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Firstorder superradiant phase transition 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 · Titas Chanda · Marcello Dalmonte 
Submission information  

Preprint Link:  https://arxiv.org/abs/2302.09901v2 (pdf) 
Code repository:  https://github.com/zenobacciconi/cavity_ladder 
Date submitted:  20230227 16:40 
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 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 superradiant 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 superradiant phases.
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Reports on this Submission
Anonymous Report 1 on 2023315 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.09901v2, delivered 20230315, doi: 10.21468/SciPost.Report.6908
Strengths
 careful study of a Dickestyle model showing a firstorder transition in equilibrium
 meanfield (+Gaussian fluctuations) compared to DMRG
 calculation of photon excitations
Weaknesses
 model most likely highly unrealistic
 some semantic issues (see report)
Report
This paper reports a careful study of a model of noninteracting fermions coupled to a single photon mode of a cavity. The coupling is "magnetic" and such systems have previously been shown to escape a famous nogo theorem which otherwise forbids equilibrium phase transitions in such models.
The study is technically of high quality and includes a comparison to DMRG. A nice aspect is that the study also computes fluctuations beyond mean fields.
I have a number of questions/comments on the paper.
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.
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. Currentcurrent interactions are extremely small (suppressed by powers of the finestructure 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). Can it help to look for moiré systems or some suitably formed molecular networks?
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. 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?
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?
In conclusion, while this is careful study by experts in the field, I am not convinced that it fulfills the acceptance criteria of scipost physics which asks for a breakthrough/groundbreaking result. Here one also has to take into account that the studied model is very unrealistic. The author seem to imply that the "groundbreaking" aspect of their study is the firstorder nature of their transition but there are plenty of Dickestyle models with first order transitions.
Anonymous on 20230330 [id 3524]
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 firstorder nature of their transition but there are plenty of Dickestyle models with first order transitions.>>
Even though the spirit is similar to Dickestyle 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 “Firstorder photon condensation in magnetic cavities: A twoleg 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. Currentcurrent interactions are extremely small (suppressed by powers of the finestructure 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 systemdependent 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 quasiexact 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 quasiexact 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 nonlinearity 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.