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Fragmentation-induced localization and boundary charges in dimensions two and above
by Julius Lehmann, Pablo Sala de Torres-Solanot, Frank Pollmann, Tibor Rakovszky
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Submission summary
| Authors (as registered SciPost users): | Julius Lehmann · Tibor Rakovszky · Pablo Sala de Torres-Solanot |
| Submission information | |
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| Preprint Link: | scipost_202209_00033v2 (pdf) |
| Date submitted: | 2023-01-15 20:12 |
| Submitted by: | Sala de Torres-Solanot, Pablo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We study higher dimensional models with symmetric correlated hoppings, which generalize a one-dimensional model introduced in the context of dipole-conserving dynamics. We prove rigorously that whenever the local configuration space takes its smallest non-trivial 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 power-law 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.
Current status:
Reports on this Submission
Report #2 by Alexey Khudorozhkov (Referee 1) on 2023-1-17 (Invited Report)
- Cite as: Alexey Khudorozhkov, Report on arXiv:scipost_202209_00033v2, delivered 2023-01-17, 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 higher-spin 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, $(2S-1)^{L^2}$, 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 half-integer S." is a bit meaningless. The bound $(2S-1)^{L^2}$ 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 $M^{L-n}$.
Author: Pablo Sala de Torres-Solanot on 2023-02-21 [id 3387]
(in reply to Report 2 by Alexey Khudorozhkov on 2023-01-17)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 $M^{L-n}$ by $M^{L^d-n}$.
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 $(2S-1)^{L^2}$ 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 $m\in[2S-3,2S]$ cannot anti-fire. However, the intersection is non-trivial 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 half-integer 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 2023-02-22 [id 3389]
(in reply to Pablo Sala de Torres-Solanot on 2023-02-21 [id 3387])Sorry, I was wrong about the bound. You are right, it only applies to $S=2$.