Flat Bands and Compact Localised States: A Carrollian roadmap

Dear Editor,

We thank the referee for meticulously going through the manuscript and for their insightful comments, which have definitely contributed to improving the quality of the work. Below we address the comments and suggestions made by the referee point by point. Associated changes in the revised manuscript have been put in at relevant places.

Reply to Report #4 by Anonymous (Referee 3)

1. p2: Could you be more precise about what is meant by "playing around with some form of local symmetries."?

   Reply: We have replaced this by a clearer statement "At the level of single particle spectrum, flat bands imply the existence of a large number of degenerate states, and hence a large number of generators of, possibly continuous, symmetries."

2. p2: "Despite this, a clear and universal Lie algebra-based understanding of it has been scarce.". Which Lie algebra?

   Reply: We have rephrased this as "Despite this, a systematic study of these symmetries in terms of Lie groups/ algebras has been scarce"

3. p3: "resultant" $\rightarrow$ resulting?

   Reply: We have made the correction in the manuscript.

4. p3: "Nevertheless, it has been shown conclusively that infinite Carrollian supertranslation invariance of the Hamiltonian and flat bands is intricately related to each other [20]." $\rightarrow$ is a bit redundant with last paragraph of page 2.

   Reply: We have omitted this statement.

5. p5: What is the mathematical definition of CLSs? Is it (2.7)? The fact the Hamiltonian tales the form (2.9)?

   Reply: In this case, yes, eq. (2.7) is the definition of the CLS. At any rate eq. (2.9) is a form of any single particle state. What sets the CLSs apart is that the quadratic form eq. (2.9) can be arrived at without going to a Bloch wavefunction basis, and the support of $\xi_j$ over the spatial lattice is compact.

6. p5: How does (2.7) relate to the "gravity" definition of supertranslations, i.e. angle-dependent translations? (in App. A for instance)

   Reply: Please refer to the first equality of eq. (2.7) and compare with the definition of supertranslation $M_f$ in eq. (A2). For gravity in asymptotically flat space-time, these $M_f$ generators define symmetries of the asymptotic boundary, which has cylindrical topology. Here $x$ coordinatizes points, as angles, on the circular foliations, whereas $t$ refers to the null-time of the boundary. Hence, the supertranslations are also named angle-dependent time translation. More often than not, in order to understand quantities in a holographic renormalization flow set-up, one needs to put a lattice cut-off on the circle, making $x$ discrete. In that case, one gets an exact equivalence between the definition of the transformation generator $\delta_f$ of (2.7) and $M_f$ of (A.2).

7. p8, bottom. Could you point at the precise relation between the Schrodinger and Carroll algebras (either explicitly in the text or reference)?

   Reply: We have added a couple of references, where these relations are spelled out, towards the end of the last paragraph of page 8.

8. p9, paragraph 1 "For more details on the interplay between energy scales and symmetry groups for Carroll invariant theories". What is the relation between taking the $t_1 \rightarrow t_2$ limit and energy scales?

   Reply: We have moved this statement toward the end of the penultimate paragraph of page 8, and for clarity, elaborated more on the connection between merging energy scales (UV/IR mixing) and the $t_1 \rightarrow t_2$ limit.

9. p9: "can be intrinsically generated using the nilpotent matrices (2.1), making our construction of flat dispersion models simple yet profound.". I agree with that statement. Is this the first time that the connection between flat band systems and nilpotent matrices is established?

   Reply: Yes, to the best of our knowledge, our work is the first to generate flat band models using a nilpotent matrix.

10. p11: "the symmetry group including conformal transformation, is generated by the BMS3 algebra [28]:" Is [28] the fist occurrence of the BMS3 algebra? See e.g. Barnich-Compère (2006) (and references therein). Reply:* We thank the referee for mentioning this. The reference to Barnich-Compère has been added. The explicit form of BMS$_3$ Lie algebra probably appeared in their work first.

11. (11) p11-12: That part of the paper refers to the Conformal Carroll algebra in 2d, while before it was only question of the Carroll algebra (including supertranslations). Do the models (2.1)-(2.12) enjoy superrotations?

    Reply: The $L_n$ generators of BMS$_3$ are not symmetries of the models discussed. This is simply because these models are gapped, whereas $L_0$, for example, is the dilatation generator. We alluded to the discussion of conformal Carroll, for drawing a similarity with the behaviours of entanglement entropy in Carroll symmetric models like ours. This points toward the fact that subsystem size independence of entanglement entropy in Carrollian theories seems to be a feature, irrespective of conformal symmetry. A side-remark: the single copy of Witt algebra generators, ie. $L_n$ of BMS$_3$ aren't strictly superrotations, as superrotations are there in 2+1 or higher dimensions.

12. p11-12: If no, to the previous question, why is (3.9) relevant? Can it be derived for a 2d Carroll field theory (not a Conformal Carroll field theory)?

    Reply: Partially answered in the above point. Moreover, when we later introduce interactions, there are phase transitions, where the system does have scale invariance. As a prelude to that, we found it useful to keep the referring to conformal Carrollian theories.

13. p15: is (3.18) exact, or only to quadratic order in $\Delta/\tau$? Is this result universal in (C)Carroll FTs? Is there a gravity/holographic counterpart of such an expression? (e.g. poles of a thermal Green's function?)

    Reply: No, the result is not exact, and was found by making a $\Delta/\tau$ expansion in the integrated up to linear order in perturbation. To the best of our understanding, the exact form of the return probability depends upon the Carroll breaking deformation term. The holographic picture of this involves deforming bulk flat space gravity by a small cosmological constant. In fact, as per our previous finding, local Carroll breaking deformations are relevant under an RG flow. Hence terms like these, however perturbatively small in the boundary theory, would signify emergence of AdS geometry deeper in the bulk. As correctly pointed out by the referee, the effect of this (either in return probability or a linear response coefficient) would indeed show up in 2 point function. These studies are very much work in progress, and would be reported elsewhere.

14. p16: Please define scar-like states. Is the claim that translation-symmetrized CLSs behave like quantum scar states? Aren't these usually associated with non-integrable systems?

    Reply: We made this remark, keeping in mind the subsystem size independence of entanglent entropy, which is one of the defining properties of a scar state. In fact scar states in flat band systems have already been studied in detail. See for example, the references mentioned in the footnote of page 16.

15. p18: notation for $C_{N/2}$?

    Reply: Probably the referee is pointing toward ${}^N C_{N/2}$. This is standard notation for "$N$ choose $N/2$".

16. p19: How is scaling dimension defined? With respect to which generator of the Carroll algebra? (see also (11))

    Reply: Scaling dimension is defined with respect to $L_0$, ie. dilatation of the conformal Carroll algebra. As mentioned above, at the phase transition points (section 5.1), the gap closes, and hence it is natural that scale invariance emerges in the system.

17. p20: "to understand how our eigenstates written " $\rightarrow$ are written

    Reply: The full line now reads: "First let us try to understand how our eigenstates, written in the CLS basis, work."

18. p21: is [53] the first time (5.12) was written?

    Reply: This is too trivial a theory to be studied from the perspective of QFT, a subject that historically was developed to address questions in high energy scattering phenomena. However, one of the present authors studied this in ref. [37] as well, again for scattering amplitudes, but from a boundary field theory perspective. Hence we included this reference here. To the best of our knowledge we didn't encounter this action, in this form, previous to these references.

19. apriori $\rightarrow$ a priori

    Reply: We have made the correction in the manuscript.

20. Sect 5.4. Make explicit the trace resulting in (5.34)?

    Reply: This is basic statistical mechanics, where each degree of freedom has the same spectrum. In this case one can construct the full grand canonical partition function using the single particle partition function. The reader may refer to any standard book like Kerson Huang for this.

21. (21) Sect 5.4.: From (5.34), can anything be said about the entropy of the system? Does it compare to (some limit of) (3.9)?

    Reply: Our main motivation behind studying the thermodynamics in Sec 5.4 was to capture the phase transition at the critical point $V = -4\tau$. In calculation of the average energy density it is not differentiable with respect to temperature across the critical point, clearly indicating the phase transition. Hence we didn't need to study any other thermodynamic property like entropy. However, one can easily see from the structure of the grand canonical partition function that the thermodynamic entropy should scale linearly with system size, $N.$ On the other hand, entanglement entropy at the ground state doesn't scale with subsystem size. Hence, we don't think of a direct way to compare the two.

22. As mentioned in the conclusion, (this work) point(s) towards a robust framework comparable to that of the study of condensed matter systems dual to gravity in AdS spacetime". In the context of AdS/CFT, an intriguing connexion between AdS gravity and the canonical Ising model has been suggested in 1111.1987, in particular that the partition function of pure Einstein gravity with $c=1$ matches that of the Ising model. Is there a flat limit" version of this statement, and would it relate to the ultra-local model addressed in this paper?

    Reply: 1111.1987 proposed an equivalence based on mapping states on gravity to states on the $c=1$ minimal model on the CFT side. This, in our opinion, is a rare case where a Hilbert space interpretation in highly quantum regime $l/G \sim 1$ of gravity is possible. There has been works towards such attempts for asymptotically flat gravity. See for example, 2307.00043, a rather recent work, where the initial steps towards Hilbert space interpretation from gravity partition function through modularity properties have been studied.

    However, in the present work and it's future follow-ups we are rather interested in RG flow mechanism from the holographic set-up. For this precise reason, we discussed the role of energy scales in determining Carrollian symmetry.

23. Define some concepts, or point at references: topological phases, DMRG, fidelity,...

    Reply: These are very standard terms which have been used generally in Physics literature for quite a long time. For the sake of the reader, who may not be well versed in these topics, we have added a few references for related reviews.

We also take this opportunity to thank the editor for pointing out the reference JHEP07(2017)142, which we have included in our discussion on flat holography, in the introduction.

We have further taken care of small typos in the manuscript and added a few references missed earlier.

Now with all the comments and suggestions made by the referee (report 4) taken care of, we hope that the revised version of the manuscript can be accepted for publication without any further delay.

Best regards

Aritra (For the authors)