(Almost) Everything is a Dicke model -- Mapping correlated light-matter systems to the exactly solvable Dicke model

The mapping is an interesting and sound result.

The article is well structured.

Many sentences are unclear and a bit too vague for proper understanding.

The limitations of the mapping are not always made as clear as should be.

In this work the authors discuss how, and in which circumstances, Dicke models generalised to include matter-matter coupling can be mapped back to separated Dicke and matter hamiltonians allow for a simple solution.

Globally, such a mapping is shown to be possible in the thermodynamic limit while assuming a low excitation number.

While the results are scientifically sound and certainly of sufficient interest for publication, they do not meet, in my opinion, any of the 4 criteria for acceptance in SciPost Physics. I therefore recommend that, after the required modifications listed below have been addressed, the paper be considered for publication in SciPost Physics Core.

1- One the issues which should be addressed is that the abstract, title and overall writing of the early sections of the article definitely seem to present the results as much broader reaching than they actually are.

A central limitation of the work is that it cannot capture the superradiant phase(s) which is certainly one of the most central aspect of Dicke-like model. The abstract and early section tend to talk exclusively about low energy sectors which naturally lead the reader to believe that the mapping should work over the full phase diagram by capturing the ground state and low energy excitation spectrum.

It would be necessary to mention early on in the paper and in the abstract that they are not really working in a low energy sector but in a low excitation number sector. The authors should be explicit about the fact that their work will exclude the treatment of any superradiant phases.

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- I also strongly feel that in view of this limitation, the title is also misleading and should therefore be changed to avoid making the vastly overreaching statement that it currently makes. The first part of that title “(Almost) everything is a Dicke model …” certainly does not convey the correct message about the content of the article and the strong limitation that excludes superradiant phases.

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- The abstract does state:

Coming from the limit of weak light-matter couplings and large number of matter entities, we map the relevant low-energy sector of a broad class of models onto the exactly solvable Dicke model.

but that is insufficient to clearly lead the reader to understand the limitation. The abstract should make it clear that the phrase “Coming from the limit of weak light-matter couplings“ is to be understood as excluding superradiance.

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On page three the exact restriction imposed is finally introduced for the first time in eq. (6):

“For our findings we further restrict ourselves onto the low-energy subspace demanding :”

This, again doesn’t not feel completely accurate, since in general there can be an important difference between low-energy and low-excitation number. It think it should be made clear here again at this point in the text that superradiant phases cannot be treated.


I know that the sentence: “We focus on the non-superradiant phase with ω0,ω ≫ g inducing two quasiparticle types, namely photons and magnons” appears at the beginning of section 2.1 but the word “focus” is not strong enough to understand that the work explicitly excludes the SR phases



Only at the beginning of section 3 do we explicitely read: “the two non-superradiant phases of the model, which obey the prerequisites in Eq. (6) “ which finally makes the exclusion clearer.


Overall, the authors should review the first few sections, the abstract and the title to make the non-superradiant point clear, from the very start.



2- Another limitation of the mapping seems to be in the anti-ferromagnetic case. Here the authors simply state that they choose a bipartite lattice “To avoid any kind of geometric frustration”. They, however, should explicitly discuss whether or not the restriction to bipartite lattices is a requirement of the mapping or not.

From the way the mapping is then implemented, it seems to be an absolute necessity to have a bipartite lattice. If it is indeed the case, the authors should be explicit about it. If not, the authors need to comment on it because the way the result is presented clearly seems to imply it is necessary.


3- There are multiple imprecise statements throughout the article concerning first and second-order phases transitions. Considering that the authors also keep making references to their not yet published article [43], it is essential that any statement relating this work to which transitions are first and second order, or to the results of [43], be more precise.

Here are some example sentences:

. According to this mean-field treatment, all quantum phase transitions between the four phases are of second order. In contrast, recent works using perturbation theory [1] and quantum Monte Carlo [43] have found that "some of the transitions" are indeed of first order in 1D.

. Therefore, the first-order nature of "certain phase transition lines" observed by exact diagonalization [1] and quantum Monte Carlo calculations [43] is in no contradiction to our findings.

. "In the realm of a second-order phase transition" the results of [2,43] and the presented effective theory do again coincide.

. In comparison to [43], the results agree for the "region of the second-order phase transition", while we are not capable of detecting the first-order phase transition found with quantum Monte Carlo calculations, as expected.

. Additionally, the ED results can be affected by the nearby phase transition, potentially happening before the closing of the gap by a first-order transition.


Most of these sentences are confusing to me as a reader. I think it should be made explicit which transitions are first and second order in the full phase diagram.

In the previous sentences, I added quotes around the many vague statements which fail to inform clearly the reader of which transition are first and second order. Words like “some of the transitions” or “certain phase transition lines” make it very hard especially without access to [43] to understand which are are which. The authors should be much more precise in their description of the various transition saying explicitly which one has been shown to be of which order.

The phrases “in the realm of a second-order transition” is used or “the results agree for the region of the second-order phase transition”, do not allow the reader to understand which specific region of the phase diagram the authors are talking about.

Therefore, they should be explicit about the order of every phase transition in figure 3 and an important effort of rewording of the various mentions of phase transitions has to be made in order to clarify the discussions and how it all relates to the unpublished results of [43].



4- The whole physical intuition section 2.2 seems discusses some points ahead of the proof and in itself is not necessarily a bad idea. However, the sentence:

As only the k = 0 mode couples to the light field, it is reasonable that Eq. (11) will decouple from the light part in HD for N → ∞.

seems to be the central point which lead the reader to gather physical intuition about their main result (namely this decoupling). I, however, do not see why the fact fact that only the k=0 mode is coupled should somehow resonably lead to such a decoupling. The sentence seems to suggest that this is obvious, but if so, it really needs to be better explained why one should expect eq. 11 (which does contains k=0 terms) to decouple. If there is no such clear intuitive way of seeing it, the authors should replace “it is reasonable that “ by a formulation such as “it will be proven that” …


5- The authors say:

"As HMM conserves the number of magnons, this part of the model is block diagonal with respect to the number of particles and can be solved, e.g., with exact diagonalization as done in [46]. "

however, ref. 46 deals with the linked coupled cluster series expansion and not with what I would usually call exact diagonalisation methods. It might be worth clarifying this issue.

On the same topic, When the authors present their results in the last sections (figures 4 an 5) they never explicitly indicate how the energy of the g independent magnon modes, coming from the matter-matter parts of the hamiltonian, are computed. The answer might be very simple but I don’t see clearly the answer when looking at (28) for example even without the (ω0 −2cJ) ̃b0† ̃b0 term.


6- Some of the formulations are problematic either for a grammatical/syntactic point of view, some of which are liste d below and should be addressed:

a- Page 2: “ While the matter part of the Dicke model consists out of an arbitrary number of spins, it breaks down to a local problem in the case of no light-matter interaction, as the local degrees of freedom are only coupled through the cavity, making it trivially solvable. “


Since local is usually employed even for problems with short range interaction, I feel like the authors should modify the sentence to use “non-interacting problem” instead of “local problem”.



b- page 3: To keep the sums over all distances finite, we restrict the coefficients to hold

the verb to hold is not correct in this context and could be replaced by “to be such that” for exemple


c - page 5: Despite the correlated processes proportional to cδ1,δ2,δ3, the general Hamiltonian H in Eq. (1) is solely made up of uncorrelated one-magnon terms in the matter part, including number operators and hopping processes.

In this sentence the word despite is wrongly used. It should be “apart from” or “to the exception of” for exemple; using despite means that the correlated processes are also uncorrelated.


d- page 14-15: “While we have no direct handle to the deviation of the energies of the finite systems in relation to the derived effective theory, we can motivate the scaling of N−1 in Eq. (40) with perturbaion theory. “

[…]

“This motivates the found scaling of the energy in Eq. (40) “

the phrase motivate the scaling is used twice in three sentences making the second time sound bizarre (or at least redundant)

e- page 6: we proof in this concluding derivation section

proof -> prove

f- page 8: as it behaves trivial in this limit.

trivial -> trivially

g- page 9 unaffected of perturbations in g

unaffected of -> unaffected by

h- page 15: While this issue can be overcome by modifying the underlying transformation

overcome -> overcomed

i- Page 16: The sentence: Introducing terms that prevent the mapping to hold would therefore act as a potential candidate to find emerging physical phenomena that can not be described by a non-interacting matter part in an effective model.

reads awkwardly: Introducing terms […] would allow to find emerging … (would, for exemple, be a better wording than: introducing terms […] would act as …)


j -In appendix c:

“of exactly two operators despite a constant term.”
here again it appears that the word despite is wrongly used in place of “apart from” or “except for”

The manuscript by Schellenberger and Schmidt discusses models involving a translationally invariant lattice of spins, with general (but bounded) hopping between sites and interactions between sites, along with coupling to a single photon mode. This class of models generalize the Dicke-Ising model, or can be seen as Dicke models generalized to include forms of interactions within the matter component. The key statement of the manuscript is that one can separately consider a Dicke sector (consisting of the zero-momentum magnon mode and the photon) and a matter sector consisting of all other magnon modes. The essential point of this is that one can understand the phase boundary to the superradiant phase purely by considering the Dicke sector.

The actual statement seems slightly more subtle than the summary above: the example of section 3.3. shows a slightly more complicated case. Here the nature of the magnetic interactions induce an order that leads to two inequivalent sublattices. This then means the Dicke sector contains instead two magnon modes. The logic here would presumably apply more generally e.g. problems with spin wave ground states as might arise from Dzyaloshinskii Moriya interactions would presumably have even more magnon modes in the Dicke sector. This would suggest there is in fact influence from the magnon Hamiltonian in determining what ground state one should consider before building the effective model. However, the main point of the paper stands: that once one has found that ground state, the Dicke sector and interacting magnon sector decouple.

To show the main results the key point in the manuscript is to show that the Dicke and magnon Hamiltonians, H_D and H_MM commute. Specifically, that (when restricted to small numbers of excitations), the commutator vanishes in the thermodynamic limit. In addition to this proof, there are numerical results shown from exact diagonalization on small systems that support this, as well as some perturbative calculations.

It is unclear to me that the manuscript meets the specific expectations for SciPost Physics, as opposed to SciPost Physics core. The work seems a useful contribution to understanding of more complicated Dicke models, but it is unclear it represents a "groundbreaking" theoretical discovery, nor that it counts as solving a long-standing research question. As the manuscript makes clear, there has been significant work on various extensions of the Dicke model in previous papers. This manuscript helps clarify some points found in previous work, but I do not see a convincing case it represents a significant step beyond that previous literature.

I would support publication in SciPost Physics Core after the minor points listed below are addressed.

1. In the introduction, the selection of papers about various extensions of the Dicke model does not seem well structured. For example, some of these papers are on situations with multiple photon modes (e.g. Refs 16, 25) but no distinction is made between single and multimode experiments. The field of extensions of the Dicke model is very large so it is not reasonable to cite everything in this field. However the references given should match better to the particular extensions being listed.

2. Equation 3 should probably have parentheses in the last term.

3. The statements in Eq. 5 and 6 could be made more precise. I think the meaning assumed here is that all these quantities approach a finite limit as N goes to infinity; if so, this should be stated more clearly.

4. After Eq. 10, when saying the two parts decouple in the thermodynamic limit, I think that this should also say it assumes finite occupations (i.e. assumes Eq. 6). This restriction is noted in all other cases where the thermodynamic limit is discussed.

5. In Figure 2 caption, the blue region (hybrid magnon-photon continuum) is not explained that clearly. The wording here describes this as the "two-magnon subspace ... which is modified by the hybrid state at k=0". However since the key point of the paper is that the Dicke sector and Magnon sector decouple, this wording seems surprising. I assume what is meant here is that this continuum is formed of states that contain one excitation in the Dicke sector and one in the magnon sector. This needs clarifying.

6. On page 6, start of Sec. 2.3, "we proof" should read "we prove"