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Hierarchic superradiant phases in anisotropic Dicke model
by DaKai He, Zhi Song
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | DaKai He |
| Submission information | |
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| Preprint Link: | scipost_202507_00064v1 (pdf) |
| Date submitted: | July 23, 2025, 2:13 p.m. |
| Submitted by: | DaKai He |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We revisit the phase diagram of an anisotropic Dicke model by revealing the non-analyticity induced by underlying exceptional points (EPs). We find that the conventional superradiant phase can be further separated into three regions, in which the systems are characterized by different effective Hamiltonians, including the harmonic oscillator, the inverted harmonic oscillator, and their respective counterparts. We employ the Loschmidt echo to characterize different quantum phases by analyzing the quench dynamics of a trivial initial state. Numerical simulations for finite systems confirm our predictions about the existence of hierarchic superradiant phases.
Current status:
Reports on this Submission
Report
Unfortunately, I have the impression that the main result reported in the manuscript is based on a flawed reasoning. The effective Hamiltonian (4) is obtained from the full Hamiltonian (1) through the Holstein-Primakoff transformation (3) and the approximation of the square root appearing therein by $\sqrt{N}$. The approximation is valid as long as only a vanishing fraction of the atoms is in the excited state. This is the case in the normal, but not in the superradiant phase. See the seminal work by Emary and Brandes (Ref. [11], Sec. III) for details. Due to this, the effective Hamiltonian (4) and the subsequent analysis for the superradiant phase is, to my impression, incorrect.
Currently, I can not recommend publication of the manuscript. However, I tend to say that the manuscript could meet all acceptance criteria after some revision if the authors would manage to resolve my concern expressed above.
Requested changes
1) Correct the effective Hamiltonian and the subsequent analysis of the superradiant phase. Ref. [11], Eq. (21) might (or not) serve as a starting point on how to deal with the superradiant phase analytically.
2) The rewriting of $H_\text{eff}$ from the form (4) to (5) could possibly be presented a bit more pedagogically for readers not familiar with Refs. [45, 46].
3) The Hilbert space of the model is infinite-dimensional due to the bosonic operators $a$, $a^\dagger$. In order for the results to be reproducable, please indicate how the Hilbert space is truncated in the numerical part of the manuscript.
4) Eq. (37), which serves as the basis for the numerical verification, is given without justification or reference. Please elaborate on how this (non-trivial claim) follows. If feasible, can anything be said about the relation between $a$ and $b$ on the one hand, and $g_1$ and $g_2$ on the other?
5) In Fig. 3, it is rather hard to validate if the numerical results are in quantitative agreement with the theoretical prediction quantitatively. I think it would be helpful to, if feasible, add dashed lines with theory predictions in the left panels.
6) Some typos to be fixed: - Line 76: there is a dot missing at the end of the sentence. - Line 77: there is a dot missing at the end of the sentence. - Line 89: there is a double dot (":") missing at the end of the line. - Line 96: "givenby". - Line 103: "an stable system". - Line 106: "oscillators is unstable".
Recommendation
Reject
Author: DaKai He on 2025-09-22 [id 5840]
(in reply to Report 1 on 2025-08-29)List of changes
Reply to the referee
We thank the referee for the accurate summary of our manuscript. First, we would like to emphasize that the layered superradiant phase mentioned in our paper is not a traditional ground-state phase transition, but a dynamical phase transition due to the change in the effective form of the Hamiltonian. Below, we respond to the referee's questions one by one: (1) We thank the referee for the insightful suggestion. We would first like to state that the mean-field approach mentioned in Eq. (21) of Ref. [11] is a classic method for solving the ground state of the Dicke model's superradiant phase. We agree with the referee that "The effective Hamiltonian (4) is obtained from the full Hamiltonian (1) through the Holstein-Primakoff transformation (3) and the approximation of the square root appearing therein by $\sqrt{N}$. The approximation is valid as long as only a vanishing fraction of the atoms is in the excited state." However, as stated in the abstract and introduction, the phase diagram presented in this work corresponds to dynamic phase transitions. Such transitions are revealed by the dynamics of the system, which involves many excited states rather than only the ground state. Specifically, our goal is to describe the dynamics starting from a trivial initial state with a small number of excited atoms. Therefore, as long as the evolving state only involves excited states with $N_b \ll \sqrt{N}$, the effective Hamiltonian in Eq. (4) is sufficiently accurate for describing the dynamics. In fact, a similar concept has been mentioned in Ref. (PhysRevE.104.034132), which points out that the dynamics of the Dicke model's superradiant phase can be effectively described by an inverted harmonic oscillator within a finite time. Thus, Eq. (4) in our manuscript can serve as the starting point for the dynamical analysis in the subsequent text. In the new version, we have added descriptions related to "dynamical phase transition" and included the relevant references. (2) We are grateful for the referee's suggestion. In the new version, we have introduced a linear transformation to make the derivation from Eq. (4) to Eq. (5) more straightforward and accessible. (3) We appreciate the referee's advice. The new version includes a discussion on the truncation of the Hilbert space. (4) We thank the referee for the suggestion. In the new version, we have added a new appendix B that details the derivation process of Eq. (37) in the manuscript, and we have also defined $a$ and $b$ in the equation in detail. (5) In Fig. 3, it is rather hard to validate if the numerical results are in quantitative agreement with the theoretical prediction quantitatively. I think it would be helpful to, if feasible, add dashed lines with theory predictions in the left panels. (6) We thank the referee for their attention to detail. The new version corrects these errors.
We have revised our manuscript in accordance with your suggestions and hope that the new version will meet with your approval.
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Anonymous on 2025-09-30 [id 5876]
(in reply to DaKai He on 2025-09-22 [id 5840])Let me (referee 1) thank the authors for their reply. I tend to say that the revised version of the manuscript meets the acceptance criteria for SciPost Physics Core.
I stand corrected regarding my initial criticism. Although I am still not entirely comfortable with replacing $\sqrt{N - b^\dagger b}$ in the Hamiltonian by $\sqrt{N}$ when studying highly excited eigenstates in the superradiant phase, this step appears to be reasonably well-accepted (see the reference [55] brought up in the reply and the revised version of the manuscript).
Two typographical errors: - Line 84: "subspaces Hamiltonian can be writen as". - Line 233: "in the for".
Anonymous on 2025-10-02 [id 5879]
(in reply to Anonymous Comment on 2025-09-30 [id 5876])We appreciate the referee's careful identification of these typographical errors, and we have corrected them in the new version.
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