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Quantum local random networks and the statistical robustness of quantum scars
by Federica Maria Surace, Marcello Dalmonte, Alessandro Silva
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Submission summary
| Authors (as registered SciPost users): | Marcello Dalmonte · Alessandro Silva · Federica Maria Surace |
| Submission information | |
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| Preprint Link: | scipost_202206_00015v1 (pdf) |
| Date submitted: | 2022-06-16 20:59 |
| Submitted by: | Surace, Federica Maria |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We investigate the emergence of quantum scars in a general ensemble of random Hamiltonians (of which the PXP is a particular realization), that we refer to as quantum local random networks. We find a class of scars, that we call ``statistical", and we identify specific signatures of the localized nature of these eigenstates by analyzing a combination of indicators of quantum ergodicity and properties related to the network structure of the model. Within this parallelism, we associate the emergence of statistical scars to the presence of ``motifs" in the network, that reflects how these are associated to links with anomalously small connectivity. Most remarkably, statistical scars appear at well-defined values of energy, predicted solely on the base of network theory. We study the scaling of the number of statistical scars with system size: by continuously changing the connectivity of the system we find that there is a transition from a regime where the constraints are too weak for scars to exist for large systems to a regime where constraints are stronger and the number of statistical scars increases with system size. This allows to define the concept of ``statistical robustness" of quantum scars.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-10-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202206_00015v1, delivered 2022-10-14, doi: 10.21468/SciPost.Report.5874
Strengths
Statistical description of scars in network models
Weaknesses
1. Misrepresentation of the models as spatially local or related to PXP
2. No connection made to closely related quantum satisfiability problems
3. Unclear what to takeaway for non-network models
Report
In this article, the authors (primarily numerically) study the emergence of quantum scars in an ensemble of random Hamiltonians. The authors define the ensemble in the computational basis: with probability p, two computational basis states that differ by a single spin flip are connected by the Hamiltonian. They find that below a critical p_c, a fraction of the eigenstates of these systems can be obtained by copying local motifs on the network, where the local motifs satisfy the local Hamiltonian. These are the statistical scars. They estimate p* for different local motifs.
The article is quite hard to follow, for example, there's a lot of talk of PXP models (Eq. 1) in the first 3 pages, but the definition of the quantum network models that the authors actually study is made in passing on page 4 with no equation. Important pieces of the physics of the statistical scars are described in figure captions alone or in single sentences buried in paragraphs (e.g. the motif discussion). The referee also did not understand what between-ness centrality was measuring.
The authors also misrepresent their Hamiltonians as spatially local! This is *not* true, in a local Hamiltonian, a bit would flip depending on the local environment in *all* configurations.Eg: in a PXP model, there would be edges connecting *all* configurations ...... 010..... and ..... 000 .... for all possibilities of .... configurations. This isn't true of the authors networks because all the edges are drawn independently. The PXP and other local models should thus have exponentially small in N probability of occurring in the random network ensemble.
This latter reason is why the referee suspects that the work is irrelevant to the study to quantum scars in local models. Multiple pages on the PXP model up front is thus misleading.
Finally, it's unclear what one learns from this study. Perhaps it's interesting that non-local network model generically have degenerate manifolds with simple descriptions of the eigenstates, but the referee suspects that this is well known in the satisfiability/network model literature. Just calling these "scars" really doesn't add much.
Requested changes
1. More concrete equations and clear discussion of the central ideas/concepts to the papers (definition of the Hamiltonian ensemble, how to tile local motifs on the graph to get the statistical scars, out-of-line definition of various network theory measures, clear discussion for what they measure, definition of periphery of the network etc.).
2. Remove assertions of spatial locality and that the network realizes the PXP model for p=1/4 (it doesn't even do this locally, see above).
3. Make connections to the satisfiability literature. In particular, were the degenerate manifolds known before? Can one think of the transition at p=p_c as a SAT-UNSAT transition?
4. Less discussion of local models like PXP up front, as these network models have nothing to do with local models.
Report #1 by Anonymous (Referee 3) on 2022-8-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202206_00015v1, delivered 2022-08-16, doi: 10.21468/SciPost.Report.5543
Strengths
The setting of the problem is interesting and intriguing, connecting the fields of graph network theory and quantum scars.
Weaknesses
-Some of the appendices are not referred and discussed properly in the main text.
-The paper appears to be originally written in the form of a letter. The authors did not change the format accordingly or expand the discussions, resulting in some bumpy reading experience.
Report
In this work, Surace et. al. study the quantum local random networks and discover that such a statistical ensemble have a high chance to have small motifs in the graph, resulting in localized wavefunctions when the probability of the connection for “local” vertices is small.
Such a claim and discovery is supported by various numerical results, including comparing the degeneracy and number of motifs, entanglement entropy, participation ratio, some graph theoretical measurements (e.g. degree, centrality and betweenness centrality).
The setting of the problem and the perspective is interesting and provides a synergetic link between the areas of graph network theory and quantum scars. Such a problem also has a potential for follow-up studies.
I therefore think that this paper has a potential to be published on SciPost. But I have some physics questions and some comments about the presentation that I hope the authors can clarify or address before I can recommend for publication.
1. The special energies have degeneracies, and most of the quantities (such as the participation ratio, degree, centrality, betweenness, entanglement entropy etc.) would depend on the particular choice of the superposition of the degenerate subspace. I am wondering if there’s any guidance or particular choice of the superposition the authors use? In particular, the authors use those quantities to infer the structure and the existence of the localized eigenstates. Does it mean that the authors try to form a superposition of the eigenstates within the degenerated eigenspace such that they are as localized as possible before calculating various quantities?
2. In Sec. III, 2nd paragraph, the authors state “When analyzing finite size QLRN, the physics of localization emerges immediately when one considers the density of states vs. ε for fixed system size N at different p ……”. Could the authors elaborate more on what they mean by “physics of localiztion”? It appears that the authors are alluding that the degeneracy implies localization. However, this is generally not true, see for examples, arXiv. 1501.00477, arXiv.1710.05927, arXiv. 1801.03103.
3. As the authors point out in In Sec. III, 3rd paragraph, it is indeed important and more interesting to understand the nontrivial localized states which are not results of some small disconnected graphs. It would be nice if the authors can state more clearly about how they rule out the trivial localized states. In particular, do the authors use some graph theoretical algorithm to identify the largest connected subgraph for each realization and study the various quantities in the largest subgraph only? If so, is the Hilbert space dimension of the largest connected cluster typically or in average grows exponentially in $N$ for all the probabilities $p$? (If so, is it still $2^N$?)
The authors also mention that at small $p$, QLRN has a phenomenon similar to the fragmentation of Hilbert spaces. Does it mean that at small enough $p$, the Hilbert space dimension of the largest connected cluster only grows polynomially? Or is it the case that the Hilbert space dimension of the largest connected cluster still grows exponentially but there are exponentially many small disconnected clusters? (The latter case would also suggest that the largest connected Hilbert space dimension grows a^N with a<2, and could change the estimation of $N_{th}$ and the critical $p_c$?)
4. In Sec. 4, could the authors elaborate on how to count the number of motif $N_{motif}$ numerically? Is it also done by some graph theoretical algorithm?
5. The graph theorical measurements might not be very familiar for the physicists. For example, could the authors define “degree” more explicitly in the main text, and expand the discussion on what it means to have a small or large degree/centrality/betweenness centrality? (See also point 8.)
I have some comments and suggestions regarding the presentation in the requested changes.
Requested changes
6. Appendices A and C are not referred to and discussed in the main text. In fact, I think Appendix C and Fig. 7 is another very useful and important information to show that other non-special-energy states are ergodic.
7. In the caption of Fig. 1, “….. indicate the special energies ($\epsilon^* = \pm 1, \pm2 ....$)”, should the $\pm2$ be $\pm \sqrt{2}$?
8. In Sec. 3, 4th paragraph, the authors defined the degree, the centrality, and betweenness centrality but then no discussions and data shown are. Some of the data are shown in Appendix E, and the authors should refer to it properly. (See also point 5.)
Author: Federica Maria Surace on 2022-12-19 [id 3157]
(in reply to Report 1 on 2022-08-16)
We thank the Referee for their careful reading and positive assessment of the manuscript, and for their constructive comments on the physical content and the presentation. Here we address their specific questions and comment:
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That is correct: most of the quantities we compute depend on the particular choice of superposition. We have not disentangled the degenerate states obtained from ED, so the eigenstates we found have support, in general, on more nodes than the simple motifs. In light of this, Figure 1c and 1d give some "global" information about the states in the degenerate subspace, for example the fact that they tend to localize in the periphery of the network, while the values of single states in the subspace are not immediately informative. It would be interesting to find a systematic way to disentangle those states, but we could not find a simple procedure that could be applied efficiently at large system size. Therefore, rather than constructing the most localized superpositions with a general procedure, we checked their presence "a posteriori" by counting the motifs in the graphs: for each motif, a localized eigenstate exists as an appropriate superposition in the degenerate eigenspace; all the properties of the corresponding eigenstate (degree, centrality, betweenness, etc.) can then be easily inferred from the structure of the motif.
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We thank the Referee for their comment: indeed, the presence of the delta peak in the density of states is not sufficient to prove localization. To claim that the origin of the peak is associated with localized states we then consider other quantities, such as the participation ratio. Nevertheless, we find it suggestive that a zero energy delta peak occurs in other random networks (Ref. 44), where similar localized states also emerge. We rephrased the text to clarify this point. We also noticed that our observation about the origin of the exponential degeneracy at ϵ=0 as a consequence of the theorem in Ref. 41 was incorrect: QLRNs are in general not invariant under inversion, so the theorem does not apply.
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To obtain the connected components of the graph we use the Python package networkx. In the range of probabilities p that we consider, our numerical analysis (see figure attached/Figure 7 in the new draft) is compatible with the scenario of weak fragmentation [PRX 10, 011047 (2020); PRX 12, 011050 (2022)]: the average number of connected components grows as a^N with a<2, while the ratio between the dimension of the largest connected component and the total Hilbert space dimension approaches 1 in the thermodynamic limit.
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To count the number of motifs we used the following algorithm. We first identified all the nodes with degree 1 (this was done with the package networkx in Python). For each of these nodes, we check the degree of the node connected to it: if it is 2, 3, or 4, we further check the other nodes connected to it. Iterating this procedure with slight variations depending on the specific type of motifs that we want to count, we can identify all the submotifs that are connected to the network through a single node. We then count the motifs by counting the pairs of submotifs that share the same "root node" (in yellow in Fig. 3 and 10).
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We thank the Referee for their comment. We have now added the definition of "degree" and expanded the discussion on the properties of the nodes.
6-8. We thank the Referee for their suggested changes, we amended the text accordingly.
Author: Federica Maria Surace on 2022-12-19 [id 3158]
(in reply to Report 2 on 2022-10-14)We thank the Referee for reading the manuscript and for their constructive criticism. We agree that the presentation was not clear enough. We nevertheless believe that a short summary of the main properties of the PXP model and its quantum scars is necessary to motivate our definition of Quantum Local Random Networks (QLRNs). To improve the clarity, we split Section 2 into two different sections: in the first one, we briefly introduce the PXP model and its quantum many-body scars, in the second one we introduce the QLRNs, that are the main subject of our investigation. We also included an explicit Hamiltonian for the QLRNs. To further improve the clarity of the manuscript, we now moved the discussion on the motifs from the caption to the main text, and we included a sentence to explain the meaning of betweenness centrality.
As the Referee correctly emphasises, the Hamiltonians of QLRNs do not in general have a representation as a sum of terms with finite support. As we specify in the text, what we mean by local" is that the Hamiltonian has non-zero matrix elements only between sites that differ by a single spin flip: such locality is reflected in the specific form of the network space, and not referred to microscopic dynamics (which, indeed, generically requires projector operators that span the entire system). This guarantees the extensivity of the Hamiltonian. We hope the text is now unambiguous.
For what concerns the relevance to the PXP model, we remark that, while QLRNs do not share the same representation as simple (i.e., with few-body terms) many-body Hamiltonian as the PXP model, they are very similar to the PXP when seen as networks: the PXP is, in fact, a subgraph of the N-dimensional hypercube graph, and QLRNs are defined as an ensemble of subgraphs of the N-dimensional hypercube graph.
The results of our work shed some light on the role of constraints (understood as the suppression of the connectivity of the network) as the origin of quantum scars. The fact that statistical scars share some features with PXP scars is not our fundamental point, but suggests that further work in this direction could lead also to a deeper understanding of the PXP model.
One of the lessons we learn from QLRNs is that by reducing the connectivity of the network (i.e., enhancing the constraints) it is possible to obtain a transition from a regime without scars to an exponential number of them. We are not aware of connections with anything well-known in the literature, other than the references already included in the manuscript. We would be grateful if the Referee could provide some references for the mentioned connections.
Reply to the requested changes: 1. We thank the Referee for the suggestions. We amended the text accordingly. 2. We now included a sentence to specify what we do and do not mean for local". 3. We are not aware of any connection with SAT-UNSAT transition. We would be grateful to the Referee if they could be more specific and explain the intuition for this possible connection. 4. We decided to maintain a discussion on the PXP model in the first section after the introduction, as it serves as a motivation for our definition of QLRNs. The discussion is now in a separate section from the definition of the QLRNs.