The Vacua of Dipolar Cavity Quantum Electrodynamics

We thank all referees for their detailed analysis and reports of our manuscript and appreciate their very positive feedback. We provide detailed replies to the individual reports below:

Report 1:
We appreciate pointing out SciPost Acceptance criterion 6 to provide reproducibility-enabling resources. For that, we have now uploaded all data presented in this paper and code snippets to reproduce the figures to an online repository at https://doi.org/10.5281/zenodo.4018821.

Regarding the other comments, we have adopted the manuscript accordingly (see list of changes below).


Report 2:
(1) We agree with the referee that for finite systems there are no phase transitions and no criticality in a strict sense. However, even though the range of available system sizes is limited in the current numerical analysis, it still provides strong evidence that the presented phase diagrams qualitatively describe the phase diagrams for large systems:


(i) All phases are well characterized by a specific set of observables, which are large throughout the phase and decrease rather abruptly at the boundaries.

(ii) The distinctly ordered phases we observe can be described by symmetry-breaking states in the thermodynamic limit and many of them connect to well-established phases in the transverse field Ising model limit.

(iii) Transitions between the phases are accompanied by sharp peaks in the fluctuations of the respective order parameters.   

While small shifts of the phase boundaries may still occur, none of our findings indicates qualitative changes in the phase diagrams that might occur for larger N. Therefore, we believe that speaking of a phase diagram is well-justified in the present context. Note that we are also very careful not to make any definite statements about the nature of the phase transitions or their critical behavior and clearly point out that such questions cannot be answered rigorously by our finite-size numerical calculations. A direct finite-size scaling analysis is currently far out of reach for any exact analysis of this model.

Finally, let us emphasize that HcQED describes the coupling of all dipoles to a single cavity mode with fixed properties, in particular, a fixed interaction region. Simply increasing the system size N does not represent a well-defined thermodynamic limit, while a rescaling of the coupling constant, g → g/\sqrt{N}, would render the dipole-field coupling non-perturbative. Therefore, the finite-size phase diagrams discussed in the present manuscript are representative for the practically relevant scenarios, where small or mesoscopic ensembles of dipoles are coupled a field mode localized within a tiny mode volume. 

We have modified the discussion in section 3 to better convey this message.

(2) We have changed the appearance of the Figures 2 and 4 and added a separate panel for the right plots to make them easier to read.

(3) We thank the referee for pointing out the unclear definition of a crossover in Appendix F, in which we wanted to point out that some observables might not show any characteristic change at all, and that the crossover boundary depends on the observable which is considered. Indeed no observable is non-analytic at a crossover and we have rephrased this definition in Appendix F.

(4) Hamiltonian (1) has been derived in Ref. [2] and in the follow-up paper Phys. Rev. A 98, 053819 (Ref. [47] in the new version of our manuscript) the choice of the dipole gauge for performing a systematic two-level approximation has been justified analytically and numerically in great detail. We are aware that the issue about the validity of the two-level approximation in different gauges has since then attracted quite some attention. However, we would like to emphasize that this debate is mainly about alternative derivations and not about the validity of the original results. Indeed, when applied to the same physical setting all the different predictions reduce to the dipole- and Coulomb gauge Hamiltonians presented in [2].


We don't think that the discussion about the choice of gauge is particularly relevant for the current analysis, but since the reader might wonder about this point we added a brief statement together with some of the main references on this topic below Eq. (1).

(5) Our analysis of the evolution from the paraelectric to the collective subradiant state shows indeed no trend towards a non-analytical behaviour in any of the observables we have computed. Since also both of these regimes do not break any symmetries, we expect this evolution to be better described by a smooth crossover, instead of a sharp phase transition.


In the non-interacting limit Ji,j=0 larger systems can be diagonalized exactly and we have added data for much larger systems up to N=200 dipoles in the latest version of the manuscript [see Fig. 9 (b-d)]. These simulations still do not show any signs of non-analytic behaviour and further support the scenario of a smooth crossover, but do not yet provide a conclusive picture about the behaviour of the crossover boundary in the limit N→∞. 


The crossover from the paraelectric to the subradiant phase remains an interesting open problem since it occurs in a regime, where both the validity of the Holstein-Primakoff approximation (valid for small couplings) and the effective spin model HS [Eq.(6) in the latest version of the manuscript] (valid for very large couplings) break down. Also, the subradiant phase is a highly entangled state which is not captured by a mean-field ansatz or a semiclassical approximation. So far we are not aware of any analytic approach which is able to reproduce the numerical findings in the crossover region for large N.


Finally, we want to again emphasize that the limit N→∞ does not represent a well-defined thermodynamic limit in our model and that our analysis is already representative for the practically relevant scenarios, in particular with the now presented large system size results for Ji,j=0.


We have also adopted the presentation of the collective subradiant phase in the manuscript to better illustrate that the crossover scenario is strongly supported by our results.


Report 3:
(1) As discussed in the manuscript, on finite-size systems symmetries cannot be broken spontaneously. Therefore, the order parameter < a > but also the polarization < p > are always strictly zero in the ground state. However, second-order moments of the order parameters (such as < aa >, < a^\dagger a > or < p^2 > and < \sigma_x \sigma_x >), become non-zero in the corresponding regimes and are typically used as "finite-size order parameters”. From our finite-size analysis, we thus expect < a > and also < p > to be non-zero in the ferroelectric phase when the models Z_2 symmetry is eventually spontaneously broken in the thermodynamic limit. 

Furthermore, it is important to note, that the polaron transformation illustrates, that superradiant phases with non-zero < p >=< S_x > directly yield coherent states for the photons with < a > = g/\omega_c < S_x >.


We have rephrased section 3 of the manuscript to further emphasize and clarify this important point.


(2) Since the Z_2 symmetry operator of the model is a combination of dipole and photon operators, photonic and dipolar order are always related to each other, which is also obvious from the polaron transformation.
However, unlike in ferroelectric phases where the coherent photon state with < a > = g/\omega_c < S_x > is directly related to the polarization of the dipoles, we illustrate in our manuscript in detail that more complex scenarios are possible. In particular, the anti-aligned Néel phase, which breaks the Z_2 symmetry in a non-trivial way, has < S_x > = 0 (in the thermodynamic limit) and thus features a vanishing photon field. So, in this case the symmetry can be considered unbroken for the photon sector.
Even more, the 3SL normal phase on the triangular lattice features strong photon number fluctuations beyond a coherent state due to the inherent fluctuations of the dipoles, while the latter are ordered in a non-trivial way.


(3) Because of the intimate relation between properties of the photonic and the dipolar sector in the ordered phases, we mainly focus on a discussion of dipolar observables in the manuscript to obtain the phase boundaries. An analysis of the photonic sector, beyond what is already shown, will give identical phase transitions in the ordered phases.
The retarded photon Green’s function is, however, not a very good quantity to estimate phase transitions on finite-size systems. Since the energy gaps at critical points will always be finite it will not diverge at zero frequency, and a detailed finite-size scaling analysis would be necessary to see traces of this divergence.

Since such an analysis would not yield new properties of the ground state phase diagrams, it goes far beyond the scope of this manuscript.


(4) The collective subradiant phase exists in the limit J=0 and g/\omega_c>>1, as illustrated in Fig. 2 of our manuscript and discussed in Ref. [23]. In this limit, one can derive the effective model H_S from which the ground state |\psi_{cs}> with the discussed features can be derived.

We have modified sections 3 and 5 to make these points clearer in our presentation.

(5) We have changed the appearance of figures 2 and 4 and have moved the central plot into a new panel (b). This should make the figures easier to read and, in particular, the captions better understandable. We are, however, convinced that the splitting the figures, as suggested, would reduce their readability, since the data among the individual panels is related.