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Firstorder photon condensation in magnetic cavities: A twoleg ladder model
by Zeno Bacciconi, Gian Marcello Andolina, Titas Chanda, Giuliano Chiriacò, Marco Schiró, Marcello Dalmonte
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Submission summary
Authors (as registered SciPost users):  Gian Marcello Andolina · Zeno Bacciconi · Marcello Dalmonte 
Submission information  

Preprint Link:  https://arxiv.org/abs/2302.09901v3 (pdf) 
Code repository:  https://github.com/zenobacciconi/cavity_ladder 
Date submitted:  20230515 10:04 
Submitted by:  Bacciconi, Zeno 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider a model of free fermions in a ladder geometry coupled to a nonuniform cavity mode via Peierls substitution. Since the cavity mode generates a magnetic field, nogo theorems on spontaneous photon condensation do not apply, and we indeed observe a phase transition to a photon condensed phase characterized by finite circulating currents, alternatively referred to as the equilibrium superradiant phase. We consider both square and triangular ladder geometries, and characterize the transition by studying the energy structure of the system, lightmatter entanglement, the properties of the photon mode, and chiral currents. The transition is of first order and corresponds to a sudden change in the fermionic band structure as well as the number of its Fermi points. Thanks to the quasione dimensional geometry we scrutinize the accuracy of (mean field) cavitymatter decoupling against large scale densitymatrix renormalization group simulations. We find that lightmatter entanglement is essential for capturing corrections to matter properties at finite sizes and for the description of the correct photon state. The latter remains Gaussian in the the thermodynamic limit both in the normal and photon condensed phases.
Author comments upon resubmission
Dear Editor,
We thank you for handling our manuscript, and the Referee for the insightful report. We think we have addressed the Referee's main concern on lack of a ``groundbreaking" aspect. In particular, we believe that our manuscript meets Scipost Physics criteria for publication (1) and (3) for the corresponding two reasons:
 Our work shows for the first time a first order photon condensation at equilibrium in a gauge invariant model. We want to stress that the few examples of first order transitions present in the literature have been shown to be artifact of model truncations.
 The phenomenology readily understandable in the presented toy model should open new pathways in the quest for photon condensed phases in electronic systems.
Hence, we understand that the Editor can opt for the second option indicated in their recommendation, consulting at least another expert in the field in addition to the present Referee.
In our resubmitted manuscript, we have applied several changes motivated by the Referee's feedback. Below, as well as in the comment in reply to the Referee's report, please find a list of changes.
List of changes
 Use of photon condensation instead of superradiant transition in both title and text, some reference to the name equilibrium superradiant phase have been kept.
 Addition of Ref. [33] on Moire systems in the introduction (section 1).
 Addition of Refs. [42,43] on magnetic cavities in the introduction (section 1).
 Addition of Ref. [71] on 1/N corrections and a more explicit comment on meanfield treatments in the conclusions (section 5).
 Addition of Ref. [79] and a discussion on first order behaviors in magnetic systems in the conclusions (section 5).
 Addition of a comment on other first order superradiant transition propsed and addition of Refs. [77,78] in the conclusions (section 5).
 Discussion of thermodynamic limit comparison of DMRG and Gaussian fluctuations revised according to the reply to the referee report in the discussion (section 3.3).
Current status:
Reports on this Submission
Anonymous Report 3 on 2023611 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.09901v3, delivered 20230611, doi: 10.21468/SciPost.Report.7332
Strengths
It discusses the possibility of photonic condensation and a potential equilibrium superradiant phase transition in electronic systems. This transition is of first order, whereas typically in these systems, the transition is of second order.
Weaknesses
The paper should provide a better explanation as to why a phase transition exists in their case.
Report
The paper is very interesting. It is for two main reasons:
* It discusses the possibility of photonic condensation and a potential equilibrium superradiant phase transition in electronic systems. This transition is of first order, whereas typically in these systems, the transition is of second order.
Therefore, the paper deserves consideration for publication. Before that happens, I would like the authors to discuss some of the following points:
* Meanfield approximation: I would expect this approximation to be "exact" in the thermodynamic limit. Previous works by the authors have made this claim, e.g., their Refs. 25 and 29. By "exact," I mean that observables of both matter and light are exact in meanfield (even in the presence of lightmatter entanglement, as in statistical mechanics). Is this the case? (It may be inferred from their study with system size.)
* Relationship with nogo theorems: If I understand the paper correctly, the authors demonstrate that the lightmatter system undergoes a quantum phase transition (QPT) as a function of the coupling parameter g. This type of transition is reminiscent of the equilibrium superradiant phase transition. As mentioned in the article, the literature is full of nogo theorems that prohibit this transition (some of these nogo theorems have been developed by the authors themselves). It seems that these nogo theorems do not apply here because the vector potential is not constant. However, the authors should explain why they do observe the transition while in other articles with similar systems (also with Peirlstype coupling), it is concluded that the transition cannot occur. Some of these articles are:
 Ref 25 and its Section V, discussing the absence of a Superradiant Phase transition.
 The Tightbinding model in their Ref 55.
 The reference https://arxiv.org/abs/2302.08528 (Weber et al. "Cavityrenormalized quantum criticality in a honeycomb bilayer antiferromagnet") explicitly states (even in the abstract) that "While the position and universality class are not changed by a single cavity mode," and this fact is explained in the article by referring to the nogo theorems. By the way, I think this article is not cited.
In fact, in this manuscript, it is stated:
"Our findings show that in the normal phase, when the photonic order parameter vanishes, collective coupling to a single cavity mode cannot change the properties of a thermodynamically large system."
However, in this system, the transition does occur, and therefore, the modification of matter properties happen.
In my opinion, the authors could use their model to understand the differences with other examples and the existence of the transition and modification of matter properties.
If this discussion is included, the paper deserves to be published.
Anonymous Report 2 on 202365 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.09901v3, delivered 20230605, doi: 10.21468/SciPost.Report.7304
Strengths
A new proposal for a relatively barebones model that has a photoncondensation transition by avoiding nogo theorems.
Weaknesses
As mentioned before, the parameter values are probably not within reach under currently attainable experimental conditions.
Report
As this is a second round of reports and the manuscript has already been vetted regarding its validity, I can keep this short. Overall I do think that the manuscript meets the acceptance criteria for SciPost Physics. In particular, I vouch for interesting novel effect over being extremely realistic on the following grounds: The emergent field of cavity quantum materials is currently still very much in its exploratory stages. As such, even overly simplified models with parameters tuned to seemingly unrealistic values in order to achieve certain effects can have their merits in pushing forward the field by adding new ideas that can inspire followup studies. As such, I am for this paper.
Requested changes
Citations:
 cavity superconductivity (refs. 8,9): an earlier theoretical proposal was https://www.science.org/doi/10.1126/sciadv.aau6969
 cavity ferroelectricity (ref. 10): here https://www.pnas.org/doi/10.1073/pnas.2105618118 should also be cited
Author: Zeno Bacciconi on 20230614 [id 3728]
(in reply to Report 2 on 20230605)We thank the Referee for their report and for the suggested references that we have added in the new version of the manuscript. We would like to take the occasion to comment on the main weakness pointed out also by Referee 1 on the experimental relevance of the model. We recently become aware of an experiment (Ghirri et. al https://arxiv.org/abs/2302.00804) reporting ultrastrong magnetic coupling between magnonic excitations and a cavity mode confined on a superconducting resonator. Although the nature of matter degrees of freedom in the experiment (spins) is different with respect to our work (itinerant electrons), we think that the experimental cavity setup can be promising. We have added a sentence in the introduction to cite this work (now Ref. [47] in the manuscript) and slightly changed the discussion at the end of section 2 accordingly.
Anonymous Report 1 on 2023515 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.09901v3, delivered 20230515, doi: 10.21468/SciPost.Report.7195
Strengths
 careful study of a model with longranged interactions showing a firstorder transition in equilibrium
 meanfield +Gaussian fluctuations compared to DMRG
 calculation of photon excitations
Weaknesses
 Model most likely highly unrealistic as currentcurrent interactions are typically very small and very large magnetic fluxes are needed to realize the phase diagram.
Report
The authors have considered and answered all comments/questions of my previous report. Only the precise origin of discrepancies in 1/N still remains unclear (numerics or boundary conditions) but this is, perhaps, a detail.
It is not fully clear to me whether the study meets the acceptance criteria of scipost physics taking into accout that the model is unrealistic and that firstorder transitions are in general not uncommon. Nevertheless, I support publication taking the overall quality of the paper into account.
Author: Zeno Bacciconi on 20230614 [id 3727]
(in reply to Report 1 on 20230515)We thank the Referee for their report. We would like to take the occasion to comment on the main weakness pointed out also by Referee 2 on the experimental relevance of the model. We recently become aware of an experiment (Ghirri et. al https://arxiv.org/abs/2302.00804) reporting ultrastrong magnetic coupling between magnonic excitations and a cavity mode confined on a superconducting surface. Although the nature of matter degrees of freedom in the experiment (spins) is different with respect to our work (itinerant electrons), we think that the experimental cavity setup can be promising and is worth a more detailed investigation. We have added a sentence in the introduction to cite this work (now Ref. [47] in the manuscript) and slightly changed the discussion at the end of section 2 accordingly.
Author: Zeno Bacciconi on 20230614 [id 3729]
(in reply to Report 3 on 20230611)We thank the Referee for their report. We now reply to the questions raised by the Referee point by point:
<< * Meanfield approximation: I would expect this approximation to be "exact" in the thermodynamic limit. Previous works by the authors have made this claim, e.g., their Refs. 25 and 29. By "exact," I mean that observables of both matter and light are exact in meanfield (even in the presence of lightmatter entanglement, as in statistical mechanics). Is this the case? (It may be inferred from their study with system size.)>>
Neglecting lightmatter entanglement is justified if one is interested in phase diagram properties (in the thermodynamic limit) as the only sizable effect to the matter state can arise from a macroscopic classical photonic state which by definition does not carry lightmatter entanglement. However the photon meanfield approximation is not exact when one is interested in observables of the cavity mode that are not extensive, e.g. the variance of the cavity quadrature X. In the present work we provided both numerical evidence with DMRG simulations and an analytical understanding with the treatment of gaussian fluctuations of the existence of lightmatter entanglement in the ground state and its relevance for some cavity observables in the thermodynamic limit (Fig. 6).
<< * Relationship with nogo theorems: If I understand the paper correctly, the authors demonstrate that the lightmatter system undergoes a quantum phase transition (QPT) as a function of the coupling parameter g. This type of transition is reminiscent of the equilibrium superradiant phase transition. As mentioned in the article, the literature is full of nogo theorems that prohibit this transition (some of these nogo theorems have been developed by the authors themselves). It seems that these nogo theorems do not apply here because the vector potential is not constant. However, the authors should explain why they do observe the transition while in other articles with similar systems (also with Peirlstype coupling), it is concluded that the transition cannot occur. Some of these articles are:
The crucial difference with present nogo theorems and the references cited by the referee is that in these setups the coupling is purely electric ($\nabla \times A=0$) while in our work the cavity mode is magnetic ($\nabla \times A\neq0$). As this point seems to be unclear in the present version the manuscript, we have added a more explicit discussion in the introducion and in the conclusion.
<< In fact, in this manuscript, it is stated: "Our findings show that in the normal phase, when the photonic order parameter vanishes, collective coupling to a single cavity mode cannot change the properties of a thermodynamically large system." >>
We thank the Referee for pointing out this sentence as it indeed seems misleading. The message we want to convey is that, in the thermodynamic limit, the only way for single cavity mode to change the matter state in the collective strong coupling regime ($A_0\propto 1/\sqrt{N}$) is via a macroscopic coherent state. Hence, within the phases, quantum fluctuations of the cavity mode and lightmatter entanglement only produce harmless $O(1)$ corrections. Differently at a quantum critical point, as discussed in the reference suggested by the referee, quantum fluctuations of the cavity mode will be in general important. Hence we have rephrased the sentence cited by the Referee in order to make the message more clear and added the suggested reference.