Time-reversal symmetry breaking and emergence in driven-dissipative Ising models

Dear Editor and Referees,

We thank the editor for considering our work to be published in SciPost and for suggesting only a minor revision. We also thank the Dr. Rose as well as the anonymous Referees for their kind words about the manuscript, and for taking time to provide valuable feedback. The authors agree with all points made, and have made changes throughout the paper to address these comments and concerns, specifically regarding the clarity of the presentation. We hope that you find the changes satisfactory.

Sincerely,
Daniel Paz and Mohammad Maghrebi

We go through each of the Referees’ suggestions one-by-one and organize them by Referee report number followed by suggestion number; for example, 2.3 denotes the second Referee’s question number 3. Our responses will be preceded by “A:”.

1.1) Please specify explicitly that \Theta (t) is the Heaviside step function.
A: We have included a few words specifying that it is indeed the Heaviside step function.

1.2) Please clarify the applicability of the modified TRS and FDR relations in section 2.
A: We have emphasized after the introduction of FDR* in Eq. (5), that FDR* and TRS* only apply to observables that have overlap with the order parameter, i.e. observables who experience the spontaneous symmetry breaking and take a finite value in the ordered phase. They do not apply to all observables in the system, unlike the usual FDR and TRS in equilibrium systems.

1.3) Please define \mathcal{F}_\omega between Eqs. 22 and 23.
A: An explicit definition of the Fourier transform used on the correlation/response functions has now been included after Eq. (3) and between what are now Eqs. (24) & (25).

1.4) Typo: Eq. 5 is referenced in the second paragraph of section 4.1, when it appears to mean Eq. 6.
A: We thank the Referee for catching this typo! It has been corrected in the updated version.

1.5) Please clarify that while Eq. 29 has been calculated throughout the normal phase, they rely on the modified TRS that is valid only near the phase transition.
A: We have included a statement after this equation explaining that the effective temperatures are only meaningful at the phase boundary as the derivation of (what is now) Eq. (28) neglects non-critical corrections. In addition, the low-frequency dynamics away from the phase boundary involves multiple modes, in contrast to the phase transition where the effective temperature is purely characterized by the soft mode. However, we also mention that despite Eq. (28) being derived using TRS*, and therefore only physically significant at the phase boundary, one can still apply this equation to the correlation and response functions anywhere in the phase diagram to extract a low-frequency effective temperature, which is what we have done in Eq. (31) of the new manuscript.

1.6) Typo: Second sentence of section 5.1, "different sets of operators".
A: We thank the Referee for pointing out this typo -- we have now corrected it in the updated version.

1.7) It is not clear why the gapped field is not relevant to the effective temperature in section 5.1. Is it due to the long-wavelength limit? If so, the order of statements seems off. Likewise for Eqs. 44 and 45. Please clarify why this can be discarded.
A: We have revised the paragraphs before Eqs. (44) & (46) in the new version to help clarify this point. The previous wording could have been interpreted incorrectly to suggest that one statement leads to another, where in fact they are the same statement just from two different perspectives. More explicitly, saying that the effective temperature is captured by the soft mode is equivalent to saying that the dominant contributions to the effective temperature come from the correlation and response functions of the soft mode. In both statements, we can neglect the gapped field because it does not participate in the low-frequency dynamics, and because it merely introduces negligible non-critical corrections to correlation and response functions.

1.8) Typo: The last paragraph of the conclusion should start with "A".
A: We have corrected this typo in the updated manuscript.

1.9) Please clarify the relevance of the $\mathbb{Z}_2$ symmetry to the calculations.
A: We thank the Referee for bringing up this point. The second paragraph of Sec. 3 as well as the discussion following Eq. (45) now contain an extended discussion on how the Z2 symmetry impacts our calculations. In addition, we have also added a discussion on the Z2 symmetry of the bosonic hopping model at the end of the first paragraph of Sec. 6. While the Z2 symmetry may not be fundamentally required to obtain a modified FDR near driven-dissipative phase transitions, we have chosen models with this symmetry as it leads to the minimal field theory characterizing a phase transition. A Z2 symmetric field theory only requires one real field to describe the soft mode near the phase transition, and it also prevents a linear term from appearing in the action (which could change the results of our calculation). Other symmetries would require a more involved analysis due to the greater number of fields needed to describe the critical behaviour; this could complicate the form of the modified FDR (if it can be shown to exist!). We state this possibility as an interesting avenue for future research in the Conclusion and Outlook.

1.10) Please clarify that the results are suggestive of what may hold for generic driven-dissipative Ising-type systems, or otherwise justify this generic statement.
A: In the discussion above Eq. (5) and in the second paragraph of the conclusion, we have included a few words specifying the necessary ingredients a model should have to expect our results to apply. The necessary ingredients are that the model has a time-reversal invariant Hamiltonian, the model is quadratic or mean-field like, and the Liouvillian has a Z2 symmetry that is spontaneously broken at the phase transition. The discussion explaining why a time-reversal invariant Hamiltonian is necessary as well as the role of the Z2 symmetry can be found in Sec. 5.1. In essence, the time reversal symmetry guarantees that only dissipation couples odd and even fields. The model needs to be quadratic or mean-field like as non-linear interactions will have a nontrivial effect if the interaction is relevant (in the RG sense) at the phase transition. Finally, we require Z2 symmetry as discussed in response (1.9) and Secs. 3 and 5.1, other symmetries could lead to more complicated forms of FDR*. Our results should hold for any model that satisfies these requirements, as near the phase transition they will have an effective action described by Eqs. (38) & (39).

The models investigated in our work come with one operator/field that overlaps with the order parameter in equilibrium (e.g. x) and is even under TR, and another operator/field that also overlaps with the order parameter but it is odd under TR (e.g. y). Some quadratic driven-dissipative Ising models, such as the open Dicke model in the large N limit, will have two even fields and two odd fields that overlap with the order parameter as it has both spin and bosonic degrees of freedom. While this will complicate Eqs. (38) & (39), we believe that this system would yield a similar FDR* at the phase transition for the same reasons as the models considered in the manuscript. Verifying this explicitly is an avenue for further research as discussed in the Conclusion and Outlook.

2.1) It would possibly be useful to also introduce the Fourier space versions of eqns 4-6 as these could be more familiar to some readers.
A: We have added the Fourier space version of the equilibrium FDR, which can be found in Eq. (3). The Fourier transform of the TRS relations does not change the equations at all, which we now mention after the introduction of TRS in Eq. (4). The Fourier space version of TRS* is simply the Fourier transform of TRS* in the time-domain just as in equilibrium. The FDR* also takes a distinct form in frequency space as is now explicitly shown in Eq. (6) in Sec. 2 for the benefit of the reader.

2.2) Should eqn 10 read |\rho(0)>>?
A: We thank the Referee for pointing out this typo -- we have now corrected it in the updated version of the manuscript.

2.3) It could be useful to give a few lines more explanation of the physical intuition behind eqn 15.
A: Thanks to the Referee’s suggestion, we have now included a discussion about the interpretation of various matrix elements in the now Eq. (17) of the revised paper.

2.4) In Fig 2 is the disagreement at short times due to finite system size? It may be useful to show a couple of values of N here to illustrate this.
A: The disagreement at short times is actually due to the applicability of the FDR. As stated in the derivation, Eq. (5) only is valid at long times/low frequencies. This leads to the short-time disagreement. In fact, it is rather surprising how quickly the convergence occurs and the slow, critical dynamics sets in. We have included a few words emphasizing the origin of the disagreement in the last paragraph of Sec. 4.2 .

2.5) In the discussion around eqn 35 the authors find that the two ways of taking the limit \Gamma -> 0 give different results. How does this behaviour show up in finite sized numerics? Is there a smooth change between the different parities?
A: At any finite size, the system is gapped even at the critical point. (In other words, phase transitions only ever occur in the thermodynamic limit.) Therefore, at a finite system size and in the limit \Gamma\to 0, the steady state commutes with the Hamiltonian. As you send N -> \infty the lower bound on the gap will continuously shrink to zero. At this point, you reach the different limits discussed in the manuscript. We have included an explanation mentioning the case of a finite-size system in the last paragraph of Sec. 4.1.

2.6) It would be useful for the authors to add a few more comments to the conclusions about how general they think the results obtained here are. Does any Markovian model with a phase transition show this kind of FDR?
A: In the second paragraph of the conclusion, we have included the requirements a model should satisfy to exhibit a modified FDR similar to the FDR* shown in this work. The necessary ingredients are that the model has a time-reversal invariant Hamiltonian, the model is quadratic (or interactions are not relevant in the sense of RG), and the Liouvillian has a Z2 symmetry that is spontaneously broken at the phase transition. Our results could very well hold beyond these conditions (especially in the presence of nonlinear interactions); however, a full picture requires further investigation. For a more detailed response, please see our comments under 1.10.

3.1) There is a commutator in definition of response, valid in quantum systems, but Eq 2 is said to be valid in classical systems as well; please clarify the definition of response function for classical systems.
A: Equation (2) is both the classical FDR as well as the finite-temperature/low-frequency limit of the quantum FDR. The classical response function is not necessary for any calculations considered in our work, but we have included a reference after Eq. (1) where interested readers can find the classical definitions of the correlation and response function.

3.2) Clarify eq 10.
A: We thank the Referee for pointing out this oversight. In the updated version of the manuscript we now mention that the inner product in the vectorized space is equal to the Hilbert-Schmidt norm in the operator space.

3.3) Eq 24 is valid in limit ω→0 only, similarly Eq 2 is valid as ℏ→0; is this what is meant by saying this holds only at large times? My understanding is that the authors look for large times because such correlations / response functions are controlled by the soft mode, hence this characterises criticality. Is this correct? Otherwise, why look only at the long-time dynamics? How would other modes affect FDR* and TRS*? Would they contribute to higher powers of ω? The equilibrium FDR includes a non-linear function of
ω, could one see differences also for ω further from 0 and attribute this to the soft modes?
A: The Referee’s understanding is correct. We primarily focus on long-times/low-frequencies because the correlation and response functions are characterized by a single soft-mode at criticality. To make this point clearer, we mention that the FDR and FDR* only apply at low-frequencies/long times anytime we introduce these equations. Regarding the Referee’s interesting question about higher frequencies, it is possible that higher frequency modes can indeed affect FDR* and TRS*. While we have not investigated this point as we have only focused on low-frequency behaviour, it is a tentative avenue for future work. We now mention this as a possible future direction briefly in the Conclusion and Outlook.

3.4) I don’t understand the use of the symbol ≃ on page 9: are these equalities? If not, in what sense are the left-hand and right-hand sides related, is it in a limit sense? What limit - is it the long-time limit?
A: We apologize for the lack of clarity. The notation \simeq is supposed to represent an equivalence up to non-critical corrections. We have defined what \simeq means right after Eq. (5) of the new manuscript, where we first introduce the notation. In addition, later in the paper we mention that the equations containing \simeq are neglecting non-critical corrections.

3.5) For clarity, please do say where the phase transitions is (Γ=Γc expressed in terms of Δ and J?) early enough in the discussion.
A: The phase transition points \Gamma_c is now defined in the third paragraph of Sec. 3, during the discussion of the driven-dissipative Ising model.

3.6) Finally, why “resurrection” in the title? It seems nowhere else in the paper the concept of resurrection is mentioned.
A: We thank the Referee for bringing up this point. After considering your question, we have decided that “resurrection” is not the correct word and have changed it to “emergence”. This choice reflects the idea that a modified form of the FDR and TRS relations emerges near the phase transition despite being microscopically broken.