Lett. 135, 133601 (2025); Comment by M. Hörmann, A. Langheld, J. Leibig, A. Schellenberger, and K. P. Schmidt, arXiv:2511.08452.
  29. P. Kirton, M. M. Roses, J. Keeling, and E. G. D. Torre, Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa, Adv. Quantum Technol. 2, 1800043 (2019).
  30. F. Sitter, Introduction to Ferromagnetism (McGraw-Hill Book Company, Inc., New York, 1937).
  31. J. Román-Roche, A. Gómez-León, F. Luis, and D. Zueco, Linear response theory for cavity QED materials at arbitrary light-matter coupling strengths, Phys. Rev. B 111, 035156 (2025).
  32. G. A. Baker Jr., Ising model with a long-range interaction in the presence of residual short-range interactions, Phys. Rev. 130, 1406 (1963).
  33. R. L. Stratonovich, On a method of calculating quantum distribution functions, Dokl. Akad. Nauk SSSR 115, 1097 (1957).
  34. J. Hubbard, Calculation of partition functions, Phys. Rev. Lett. 3, 77 (1959).
  35. A. W. Sandvik, Multichain mean-field theory of quasi-one-dimensional quantum spin systems, Phys. Rev. Lett. 83, 3069 (1999).
  36. J. Keeling, M. J. Bhaseen, and B. D. Simons, Fermionic superradiance in a transversely pumped optical cavity, Phys. Rev. Lett. 112, 143002 (2014).
  37. C.-M. Halati, A. Sheikhan, and C. Kollath, Cavity-induced artificial gauge field in a Bose-Hubbard ladder, Phys. Rev. A 96, 063621 (2017).
  38. Y.-Q. Zhang, L.-H. Yu, J.-Q. Liang, G. Chen, S. Jia, and F. Nori, Quantum phases in circuit QED with a superconducting qubit array, Sci. Rep. 4, 4083 (2014).
  39. R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Quantum criticality in an Ising chain: Experimental evidence for emergent E8 symmetry, Science 327, 177 (2010).

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Phys. Rev. Research 8, 023150– Published 11 May, 2026
  • Received 3 December 2025
  • Accepted 11 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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