pith. sign in

arxiv: 2503.09982 · v2 · submitted 2025-03-13 · 🪐 quant-ph

General phase diagram features of superradiant phase transitions

Wen Zhao , Junlong Tian , Jie Peng This is my paper

Pith reviewed 2026-05-23 00:01 UTC · model grok-4.3

classification 🪐 quant-ph
keywords superradiant phase transitionphase diagrammean-field approximationfinite temperaturelight-matter interactionnormal phasecoupling parametersdisorder
0
0 comments X

The pith

Superradiant phase transitions occur only once along any radial line from the origin in coupling-parameter space under mean-field theory at finite temperature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a broad class of light-matter systems that include multiple qubits and modes, anisotropic couplings, one- and two-photon processes, Stark shifts, cavity hoppings, and qubit interactions. In all these cases the mean-field treatment at finite temperature places the origin of the multi-dimensional coupling space inside the normal phase, with the superradiant transition crossed at most once when any straight radial line is followed outward. This single-crossing property reduces the task of locating the phase boundary and computing transition characteristics to a one-dimensional radial scan. The authors illustrate the result on concrete models, showing that multimode collective effects can produce the transition even in the strong-coupling regime and that disorder merely shifts the boundary location.

Core claim

In systems composed of multiqubits and multimodes with anisotropic couplings, one- and two-photon interactions, Stark shifts, inter-cavity hoppings, qubit-qubit interactions and similar terms, the mean-field method at finite temperature shows that the origin always lies in the normal phase and that a superradiant phase transition occurs only once along the radial direction of any chosen coupling parameter vector. This structure permits the phase boundary and the properties of the transition to be obtained by a concise radial procedure.

What carries the argument

Mean-field free-energy minimization at finite temperature performed along radial rays in the multi-parameter coupling space.

If this is right

  • Phase boundaries and transition properties can be computed by scanning a single radial coordinate rather than searching the full multi-dimensional parameter space.
  • Multimode collective behavior allows the superradiant transition to appear inside the strong-coupling regime.
  • Disorder in the couplings shifts the radial location of the phase boundary without altering the single-crossing structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The radial-only crossing rule may reduce the computational cost of mapping phase diagrams when the number of independent couplings is large.
  • Experimental protocols can be simplified by choosing parameter paths that cross the transition at most once.

Load-bearing premise

The mean-field approximation at finite temperature remains accurate and produces the single radial transition property for every listed combination of interactions.

What would settle it

A mean-field or exact calculation in any of the models that shows the system entering the superradiant phase and then returning to the normal phase along one radial line would disprove the claimed general feature.

Figures

Figures reproduced from arXiv: 2503.09982 by Jie Peng, Junlong Tian, Wen Zhao.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The phase diagram of two-mode Dicke model [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Phase diagram of the on-site effective Hamilto [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase diagram of the model in the classical oscil [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The energies in the ground state (GS) and the first [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Various light-matter interactions lead to diverse phase diagram structures in superradiant phase transition (SPT) studies. Such systems consist of multiqubit and multimode with anisotropic couplings, one- and two-photon interactions, Stark shifts, inter-cavity hoppings, qubit-qubit interactions and so on. We find a general phase diagram feature that the origin is in normal phase (NP) and SPT happens only once along the radial direction of a chosen coupling parameter vector with the mean-field method at finite temperature. We can calculate the phase boundary and SPT properties by a concise method. We illustrate it with specific models and find SPT can be achieved in strong coupling regime by means of multimode collective behavior. We also find the disorder will shift the phase boundary. These general features facilitate SPT studies and their diverse applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript identifies a general structural feature of superradiant phase transitions (SPT) across a family of light-matter models (multiqubit/multimode, anisotropic couplings, one- and two-photon interactions, Stark shifts, inter-cavity hoppings, qubit-qubit interactions, and disorder). Within the mean-field treatment at finite temperature, the normal phase occupies the origin of coupling-parameter space and a single SPT occurs along any radial ray; the authors supply a concise method for locating the phase boundary and computing SPT properties, illustrate the result on concrete models (including multimode collective enhancement into the strong-coupling regime), and note that disorder merely shifts the boundary.

Significance. If the homogeneity argument holds, the result supplies a model-independent organizing principle that reduces the dimensionality of phase-diagram searches and clarifies why multimode systems can reach strong-coupling SPT. The explicit scoping to mean-field finite-T and the radial-only claim are strengths; the concise boundary method, if parameter-free or directly derivable from the free-energy functional, would be a practical contribution.

minor comments (2)
  1. The abstract and introduction would benefit from an explicit statement of the homogeneity property (e.g., scaling all light-matter couplings by a common factor λ) that underpins the single-crossing result; this would make the scope of the claim immediately clear to readers.
  2. Figure captions should state the temperature and the precise mean-field ansatz used for each panel so that the radial-only property can be verified by inspection.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the homogeneity property under mean-field finite-temperature treatment, and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation follows from mean-field homogeneity

full rationale

The claimed general feature (NP at origin, single radial SPT) is presented as a direct mathematical consequence of the homogeneity of the finite-temperature mean-field self-consistency condition or free-energy functional once couplings are scaled uniformly. This structure is independent of specific model details (anisotropy, two-photon terms, Stark shifts, etc.) and does not reduce to a fitted parameter, self-citation chain, or ansatz smuggled via prior work. No equations or claims in the abstract or skeptic summary exhibit the enumerated circular patterns; the concise method is scoped explicitly to this mean-field property without invoking external uniqueness theorems from the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the mean-field approximation for the broad class of multiqubit-multimode systems with the enumerated interactions; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Mean-field theory at finite temperature accurately captures the phase structure for all the listed light-matter models.
    The general feature is explicitly derived with the mean-field method.

pith-pipeline@v0.9.0 · 5661 in / 1265 out tokens · 43147 ms · 2026-05-23T00:01:55.531833+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages

  1. [1]

    The stability condition of the system is 1 − 4J/ω − U/ω > 0, which will be discussed later

    and inter-cavity hopping, respectively. The stability condition of the system is 1 − 4J/ω − U/ω > 0, which will be discussed later. Employing the decoupling ap- proximation [45] aja† j+1 ≈ ajψ∗ + a† j+1ψ − | ψ|2 where the coherent field ⟨aj⟩ = ψ is the expectation value of aj in terms of the stable state, we obtain a decoupled Hamiltonian: H ≈ MX j Heff j...

    work page

  2. [2]

    R. H. Dicke, Phys. Rev. 93, 99 (1954)

    work page 1954

  3. [3]

    Hepp and E

    K. Hepp and E. H. Lieb, Ann. Phys. (N. Y.) 76, 360 (1973)

    work page 1973

  4. [4]

    Y. K. Wang and F. T. Hioe, Phys. Rev. A 7, 831 (1973)

    work page 1973

  5. [5]

    Emary and T

    C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003)

    work page 2003

  6. [6]

    Bakemeier, A

    L. Bakemeier, A. Alvermann, and H. Fehske, Phys. Rev. A 85, 043821 (2012)

    work page 2012

  7. [7]

    Ashhab, Phys

    S. Ashhab, Phys. Rev. A 87, 013826 (2013)

    work page 2013

  8. [8]

    Hwang, R

    M.-J. Hwang, R. Puebla, and M. B. Plenio, Phys. Rev. Lett. 115, 180404 (2015)

    work page 2015

  9. [9]

    Hwang and M

    M.-J. Hwang and M. B. Plenio, Phys. Rev. Lett. 117, 123602 (2016)

    work page 2016

  10. [10]

    M. Liu, S. Chesi, Z.-J. Ying, X. Chen, H.-G. Luo, and H.-Q. Lin, Phys. Rev. Lett. 119, 220601 (2017)

    work page 2017

  11. [11]

    Forn-D´ ıaz, J

    P. Forn-D´ ıaz, J. Lisenfeld, D. Marcos, J. J. Garc´ ıa-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. Lett. 105, 237001 (2010)

    work page 2010

  12. [12]

    Niemczyk, F

    T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. H¨ ummer, E. Solano, A. Marx, and R. Gross, Nat. Phys. 6, 772 (2010)

    work page 2010

  13. [13]

    Ballester, G

    D. Ballester, G. Romero, J. J. Garc´ ıa-Ripoll, F. Deppe, and E. Solano, Phys. Rev. X 2, 021007 (2012)

    work page 2012

  14. [14]

    Bayer, M

    A. Bayer, M. Pozimski, S. Schambeck, D. Schuh, R. Hu- ber, D. Bougeard, and C. Lange, Nano Lett. 17, 6340 (2017)

    work page 2017

  15. [15]

    Casanova, G

    J. Casanova, G. Romero, I. Lizuain, J. J. Garc´ ıa-Ripoll, and E. Solano, Phys. Rev. Lett. 105, 263603 (2010)

    work page 2010

  16. [16]

    Yoshihara, T

    F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Nat. Phys. 13, 44 (2017)

    work page 2017

  17. [17]

    Dimer, B

    F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Phys. Rev. A 75, 013804 (2007)

    work page 2007

  18. [18]

    M. P. Baden, K. J. Arnold, A. L. Grimsmo, S. Parkins, and M. D. Barrett, Phys. Rev. Lett. 113, 020408 (2014)

    work page 2014

  19. [19]

    Huang and L

    J.-F. Huang and L. Tian, arXiv: 2208.12524

    work page arXiv

  20. [20]

    Baumann, C

    K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Nature 464, 1301 (2010)

    work page 2010

  21. [21]

    Baumann, R

    K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, Phys. Rev. Lett. 107, 140402 (2011)

    work page 2011

  22. [22]

    De Bernardis, T

    D. De Bernardis, T. Jaako, and P. Rabl, Phys. Rev. A 97, 043820 (2018)

    work page 2018

  23. [23]

    M. O. Ara´ ujo, I. Kreˇ si´ c, R. Kaiser, and W. Guerin, Phys. Rev. Lett. 117, 073002 (2016)

    work page 2016

  24. [24]

    S. J. Roof, K. J. Kemp, M. D. Havey, and I. M. Sokolov, Phys. Rev. Lett. 117, 073003 (2016)

    work page 2016

  25. [25]

    Viehmann, J

    O. Viehmann, J. von Delft, and F. Marquardt, Phys. Rev. Lett. 107, 113602 (2011)

    work page 2011

  26. [26]

    Zhang, L

    Y. Zhang, L. Yu, J. Q. Liang, G. Chen, S. Jia, and F. Nori, Sci. Rep. 4, 4083 (2014)

    work page 2014

  27. [27]

    Bamba, K

    M. Bamba, K. Inomata, and Y. Nakamura, Phys. Rev. Lett. 117, 173601 (2016)

    work page 2016

  28. [28]

    Tighineanu, R

    P. Tighineanu, R. S. Daveau, T. B. Lehmann, H. E. Beere, D. A. Ritchie, P. Lodahl, and S. Stobbe, Phys. Rev. Lett. 116, 163604 (2016)

    work page 2016

  29. [29]

    R. G. DeVoe and R. G. Brewer, Phys. Rev. Lett.76, 2049 (1996)

    work page 2049

  30. [30]

    Genway, W

    S. Genway, W. Li, C. Ates, B. P. Lanyon, and I. Lesanovsky, Phys. Rev. Lett. 112, 023603 (2014)

    work page 2014

  31. [31]

    Puebla, M.-J

    R. Puebla, M.-J. Hwang, J. Casanova, and M. B. Plenio, Phys. Rev. Lett. 118, 073001 (2017)

    work page 2017

  32. [32]

    F. M. Gambetta, I. Lesanovsky, and W. Li, Phys. Rev. A 100, 022513 (2019)

    work page 2019

  33. [33]

    M. L. Cai, Z. D. Liu, W. D. Zhao, Y. K. Wu, Q. X. Mei, Y. Jiang, L. He, X. Zhang, Z. C. Zhou, and L. M. Duan, Nat. Commun. 12, 1126 (2021)

    work page 2021

  34. [34]

    Y. Chen, H. Zhai, and Z. Yu, Phys. Rev. A 91, 021602 (2015)

    work page 2015

  35. [35]

    Zhang, Y

    X. Zhang, Y. Chen, Z. Wu, J. Wang, J. Fan, S. Deng, and H. Wu, Science 373, 1359 (2021)

    work page 2021

  36. [36]

    Baksic and C

    A. Baksic and C. Ciuti, Phys. Rev. Lett. 112, 173601 (2014)

    work page 2014

  37. [37]

    Xie, X.-Y

    Y.-F. Xie, X.-Y. Chen, X.-F. Dong, and Q.-H. Chen, Phys. Rev. A 101, 053803 (2020)

    work page 2020

  38. [38]

    Jiang, B

    X. Jiang, B. Lu, C. Han, R. Fang, M. Zhao, Z. Ma, T. Guo, and C. Lee, Phys. Rev. A 104, 043307 (2021)

    work page 2021

  39. [39]

    Zhang, Z.-X

    Y.-Y. Zhang, Z.-X. Hu, L. Fu, H.-G. Luo, H. Pu, and X.-F. Zhang, Phys. Rev. Lett. 127, 063602 (2021)

    work page 2021

  40. [40]

    Felicetti, J

    S. Felicetti, J. S. Pedernales, I. L. Egusquiza, G. Romero, L. Lamata, D. Braak, and E. Solano, Phys. Rev. A 92, 033817 (2015)

    work page 2015

  41. [41]

    Duan, Y.-F

    L. Duan, Y.-F. Xie, D. Braak, and Q.-H. Chen, J. Phys. A: Math. Theor. 49, 464002 (2016)

    work page 2016

  42. [42]

    Garbe, I

    L. Garbe, I. L. Egusquiza, E. Solano, C. Ciuti, T. Coudreau, P. Milman, and S. Felicetti, Phys. Rev. A 95, 053854 (2017)

    work page 2017

  43. [43]

    Chen and Y.-Y

    X.-Y. Chen and Y.-Y. Zhang, Phys. Rev. A 97, 053821 (2018)

    work page 2018

  44. [44]

    Cong, X.-M

    L. Cong, X.-M. Sun, M. Liu, Z.-J. Ying, and H.-G. Luo, Phys. Rev. A 99, 013815 (2019)

    work page 2019

  45. [45]

    C. F. Lo, Sci. Rep. 10, 18761 (2020)

    work page 2020

  46. [46]

    A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, Nat. Phys. 2, 856 (2006). 9

    work page 2006

  47. [47]

    Lei and R.-K

    S.-C. Lei and R.-K. Lee, Phys. Rev. A 77, 033827 (2008)

    work page 2008

  48. [48]

    Schir´ o, M

    M. Schir´ o, M. Bordyuh, B. ¨Oztop, and H. E. T¨ ureci, Phys. Rev. Lett. 109, 053601 (2012)

    work page 2012

  49. [49]

    Schir´ o, M

    M. Schir´ o, M. Bordyuh, B. ¨Oztop, and H. E. T¨ ureci, J. Phys. B: At., Mol. Opt. Phys 46, 224021 (2013)

    work page 2013

  50. [50]

    Y. Wang, M. Liu, W.-L. You, S. Chesi, H.-G. Luo, and H.-Q. Lin, Phys. Rev. A 101, 063843 (2020)

    work page 2020

  51. [51]

    C. H. Alderete and B. M. Rodr´ ıguez-Lara, J. Phys. A: Math. Theor. 49, 414001 (2016)

    work page 2016

  52. [52]

    Shen, J.-W

    L.-T. Shen, J.-W. Yang, Z.-R. Zhong, Z.-B. Yang, and S.-B. Zheng, Phys. Rev. A 104, 063703 (2021)

    work page 2021

  53. [53]

    R. R. Soldati, M. T. Mitchison, and G. T. Landi, Phys. Rev. A 104, 052423 (2021)

    work page 2021

  54. [54]

    A. Devi, S. D. Gunapala, M. I. Stockman, and M. Pre- maratne, Phys. Rev. A 102, 013701 (2020)

    work page 2020

  55. [55]

    Y.-F. Xie, L. Duan, and Q.-H. Chen, Phys. Rev. A 99, 013809 (2019)

    work page 2019

  56. [56]

    J. Peng, E. Rico, J. X. Zhong, E. Solano, and I. L. Egusquiza, Phys. Rev. A 100, 063820 (2019)

    work page 2019

  57. [57]

    S. Cui, F. H´ ebert, B. Gr´ emaud, V. G. Rousseau, W. Guo, and G. G. Batrouni, Phys. Rev. A 100, 033608 (2019)

    work page 2019

  58. [58]

    S. Cui, B. Gr´ emaud, W. Guo, and G. G. Batrouni, Phys. Rev. A 102, 033334 (2020)

    work page 2020

  59. [59]

    Z.-J. Ying, L. Cong, and X.-M. Sun, J. Phys. A: Math. Theor. 53, 345301 (2020)

    work page 2020

  60. [60]

    He, L.-W

    S. He, L.-W. Duan, Y.-Z. Wang, C. Wang, and Q.-H. Chen, Phys. Rev. A 106, 033712 (2022)

    work page 2022

  61. [61]

    Felicetti and A

    S. Felicetti and A. Le Boit´ e, Phys. Rev. Lett.124, 040404 (2020)

    work page 2020

  62. [62]

    J. Peng, E. Rico, J. X. Zhong, E. Solano, and I. L. Egusquiza, arXiv:1904.02118

    work page arXiv 1904

  63. [63]

    A. L. Grimsmo and S. Parkins, Phys. Rev. A 87, 033814 (2013)

    work page 2013