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Fragmentationinduced localization and boundary charges in dimensions two and above
by Julius Lehmann, Pablo Sala de TorresSolanot, Frank Pollmann, Tibor Rakovszky
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Submission summary
Authors (as registered SciPost users):  Julius Lehmann · Tibor Rakovszky · Pablo Sala de TorresSolanot 
Submission information  

Preprint Link:  scipost_202209_00033v2 (pdf) 
Date submitted:  20230115 20:12 
Submitted by:  Sala de TorresSolanot, Pablo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approaches:  Theoretical, Computational 
Abstract
We study higher dimensional models with symmetric correlated hoppings, which generalize a onedimensional model introduced in the context of dipoleconserving dynamics. We prove rigorously that whenever the local configuration space takes its smallest nontrivial value, these models exhibit localized behavior due to fragmentation, in any dimension. For the same class of models, we then construct a hierarchy of conserved quantities that are powerlaw localized at the boundary of the system with increasing powers. Combining these with Mazur's bound, we prove that boundary correlations are infinitely long lived, even when the bulk is not localized. We use our results to construct quantum Hamiltonians that exhibit the analogues of strong zero modes in two and higher dimensions.
List of changes
We have updated the text to include: The lower bound on the number of frozen states applicable to every spin $S$, previously contained in Appendix B, in Section 3.2. Section 3.3, with the updated title ``Strong fragmentation of the configuration space'', now includes Appendices C and D regarding the argument about the strong fragmentation of the configuration space and the lower for frozen states respectively. The content of the former now appears as proof of Theorem 2. We have also corrected several typos appearing in the text.
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Reports on this Submission
Report 2 by Alexey Khudorozhkov on 2023117 (Invited Report)
 Cite as: Alexey Khudorozhkov, Report on arXiv:scipost_202209_00033v2, delivered 20230117, doi: 10.21468/SciPost.Report.6549
Report
The authors addressed the main raised issue.
Indeed, the proof from Appendix H of SciPostPhys 13.4.098 straightforwardly generalizes to higherspin models with the "discrete Laplacian" Hamiltonian on the hypercubic lattice (and perhaps on any translationally invariant graph). Now, I am convinced that such models do not exhibit any strictly local conserved quantities and that the strong fragmentation originates from the restricted dynamics of the model, rather than from the local integrals of motion.
I have another minor issue about the revised manuscript and a possible typo:
1. The bound for the number of frozen states I gave in the previous review, $(2S1)^{L^2}$, is valid not only for $S=2$ (as the authors claim in the revised manuscript), but for any $S$. What's more, for any $S\geq 2$, this bound is larger than the bound calculated from tiling the plane with specific configurations, $(2S+1)^{\frac{5}{9}L^2}$. Therefore, the claim "Nevertheless, we can also derive a less tight but rigorous lower bound which holds for all halfinteger S." is a bit meaningless. The bound $(2S1)^{L^2}$ is no less rigorous and is a better bound for any $S\geq 2$. The $(2S+1)^{\frac{5}{9}L^2}$ bound is only tighter for $S=1/2, 1, 3/2$.
2. In the proof of Theorem 2: "If a configuration has n frozen sites, i.e. sites whose state cannot evolve, then it is connected at most to $M^{L−n}$ other configurations.". Did the authors perhaps mean $M^{L^d n}$? Otherwise, it is not clear to me why it is $M^{Ln}$.
Author: Pablo Sala de TorresSolanot on 20230221 [id 3387]
(in reply to Report 2 by Alexey Khudorozhkov on 20230117)We thank the referee for noticing the typo pointed out in the second comment. We have corrected it in the new version of the draft substituting $M^{Ln}$ by $M^{L^dn}$.
We also thank the referee for the other valuable comment. However, we remain uncertain about some aspects of the bound. We do not understand why the referee claims the bound $(2S1)^{L^2}$ as the argument they gave in their previous reply ''Any state with no empty or maximally occupied sites is frozen'' seems to apply only to $S=2$. In general, any local state with $m\in[0,3]$ cannot fire and any $m\in[2S3,2S]$ cannot antifire. However, the intersection is nontrivial only for $S=2$ (giving rise to the bound suggested by the referee) as well as for $S=5/2$ and $S=3$. In fact, it is easy to find configurations that are not frozen and have no site with $m=0$ or $m=2S$: For example take $S=3$ (i.e, $m$ takes values in $\{0,1,\dots,5,6\}$) and make $m=4$ everywhere. Then any site can fire. Similarly for halfinteger spins, e.g. $S=5/2$ taking values $\{0,1,\dots,4,5\}$, choosing $m=3$ everywhere has the same effect. This generalizes to any spins $S>2$ and as such, we do not see the argument for the proposed bound.
Alexey Khudorozhkov on 20230222 [id 3389]
(in reply to Pablo Sala de TorresSolanot on 20230221 [id 3387])Sorry, I was wrong about the bound. You are right, it only applies to $S=2$.