Hierarchic superradiant phases in anisotropic Dicke model

reviewer report for "Hierarchic superradiant phases in anisotropic Dicke model"

The authors consider a generalized (anisotropic) Dicke model (1), where counter-rotating-wave and and rotating-wave terms may have different amplitudes. By using a standard Holstein-Primakoff transform in the thermodynamic limit, they arrive at the usual quadratic harmonic oscillator system. Diverting from the standard treatment, they perform a rotation (5) and introduce non-Hermitian Nambu spinors with the aim to decouple the new modes. In the resulting Hamiltonian (6), the block matrices are non-hermitian and display a formal similarity to exceptional points where both eigenvalues and eigenvectors become degenerate. The exceptional points define additional phase boundaries in the superradiant phase, where sub-phases can be classified by the stability of the Hamiltonian. To highlight the physical relevance of the exceptional points, the authors consider the Loschmidt echo for a particular initial state and indeed find from extensive finite-size numerical simulations that this experimentally accessible quantity also shows the phase boundaries defined by the exceptional points with distinct dynamical features (Figs 2,3). They conclude that these hidden exceptional points signal significant changes in complete sets of eigenstates.

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In general, I found the paper interesting to read. I have a few objections though and would recommend the authors to revise their paper, taking into account the comments below.

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1.) I am missing a discussion of the origin of negative energies e.g. in (16). It is perfectly acceptable to theoretically discuss Eq. (1) but the light-matter minimal coupling procedure would normally yield a quadratic term that prohibits the superradiant phase transition in standard setups, see e.g. Refs https://doi.org/10.1038/ncomms1069 and https://doi.org/10.1103/PhysRevA.86.053807 or similar, and always provides a lower bound to the spectrum of the Hamiltonian, unlike (15-16). Since the authors care about an experimental verification, I think some link to such discussions could be relevant.

2.) What freedom does one have when introducing the spinor form in (6)? As it is written, it just looks like a convenient way to identify the phase boundaries of the model (17,18). I suppose that these could have also been extracted from (4) by a two-mode Bogoliubov transform, and we could have inferred the phase boundaries also from the vanishing of the \Omega_i. What is the extra benefit of the exceptional point picture and can it be generalized to arbitrary bosonic/fermionic quadratic models and/or multiple modes?

3.) Below (17,18), I find the discussion of stability rather confusing. A Hamiltonian without a lower spectral bound like (16) would also be unstable in the sense that a thermal reservoir would induce more and more particles in the system. So please clarify stability here.

4.) How sensitive is the highlighting of phase boundaries to the chosen initial state in the Loschmidt echo? I could imagine that the current initial state has been chosen to highlight the importance of the counter-rotating terms. Another interesting choice could be to investigate a single reservoir version of Dicke superradiant decay (all atoms in their excited state, the cavity mode in its ground state).

5.) Finite-dimensional quantum systems will evolve (quasi-)periodically (e.g. https://doi.org/10.1103/PhysRevA.18.2379). I do not see that captured in (38). Please clarify the timescales over which the approximations can be expected to be valid.

6.) I also find the summary where EP-phase separations are proclaimed very different from quantum phase transitions a bit confusing. Also in paradigmatic models of quantum phase transitions like the 1d Ising model in a transverse field or Dicke model all eigenstates undergo sudden changes at the critical point, so I do not see a strong contrast here. It is true that mostly one focuses on the ground state but finite-temperature quantities may also be affected. Maybe the exceptional point picture rather highlights the role of the dynamics?

MINOR COMMENTS + abstract: rather not use/define abbreviations in the abstract (EP) + incomplete sentence between (8) and (9) + above (10): "right eigenvectors"? + before (17): "positive factor \Omega_i": From the definition below, these may become negative + below (18): "The inverted harmonic ..." sentence has no meaning + further below: "ground state has highest energy ... excited states have lower energies" is a contradiction, I suggest to use "zero-particle state" and "many-particle states" or similar + in the enumeration (i)-(iv), it may be helpful to refer to the NP, SP1, ... in Fig. 1b + Eqns (26,27) could also be omitted by referring to (22,23) + above (38): "the" missing before "evolved state" + below (40): "In each region of superradiant ..." I thought this is also true for the normal phase? + below (44): typo "trancated" + (A3) and below: put brackets around \Theta to avoid ambiguities + below (B7): sentence not completed after equation, maybe link to Eq. (38) in the main text? + references: some names of researchers (Dicke, Rayleigh, Bragg, Landau) and journals (Physical Review Letters) are not capitalized

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