SciPost Submission Page
Symmetry-enforced minimal entanglement and correlation in quantum spin chains
by Kangle Li, Liujun Zou
Submission summary
| Authors (as registered SciPost users): | Liujun Zou |
| Submission information | |
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| Preprint Link: | scipost_202503_00029v1 (pdf) |
| Date submitted: | 2025-03-19 03:31 |
| Submitted by: | Zou, Liujun |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The interplay between symmetry, entanglement and correlation is an interesting and important topic in quantum many-body physics. Within the framework of matrix product states, in this paper we study the minimal entanglement and correlation enforced by the $SO(3)$ spin rotation symmetry and lattice translation symmetry in a quantum spin-$J$ chain, with $J$ a positive integer. When neither symmetry is spontaneously broken, for a sufficiently long segment in a sufficiently large closed chain, we find that the minimal R\'enyi-$\alpha$ entropy compatible with these symmetries is $\min\{ -\frac{2}{\alpha-1}\ln(\frac{1}{2^\alpha}({1+\frac{1}{(2J+1)^{\alpha-1}}})), 2\ln(J+1) \}$, for any $\alpha\in\mathbb{R}^+$. In an infinitely long open chain with such symmetries, for any $\alpha\in\mathbb{R}^+$ the minimal R\'enyi-$\alpha$ entropy of half of the system is $\min\{ -\frac{1}{\alpha-1}\ln(\frac{1}{2^\alpha}({1+\frac{1}{(2J+1)^{\alpha-1}}})), \ln(J+1) \}$. When $\alpha\rightarrow 1$, these lower bounds give the symmetry-enforced minimal von Neumann entropies in these setups. Moreover, we show that no state in a quantum spin-$J$ chain with these symmetries can have a vanishing correlation length. Interestingly, the states with the minimal entanglement may not be a state with the minimal correlation length.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Thank you for dealing with our draft, and we also thank the referees for their comments. We have addressed these comments in the response to each referee on the submission webpage, and have improved the manuscript accordingly.
For the comments from referee 1, we have highlighted the intuition behind the result and the potential applications of our conclusion in the Discussion section.
For the comments from referee 2, we explain why our work is original and valuable, and why the criticism of referee 2 is invalid.
We believe that the revised manuscript meets the high standard of SciPost Physics and is suitable to be published there.
Best regards,
Kangle Li and Liujun Zou
List of changes
1. At the end of the second paragraph in Section IV, we have added ``This is because...''. This describes the idea of applying the AKLT-like states in searching the minimal symmetry-enforced correlation length.
2. We have modified the first paragraph in Section IV C by adding ``Due to the ...'', which explains why we consider the examples below.
3. In Section V (Discussion), we have added two paragraphs (the second and third) on the intuition of understanding the minimally entangled states and its applications in studying the relation between correlation length and entanglement measures.
4. In Section V (Discussion), we have changed the part related to the convergence of entanglement entropy (Point 1 under ``which we believe are reasonable ...'' and Point 1 under ``Below we enumerate ...'' ) to add a special case that has been proved. Correspondingly, we have added a new section in the appendices (Appendix A in the revised version) to present the details of the proof. We also mention Appendix A in the second paragraph in Introduction.
5. In Appendix E we have added a sentence in Observation 2, ``and it is always ...''.
Current status:
Reports on this Submission
Report
I would like to thank the authors for their detailed response to my report, and I apologize for missing some of the aspects of the manuscript. Here I give a response to all the authors' points:
1- First, the referee seems to miss one of the main results of our work. In addition to the minimal entanglement and the inequivalence between minimal entanglement and minimal correlation length, we also show that the correlation length cannot be zero. This point does not appear in the summary of our paper by the referee.
Response: I apologize for not mentioning this result. On the other hand, I think it is quite obvious that the correlation length cannot be zero: A state with zero correlation length is a product state, and a translation invariant product state necessarily breaks SO(3) symmetry.
2- Second, the referee's main criticism of the minimal entanglement part is that the result is intuitively expected. However, even if a result is intuitively expected, its rigorous derivation is still valuable if the derivation is nontrivial. Consider another example, the entanglement area law of a ground state of a gapped local Hamiltonian is also intuitively expected, but its proof is still very important.
Response: I am well aware that mathematical proofs are important, and that derivations can be an interesting addition to the literature. In this case, however, I do not believe that the derivations are particularly interesting, since all results follow directly from the formalism of symmetric MPS. Again, the structure of SU(2)-invariant MPS with integer spin is well-known in terms of fusion trees, see e.g. [1]: Constructing the allowed fusion trees underlying the SU(2)-invariant MPS directly leads to all the results in the paper.
3- Third, the intuition of the referee is that "it is well known that MPS with integer-J spin chains allow for SPT-like order (half integer bond representations) or trivial order (integer bond representations). Given that fact, it is quite clear that the minimal representations are the ones presented in the paper." This intuition is actually incorrect. As we remark in the paper, the (0, J) representation has neither the SPT order nor the trivial order. Instead, it has a spontaneous symmetry breaking (SSB) order, which is beyond the above intuitive expectation (note that the states which do not spontaneously break the symmetry can have entanglement arbitrarily close to this state, as we discuss in the paper). Moreover, even if one has SPT order, trivial order and SSB order all in mind, it is not obvious which state has the minimal entanglement, since for each order there are infinitely many states and careful analysis is needed. Additionally, specific properties of the SO(3) group are crucial in our rigorous derivation of the minimal entanglement, and it is not entirely straightforward to generalize our results to other internal symmetry groups. Therefore, any intuition based on the fact that there are SPT order, trivial order and SSB order is unlikely to hold in general.
Response: My point was not that the minimal entanglement is directly related to the type of order (SPT, trivial or SSB) the state exhibits. I was mentioning the difference between SPT and trivial in order to make clear that the symmetry structure of SO(3)-invariant MPS is rather well known in the literature, because it is crucial for classifying MPS. Again, using SU(2) fusion trees, everything follows naturally. For that matter, I think the generalization to other symmetries such as SU(N), can be carried out by constructing the allowed SU(N) fusion trees and writing down generalized AKLT states, see e.g. [2] for a recent contribution in that context.
4- We agree that we did not fully solve the problem of the minimal correlation length, which is also explicitly commented on in the draft. However, we believe that our discussion is still physically meaningful for the following reasons.
It is true that we do not have to expect that the minimally entangled states exhibit the minimal correlation length. Nevertheless, the minimally entangled states (in particular, the AKLT-like states) provide a natural starting point to study the minimal correlation length, since it has a relatively simple tensor structure. Its correlation length can be regarded as a reference, and we can ask how the correlation length of the state changes when the tensor of the AKLT-like state is perturbed. This may be an interesting insight for studying the relation between correlation length and entanglement measures.
As for the current manuscript, we remark that the statement "the minimally entangled MPS do not exhibit the minimal correlation lengths'' is just a feature of our first-step exploration, not the most essential part of the paper. We keep the record of some numerical results to show that we can lower the correlation length by adding off-diagonal spin blocks in the tensor, which has not been explicitly discussed previously, as far as we know.
On the other hand, the data of the small correlation length we found can be interesting on its own. As mentioned, the AKLT-like states provide a good reference scale of correlation length to start with, and value we found is the smallest that we can find through global minimization using a laptop. For future research, our data can be an interesting object to compare with.
Response: I agree that it could be interesting to investigate what is the minimal correlation length for an MPS with translation and SO(3) symmetry. I would expect, however, that the correlation length can be further decreased by increasing the bond dimension of the MPS, allowing for higher-spin representations on the bonds. In that respect, it is not obvious why the minimal-entanglement states are the right starting point. As the manuscript shows, there is no relationship between minimal entanglement and minimal correlation length.
5- Summarizing the replies above, we believe that our paper is valuable and suitable for being published on SciPost Physics.
Reponse: As I stated in my previous report, I think the manuscript can be published in another journal (such as SciPost Physics Core), but I don't think the results are original or impactful enough to warrant publication in SciPost Physics. I hope I have been able to motivate my opinion.
[1] Philipp Schmoll, Sukhbinder Singh, Matteo Rizzi, Román Orús, A programming guide for tensor networks with global SU(2) symmetry, Annals of Physics 419, 168232 (2020)
[2] Loïc Herviou, Anthony Rey, Frédéric Mila, Singularity with and without disorder at AKLT points, arXiv:2411.17848
Recommendation
Accept in alternative Journal (see Report)