Flat Bands and Compact Localised States: A Carrollian roadmap

The work by Ara, Banerjee, Basu and Krishnan is very interesting and deserves in my opinion publication in a high-level journal such as SciPost. It easily meets expectations and criteria for this journal, as fas as I can tell.

In my opinion, one high added value of this work is to establish and elaborate on a new connexion between two seemingly disparate communities: that of high-energy/gravitational physics and condensed matter theory. Common grounds between these two fields are of course not new, and date back to 2007-2008 with the first occurrences of AdS/CMT. The present paper however discusses a new and potentially far-reaching occurrence of this connexion: the relevance of a particular type of symmetry (Carroll), that was argued to appear in flat band systems in particular. This symmetry has nowadays become instrumental in the endeavour of understanding gravitational systems in asymptotically flat space-times under the name of "Carroll holography" and its links to the celestial holography program.

The authors present a construction of flat-band fermionic lattice systems in one spatial dimension, characterized by Carrollian symmetry and supertranslation invariance. From nilpotent matrices, they construct ultra-local Hamiltonians whose eigenstates are Compact Localised States (CLSs). The paper further explores the dynamical and entanglement properties of such systems, introduces supertranslation-invariant interactions, and analyses quantum phase structure (both at and away from half-filling), highlighting an exotic phase with a highly degenerate ground state.

High-energy/gravitational physicists might not be not well-acquainted with some of the condensed-matter concepts the paper deals with, which could result in requiring efforts to get into some parts of the paper, which is otherwise very well-presented. Since I believe the results of this paper could be of high interest for both high-energy and cond-mat communities, and in order to bridge the gap more smoothly between them making the paper even more useful, I would suggest to elaborate on some aspects and material tackled in the paper and not necessarily background material for the gravity community.

Below is listed (in order of appearance in the manuscript) a set of questions/remarks of various levels of importance, from trivial to more technical, and from typos to general clarifications. I believe it would be useful if the authors could address these points.

Comments/questions:

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

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

(3) p3: "resultant" -> resulting?

(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]." -> is a bit redundant with last paragraph pf page 2.

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

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

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

(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 $t_1 \right arrow t_2$ limit and energy scales?

(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 connexion between flat band systems and nilpotent matrices is established?


(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).

(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?

(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)?

(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?)

(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?

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

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

(17) p20: "to understand how our eigenstates written " -> are written

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

(19) apriori -> a priori

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

(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)?


(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?

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

Ask for minor revision

The paper easily meets the journal's acceptance criteria, as it strengthens the remarkable link between seemingly exotic spacetime symmetries - Carroll symmetries, which arise in the limit of vanishing speed of light from Poincare - and condensed matter systems with specific phase structure, esepcially the emergence of flat bands. It is not trivial to find interacting theories that exhibit Carrollian symmetries and the model (4.1) introduced and studied in this paper constitutes important progress. Given the listed strengths and the (by comparison) marginal weaknesses I suggest publication of this work in SciPost Physics after the authors amended their manuscript to address the weaknesses.

1- There is a repeated typo where they write "Appendix.(A)". The "." and parentheses should be removed. 2- When they mention holographic computations of entanglement entropy they probably should add the holographic computation of entanglement entropy by W. Song et al, e.g., the one using swing surfaces. 3- I do not think Appendix A reaches the target audience. In particular, the algebra (A.1) does not look at all like the algebra (3.8) in the main text, so they should add an explanation how introducing modes in (A.1) produces (3.8). Another subtlety that they do not explain in Appendix A is how the central charges emerge. Their vector field representation (A.2) leads to c_L=0=c_M, which is not what they are using in the main text. I found also the quote of Ref. [28] misplaced when mentioning BMS_3. Instead, they should quote gr-qc/9608042 and gr-qc/0610130 where this algebra was introduced (in the second case with central extensions). 4- Given that one of the key points of the papers is to come up with an interacting model that in some limit has Carroll symmetries it seems fair to mention other constructions of interacting theories with Carroll symmetries, for instance, models with spacetime subsystem symmetries, see 2303.15590 (while the relation to Carroll was not worked out so clearly in that paper, some of the later papers made this connection transparent and could be quoted as well).

Publish (surpasses expectations and criteria for this Journal; among top 10%)

In this work, the authors studied how to use the Carrollian symmetry to construct one-dimensional lattice fermionic systems with all flat-band spectra. They discovered a striking relation between the compact localized states (CLS) and Carrollian symmetry. They showed that the CLS modes, which are essential in generating flat-band system, can be generated by the supertranslation transformation in Carrollian quantum field theory. With the site-local CLS modes, they were allowed to construct the supertranslation invariant interaction terms. In the work, they introduced four-fermion interactions into the systems and studied the quantum phase structure, both with the filling factor constrained at half and unconstrained.

The work presents some novel and important findings. It may lead to further studies of the strongly correlated systems with emergent ultra-local features from Carrollian symmetry point of view. I would like to recommend it for publication in its present form.

Publish (easily meets expectations and criteria for this Journal; among top 50%)