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Exact Mazur bounds in the pairflip model and beyond
by Oliver Hart
Submission summary
Authors (as registered SciPost users):  Oliver Hart 
Submission information  

Preprint Link:  https://arxiv.org/abs/2308.00738v1 (pdf) 
Date submitted:  20230810 00:49 
Submitted by:  Hart, Oliver 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
By mapping the calculation of Mazur bounds to the enumeration of walks on fractal structures, we present exact bounds on the latetime behavior of spin autocorrelation functions in models exhibiting pairflip dynamics and more general $p$flip dynamics. While the pairflip model is known to exhibit strong Hilbert space fragmentation, the effect of its nonlocal conservation laws on autocorrelation functions has, thus far, only been calculated numerically, which has led to incorrect conclusions about their thermodynamic behavior. Here, using exact results, we prove that infinitetemperature autocorrelation functions exhibit infinite coherence times at the boundary, and that bulk Mazur bounds decay asymptotically as $1/\sqrt{L}$, rather than $1/L$, as had previously been thought. This result implies that the nonlocal conserved operators implied by $p$flip dynamics have an important qualitative impact on bulk thermalization properties beyond the constraints imposed by local symmetries alone.
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Report
The Author performs an original analytical calculation of Mazur bounds on correlation functions in a class of models exhibiting Hilbertspace fragmentation. The results point out a discrepancy between the scalings of boundary and bulk correlations and their decay, and they show that a previous numericsbased conjecture regarding the scalings is incorrect and, in fact, currently available numerical resources are sufficient to establish such a conjecture. This work is a nice addition to the literature tackling the nonthermal behaviour in fragmented models with original, exact analytical methods.
The paper is written rather technically and, in my opinion, addresses an interesting specific problem in the field of Hilbertspace fragmented models. It certainly warrants publication in some form. However, I have checked the criteria for acceptance in SciPost Physics and am unfortunately not convinced that any one of the required are satisfied:
 Detail a groundbreaking theoretical/experimental/computational discovery;
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block;
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work;
 Provide a novel and synergetic link between different research areas.
I should stress that these criteria are strict so my judgement does not diminish the relevance of the paper in solving a particular question of a specific community. In fact, I found the methods used quite intriguing and believe they have an abovethenorm degree of originality. I admire in particular how the Author manages to point out a peculiar crossover between two different scaling behaviours, which is extremely hard to observe with currently accessible numerical resources. This work thus certainly could be published in SciPost Physics Core according to its required criteria! I would recommend publication there, after remarks below have been addressed.
The main criticism of mine concerns the role of locality for thermalization or lack thereof [see remark (5) below]. In particular, the Author regularly emphasizes the role played by nonlocal conserved labels of Hilbert space fragments in the lack of thermalization. However, the redundacny of such objects is never really checked, since bounds on correlation functions are computed using the entire basis of projectors on Krylov sectors. I do not see how this excludes possible existence of local conserved quantities that would saturate the Mazur bound. Specifically, since there is a clear discrepancy between behaviours of the bulk and the boundary correlators (a discrepancy which, as pointed out by the Author, vanishes in a periodic chain described by a model with a similar phenomenology), one should perhaps exclude the possibility of boundarylocalized modes preventing thermalization before claiming the role of nonlocal conserved quantities in that respect. Additional critique is regarding acknowledgement of other works. I would suggest the Author to consider checking more of the literature that relates the concepts of Mazur bounds, locality, and thermalization. Please find a short nonexhaustive list of recommendations in points (6) and (7) below.
Requested changes
(1) The Introduction is occasionally a bit cryptic. For example, the Author states “It is now understood that Hilbert space fragmentation can arise in more generic settings from the simultaneous enforcement of symmetries and strict spatial locality …” — What exactly is meant by simultaneous enforcement of symmetries and strict spatial locality? Locality of what?
(2) Typo on page 3: “This suggests that, in the bulk, the nonlocal patterns conserved by the dynamics do not change the scaling that [one?] obtains from the local continuous symmetries of the pairflip model.”
(3) Between eqs. (14), (15), please define $\hat{S}^z(\alpha)$. (I guess it is the diagonal element of the $\hat{S}^z$.)
(4) If I understand correctly the discussion of labels of Krylov sectors in Section 2.2, the patterns that remain after a decimation can always be moved to one side of the system, say, left. If one then cuts that left part of the system containing only the label away, that cut part is frozen. On the right side of the system, on the other hand, remain only the contiguous sequences of colours that can be flipped and reshuffled. This reminds of a typical situation of Hilbertspace fragmented systems, where any Krylov sector can be constructed by joining a frozen configuration of a subsystem (that frozen configuration determines the label of the Krylov sector) to a vacuum, or some other state containing movable particles, on the rest of the system — see for instance reference [16]. Perhaps the Author could comment on this analogy, if correct.
(5) The Author repeatedly refers to the “effect of nonlocal conserved quantities” [i.e., the ones in eq. (10)] on the dynamics. While the Mazur bounds are calculated using a basis that is equivalent to the nonlocal operators in eq. (10), it is still not clear whether and how many of these quantities are redundant. In other words, I am not sure that the calculations presented herein exclude the possibility that there exist some (perhaps small set of) local objects that are conserved and prevent thermalization. Especially so, since there is a discrepancy between the bulk and the boundary scaling of the correlation functions. — Since this discrepancy vanishes in the tJ model with periodic boundary conditions, should one expect some kind of a boundarylocalized operator [e.g., à la J. Stat. Mech. (2017) 063105] to be behind the sloweddown thermalization or nonthermalization of the correlators? While this question probably cannot be answered in this work, I would perhaps suggest the Author to at least refrain from statements that would imply nonlocal quantities affecting the correlation functions of local observables, as their redundancy has not been checked. Perhaps a more neutral statement that “fragmentation” affects thermalization is more proper, although its message is not so surprising.
(6) Since the alleged effect of nonlocal conservation laws is emphasized, it would be fitting to mention the fundamental role of locality/extensivity of conservation laws that affect thermalization — see, e.g., refs. Commun. Math. Phys. 351, 155 (2017); Phys. Rev. Lett. 115, 157201 (2015); J. Stat. Mech. (2016) 064008, etc.
(7) I believe that more works should be acknowledged, in particular concerning Mazur bounds and on the concepts of thermalization and correlation functions. Some suggestions:
 Mazur bounds on transport coefficients: Phys. Rev. B 55, 11029 (1997); Phys. Rev. Lett. 106, 217206 (2011); Phys. Rev. B 83, 035115 (2011); Commun. Math. Phys. 318, 809 (2013); Phys. Rev. Lett. 111, 057203 (2013); Nucl. Phys. B 886, 1177 (2014); Phys. Rev. Lett. 122, 150605 (2019).
 Some references on hydrodynamic projections (Mazur bounds can be thought of as truncations of the latter): SciPost Phys. 3, 039 (2017); J. Stat. Phys. 186, 25 (2022); Commun. Math. Phys. 391, 293–356 (2022).
 Mazur bound used in establishing nonthermal behaviour of correlation functions: Phys. Rev. B 102, 041117(R); SciPost Phys. 9, 003 (2020).
 Some references concerning lack of thermalization signalled by slow correlation decay in structural glasses and related models: Phys. Rev. X 10, 021051 (2020); Phys. Rev. B 92, 100305(R) (2015); Phys. Rev. Lett. 121, 040603 (2018); Phys. Rev. B 108, L100304, (2023); arXiv:2306.12467 (2023).
 Lack of thermalization of boundary spins due to edge modes: J. Stat. Mech. (2017) 063105, and refs. therein.
(8) Enumeration of equations needs to be fixed — see eqs. (38) and (52) and neighbouring ones.
(9) Section 4.1.2, the Author states “… only the first (last) dot in the label is able to touch the leftmost (rightmost) site under pflip dynamics. Hence, the fact that nonzero autocorrelation is able to develop follows from the nonzero “return probability” for the first and last dots in the label.” I do not see the implication from the first to the second sentence. Perhaps this should be explained in an additional sentence/figure, or, if trivial, rephrased so that it becomes obvious.
(10) Related to the previous point: Section 4.3.1 uses the same implication. I would appreciate some explanation, to make the rather technical part of the work easier to read.
(11) Could more numerics be performed, independently of the analytical methods used herein, in order to benchmark some of the steps of the analytical calculation (to make a reader who wishes to go through technical details fast trust the analytical calculation)?
(12) Is there any physical reason or intuitive explanation for the crossover between the two scaling behaciours in fig. 6?
Strengths
1 computation of Mazur bounds that were previously unavailable
Weaknesses
1 Physical motivation for the model lacking
2 Relevant boundary charges are local
Report
The author computes local and nonlocal charges for a "pflip" spin model previously studied numerically in Ref. [18] and uses them compute Mazur bounds for certain \emph{local} observables. The ones close to the boundary are finite in the thermodynamic limit L \to \infty and the ones in the bulk decay as 1/\sqrt{L}. However, the author refers to them as nonlocal, but if they give finite Mazur bounds for strictly local observables, they must be local operators and indeed there seem to be conservation laws localized near the boundary by inspection of Eq. (10).
Requested changes
1 Discuss as above local vs nonlocal more precisely
2 More physical motivation for the model is needed beyond that it has been studied in the previous literature