First-order superradiant phase transition in magnetic cavities: A two-leg ladder model

We thank the referee for the honest and factual report. We first reply on what is, in our opinion, the critical novelty element of our study. In the concluding remarks the referee writes:

<< The author seem to imply that the "groundbreaking" aspect of their study is the first-order nature of their transition but there are plenty of Dicke-style models with first order transitions.>>

Even though the spirit is similar to Dicke-style models we see two crucially defining differences in the presented ladder model. - The examples we have in mind of first order transitions (Refs [2,3]) have been proven, in a recently appeared work by some of us [1], to be artifacts of broken gauge invariance. Hence to the best of our knowledge, the present work is the first example of a first order photon condensation in a gauge invariant model.
-The nature of the matter degree of freedom is fundamentally different as our study deals with itinerant electrons and not localized emitters.

We now reply at the listed comments of the referee report:

<< 1) The paper uses the term "superradiant" 50 times throughout the paper including the title. While I understand the historical context of this term, nothing is "radiating" in the phases discussed in the paper.The formal justification is that the photon field develops a static expectation value but with the same justification one could identify any magnet (and in some gauges even systems with a static electric field, like a charge) as a "superradiant" state. The authors are fully aware of this issue and do identify their "superradiant" state correctly as a Condon phase (i.e., a phase with spontaneously created static magnetic field). Nevertheless, I think one should just avoid the term "superradiant"when absolutely nothing is radiating. >>

As the referee pointed out, the term “superradiant” has an historical motivation and only in the last years the term Condon phase has been reintroduced for such transitions. Given the terminology used in some recent related works [1] we would like to comply with the referee's suggestion and use the term “photon condensation” instead of “superradiant” transition. The title would change to “First-order photon condensation in magnetic cavities: A two-leg ladder model”.

<< 2) For me a main issue is (again this is something the authors explicitely mention in their paper) that the model and the investigated physical regime appear to be completely unrealistic. Current-current interactions are extremely small (suppressed by powers of the fine-structure constant) and I do not see how even a perfect cavity could help (there is no resonant enhancement anywhere and the induced magnetic fields are just static). >>

Even though we agree with the referee that our model is difficult to realize experimentally, as also clearly stated in the manuscript, we preferred to focus on more fundamental aspects which are well captured by the presented simple model. In particular we remark that it does not spoil basic conservation laws (gauge invariance) and focuses on first order photon condensation; investigating a previously unexplored mechanism for the appearance of a Condon phase in a system of itinerant electrons.

<< Can it help to look for moiré systems or some suitably formed molecular networks?>>

We note that Condon instabilities have already been explored in Moirè structures [4], but the focus has been on instabilities of the normal phase hence second order transitions. As a general statement, we agree that looking for a system with a huge unit cell area should, in principle, increase the effective magnetic coupling. However we kept ourselves from discussing the details of possible realization in physical systems as this would require a system-dependent analysis which is not the scope of this work. Moreover we want to stress that the quasi one dimensional nature of the model, even though far from possible 2 dimensional material realizations, allowed us to do a careful quasi-exact numerical treatment of the full problem with DMRG.

<< 3) The authors could, perhaps, add a discussion that mean field is exact for such type of models in the thermodynamic limit and that one can furthermore calculate fluctuation effects systematically in a 1/N expansion of precisely the form used by the authors. >>

We agree with the referee that photon mean field treatments for such models are exact when one is interested in thermodynamic properties, such as for scanning the matter phase diagram or calculating extensive matter observables. We might have not stressed it enough and we thank the referee for pointing this out. We plan to add a more explicit sentence in the discussion part and in the conclusions in a refined version of the manuscript. It is indeed one of the results of the present study to corroborate this idea with quasi-exact numerical calculations (DMRG).

<< In this context it is surprising, that there still seems to be some mismatch of the 1/N results to DMRG extrapolated to large N. Does this have a trivial origin in 1/N effects arising from the difference of boundary conditions? Or is it a numerical issue? >>

The very small mismatch (around 5x10^-3) in the extrapolated thermodynamic limit between DMRG (with opened boundary condition) and photon mean field + gaussian fluctuations (with periodic boundary conditions) is discussed relative to photonic observables which are converged within the DMRG numerics. However the extrapolation to the thermodynamic limit is done with a “naive” 1/L correction ansatz. Given the high degree of non-linearity of the coupling and the non additivity of the Hamiltonian we cannot exclude the presence of more subtle corrections which we cannot address with the presented data. We cannot also exclude the role of boundary conditions, even though we find it very unlikely that this should have an effect on the extrapolated values. As this point is not addressed in the present version of the manuscript we plan to add a more clear comment in this sense.

<< 4) The authors compare two models (square and triangular ladder) which both show the same behavior. A different behavior should occur if one goes for a system where an tiny flux already opens a gap at the Fermi energy. I expect this to happen in almost any 1d ladder model with a q=0 Dirac point (as, e.g., realized in metalic carbon nanotubes, but most likely also in some simple ladder systems). Did the authors consider this option? >>

We thank the referee for this insightful comment. We indeed tried different configurations of the ladder system, mainly by tuning the hopping parameters and filling of the ladders that in the presented results are instead fixed. We preferred to focus on two cases where the transition is first order to better convey the message but, as briefly mentioned in the conclusions, transitions of second order (small flux) are possible (e.g. t2=0 and t1>2*t0 at half filling is insulator to insulator, t2=0 and t1=t0 but quarter filling is metal to metal). Even though there are cases of the presented ladder models with q=0 dirac points (e.g. t0=1 t1=-t2>2), a finite magnetic flux as naively described in the manuscript does not open a gap but rather shifts the chemical potential. Of course we do not exclude that other ladder models where the q=0 Dirac point and/or the coupling to magnetic fields have a different nature could have a gap opening for small external magnetic fields but we think this interesting option should be addressed in a separate study.