SciPost Submission Page
A theorem on extensive ground state entropy, spin liquidity and some related models
by Sumiran Pujari
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users): | Sumiran Pujari |
Submission information | |
---|---|
Preprint Link: | https://arxiv.org/abs/2407.06236v6 (pdf) |
Date submitted: | 2024-12-13 07:16 |
Submitted by: | Pujari, Sumiran |
Submitted to: | SciPost Physics Core |
Ontological classification | |
---|---|
Academic field: | Physics |
Specialties: |
|
Approach: | Theoretical |
Abstract
The physics of the paradigmatic one-dimensional transverse field quantum Ising model $J \sum_{\langle i,j \rangle} \sigma^x_i \sigma^x_j + h \sum_i \sigma^z_i$ is well-known. Instead, let us imagine "applying" the transverse field via a transverse Ising coupling of the spins to partner auxiliary spins, i.e. $H= J_x \sum_{\langle i,j \rangle} \sigma^x_i \sigma^x_j + J_z \sum_i \sigma^z_i \sigma^z_{\text{partner of }i}$. If each spin of the chain has a unique auxiliary partner, then the resultant eigenspectrum is still the same as that of the quantum Ising model with $\frac{h}{J} = \frac{J_z}{J_x}$ and the degeneracy of the entire spectrum is $2^{\text{number of auxiliary spins}}$. We can interpret this as the auxiliary spins remaining paramagnetic down to zero temperature and an extensive ground state entropy. This follows from the existence of extensively large and mutually \guillemotleft anticommuting\guillemotright $\;$ sets of $local$ conserved quantities for $H$. Such a structure will be shown to be not unnatural in the class of bond-dependent Hamiltonians. In the above quantum Ising model inspired example of $H$, this is lost upon the loss of the unique partner condition for the full spin chain. Other cases where such degeneracy survives or gets lost are also discussed. Thus this is more general and forms the basis for an exact statement on the existence of extensive ground state entropy in any dimension. Furthermore this structure can be used to prove spin liquidity non-perturbatively in the ground state manifold. Higher-dimensional quantum spin liquid constructions based on this are given which may evade a quasiparticle description.
Current status:
Reports on this Submission
Strengths
This is a very clear written paper. Much care was taken by the author to be detailed and careful. The result is very general and rigorous.
Report
There are not many weaknesses in this manuscript and it is quite noteworthy. This work should, in my opinion, be commended.
It could perhaps be mentioned that the theorem was used but not in the very general form introduced by the author which deserves publication. In Europhysics Letters 84, 36005 (2008), diluted orbital compass models (in which the non-commuting bonds were two-fold coordinated) were solved by a gauge transformation. This led to decoupled transverse field Ising chains. The more general idea behind that solution was that of "bond algebras." The gauge transformation was a particular way of implementing those dualities.
Requested changes
Reference to the above noted earlier gauge transformation for the particular case of diluted orbital compass models should be made. The paper is otherwise quite complete and, as told, reads very well.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Sumiran Pujari on 2025-02-03 [id 5181]
(in reply to Report 3 on 2025-01-31)
I am genuinely pleased by the Referee's positive evaluation! I am grateful to the Referee for pointing out the Europhysics letters reference [*] of which I was unaware. It will be cited appropriately in the revised submission.
[*] Incidentally, it turns out in a previous work of mine (Ref. [17]) which led to the more general setting and discussion of this work, I rediscovered the result of the above reference coming from a different perspective. Luckily it is not published yet, so the scientific record will be put straight there as well.
Strengths
Innovative Approach: The use of anticommuting conserved quantities introduces a novel method for generating extensive ground state degeneracy.
Rigorous Mathematical Treatment: The mathematical framework is detailed, with well-presented derivations and proofs.
Connection to Established Models: The work builds on recognized models like the Kitaev honeycomb model and the SYK model, providing theoretical context
Weaknesses
Insufficient Physical Insight: The focus on mathematical formalism sometimes overshadows physical implications and testable predictions.
Key Proof Needs Clarification: The central proof emphasizes local conserved quantities rather than the equivalence of energy spectra, which needs further elaboration.
Limited Real-World Relevance: The spin-liquid ground states arise directly from the model’s construction, lacking a clear connection to physical materials.
Unstructured Conclusion: The "Summary and Outlook" lacks synthesis, focusing more on open questions than summarizing key contributions.
Report
The submitted manuscript investigates quantum spin models exhibiting extensive ground state entropy, introducing a novel mechanism based on anticommuting sets of local conserved quantities. The author constructs and explores both one- and two-dimensional models, analyzing their implications for quantum spin liquids, spin liquidity, and ground state degeneracy. The work is grounded in theoretical physics and rigorous mathematical formalism, with connections to established models such as the Kitaev honeycomb model and the Sachdev-Ye-Kitaev (SYK) model.
The manuscript has several strengths:
• The use of anticommuting conserved quantities to generate extensive degeneracy is a novel and interesting theoretical approach, contributing new perspectives to the study of strongly correlated systems.
• The mathematical treatment is detailed and rigorous, with carefully presented derivations and proofs that offer a solid theoretical foundation.
However, there are areas where the manuscript could be improved:
• The models presented are highly abstract and lack a direct connection to physical systems. Although parallels to Kitaev spin liquids and the SYK model are mentioned, they remain insufficiently developed. Including more explicit discussion of possible experimental realizations or physical contexts could enhance the manuscript's relevance.
• The introduction primarily focuses on theoretical constructs without sufficiently motivating the broader implications or relevance of the results. Clarifying the significance of the proposed mechanisms in a wider physical context would improve accessibility.
• The emphasis on mathematical formalism sometimes overshadows physical insight. A clearer discussion of observable consequences or testable predictions would make the results more compelling.
• The main point of the present article is a mapping equivalence between several zero-field Ising chains with the ‘longitudinal’ coupling between backbone spins and ‘transverse’ coupling between backbone and auxiliary spins with a simple quantum Ising chain in an effective transverse magnetic field h/J = J_z/J_x. However, the relevant proof provides evidence for locally conserved quantities and degeneracy rather than the equivalence between energy spectra of both kind of quantum spins chains. This issue is the central point of the article and should be addressed in much more detail.
• All investigated quantum spin models have competing bond-dependent couplings and hence, the presence of spin-liquid ground states is not so surprising as they are direct consequence of construction of these relatively artificial quantum spins models. From this point of view, the author should provide a detailed connection to real-world physical systems without which the investigated topic is only of marginal importance.
• The "Summary and Outlook" section currently lacks structure, reading more like a list of open-ended questions than a synthesis of the work's key contributions. While posing further questions is valuable, a clearer summary of findings and a more focused discussion of future directions would strengthen the conclusion.
After incorporating these suggested revisions, I recommend the manuscript for publication in SciPost Physics Core.
Requested changes
See individual points in the review report.
Recommendation
Ask for major revision
Author: Sumiran Pujari on 2025-01-30 [id 5164]
(in reply to Report 2 on 2025-01-16)
I thank the Referee for the general appreciation of the work. Below are replies to the directed comments by the Referee.
The Referee writes:
The models presented are highly abstract and lack a direct connection to physical systems. Although parallels to Kitaev spin liquids and the SYK model are mentioned, they remain insufficiently developed. Including more explicit discussion of possible experimental realizations or physical contexts could enhance the manuscript's relevance.
Reply: In terms of experimental realizations, the only thing I can think of are artificial quantum systems. I am not an expert on quantum chemistry enough to guess what sort of materials can potentially host such couplings (given the Jackeli-Khaillulin mechanism for Kitaev bond-dependent couplings). In the revision, I can point to the possibility of realizing the spin liquid physics described here in artificial quantum platforms and possibly at play in some quantum spin ice materials if the quantum Ising couplings' quantization axes somehow gets "staggered" across the lattice. Another paper (Homeier et al, Phys. Rev. B 104, 085138, (2021)) I recently encountered in the context of realizing the Kitaev toric code experimentally may also be a possibility in this context. Making a more detailed connection than the above is beyond the scope of the paper (rather perhaps of the author); the paper is more focused on the consequences of the studied theoretical structures.
The Referee writes:
The introduction primarily focuses on theoretical constructs without sufficiently motivating the broader implications or relevance of the results. Clarifying the significance of the proposed mechanisms in a wider physical context would improve accessibility.
Reply: Thank you for the suggestion. I will attempt to highlight them better.
The Referee writes:
The emphasis on mathematical formalism sometimes overshadows physical insight. A clearer discussion of observable consequences or testable predictions would make the results more compelling.
Reply: The natural testable prediction is a residual ground state entropy akin to classical spin ices in the experimentally relevant temperature regime. Apart from this, due to 2-point spin correlators (both static and dynamic) being hyperlocal in space and time (see this comment) implies essentially featureless spin structure factors which is connected to the speculation of no well-defined quasiparticles in these models. In the revision, these points can be mentioned at some appropriate spots.
The Referee writes:
The main point of the present article is a mapping equivalence between several zero-field Ising chains with the ‘longitudinal’ coupling between backbone spins and ‘transverse’ coupling between backbone and auxiliary spins with a simple quantum Ising chain in an effective transverse magnetic field h/J = J_z/J_x. However, the relevant proof provides evidence for locally conserved quantities and degeneracy rather than the equivalence between energy spectra of both kind of quantum spins chains. This issue is the central point of the article and should be addressed in much more detail.
Reply: The equivalence of the spectra can be seen from Eq. (17) from Sec. I C when $J'_x=0$. The paper will be revised to bring this argument to Sec. I A where the degeneracy is first discussed.
The Referee writes:
All investigated quantum spin models have competing bond-dependent couplings and hence, the presence of spin-liquid ground states is not so surprising as they are direct consequence of construction of these relatively artificial quantum spins models. From this point of view, the author should provide a detailed connection to real world physical systems without which the investigated topic is only of marginal importance.
Reply: This intuition is indeed on point. However the $90^\circ$ quantum compass model has Ising order instead of spin liquidity; this shows the importance of having extensive in system size conserved quantities (such as here or in the Kitaev model). This has been mentioned in the main text but perhaps needs to be emphasized again in the summary of the paper.
Regarding real world physical systems, I agree that the models are unrealistic as also discussed in point 1 reply. But I would also say that it also highlights the reach of the anticommutation structure of the local conserved quantities to guarantee spin liquidity without knowing the details of the ground state manifold. It would be desirable to have more insights or more detailed understanding into the nature of spin liquids. I believe the paper still stands on its own without a full characterization of the spin liquids or presently unclear status of real world realizations.
The Referee writes:
The "Summary and Outlook" section currently lacks structure, reading more like a list of open-ended questions than a synthesis of the work's key contributions. While posing further questions is valuable, a clearer summary of findings and a more focused discussion of future directions would strengthen the conclusion.
Reply: In the revised version, this section will be further subsectioned into a "Summary" subsection and an "Outlook and Speculations" subsection. This should already help. I will attempt to focus more the speculations as well if I can.
Strengths
1-Topicality of quantum spin liquid research
2- Introduction of a theorem evidencing existence of extensive ground state entropy
3-Compelling arguments for a few locally conserved quantities
4-Applicability of the methodology to a special class of related quantum spin models
Weaknesses
1-Missing relevance to real-world systems and experimental validation of the bond-dependent quantum spin models
2-The paper is not put into a proper context with respect to existing literature concerning mainly with the quantum compass model and Kitaev-type models.
3-The unclear proof of the main theorem, which provides connection of the studied bond-dependent quantum spin chains in zero field with the quantum ising chain in a transverse field.
4-Missing supporting evidence of the spin liquid ground states from independent method (e.g. numerical calculations).
5-Too broad conclusion with several speculative statements
Report
Reviewer report on the manuscript 2407.06236v6: "A theorem on extensive ground state entropy, spin liquidity and some related models" by Sumiran Pujari
This manuscript deals with several one- and two-dimensional quantum spin models with bond-dependent interactions, for which spin liquid ground states with an extensive residual entropy are theoretically predicted. The issue of spin liquidity is of particular research interest in the modern condensed matter theory and the main topic of the present manuscript is thus of general interest for readers of SciPost Physics Core journal. The main strength of the manuscript is introduction of a theorem evidencing existence of extensive ground state entropy, compelling arguments for a few locally conserved quantities, and the applicability of the methodology to a special class of related quantum spin models. However, the manuscript also has several serious weak points and insufficiencies listed below.
1. First of all, the introduced quantum spin models with a rather unconventional setting of bond-dependent interactions represent, at least in my opinion, rather unrealistic toy models. While the theoretical constructs are quite robust, the manuscript lacks explicit discussion on experimental validations of the studied models suitable for experimental testing of the relevant theoretical predictions. The relevance of the studied models to real-world systems remains uncertain and hence, the manuscript should include a dedicated section on potential experimental realizations that could manifest these intriguing spin liquid ground states.
2. Although the manuscript brings a rather broad discussion of research even from a relatively unrelated or just marginally related research area (see for instance the point 9.), the manuscript only marginally discusses the connection of the studied models with some closely related quantum spin models such as quantum compass models, Kitaev honeycomb model, branched and comb-like quantum spin chains, etc. The manuscript is thus missing connection to closely related literature, some of the studied models such as the ladder model shown in Fig. 2 was even exactly solved in previous studies (see e.g. Brzezicki, W., Oleś, A.M. Physical Review B 80 (2009) 014405). From this perspective, the paper should be put more carefully into a more proper context of existing literature on this topic.
3. The main paper’s outcome is the theorem: "The resultant eigenspectrum is still the same as that of the quantum Ising model with h/J = J_z/J_x." and "The degeneracy of the spectrum is 2N∂i where N∂I is the number of the auxiliary spins." While the proof of the second statement is rather trivial, the proof of the first statement concerning the resultant eigenspectrum is quite speculative and ambiguous. The idea of this proof should be substantially expanded to prove the equivalence between both resultant energy spectra. The diagonalized form of the Hamiltonians (in fermionic representation) should be also presented.
4. The physical interpretation for some terms is completely missing. For instance, the manuscript does not explain the physical meaning of the operator M^z mod 2 entering the commutator in Eq. (7).
5. The abstract is highly imbalanced. Three-quarters of the abstract relate to the findings for the model (a) though the manuscript addresses other nine models and the content related to the model (a) does not exceed 20% of the whole manuscript.
6. The manuscript predicts spin liquids for a rather wide class of one-dimensional and two-dimensional quantum spin models with bond-dependent interactions. How about the nature of spin liquids? Is the nature of spin liquid ground states of one-dimensional and two-dimensional quantum spin models identical? Provide convincing evidence and characterization of the relevant quantum spin liquids.
7. The paper's notation and structure may be challenging for readers outside the given field. Simplified explanations or a brief summary of key results could enhance readability and accessibility for a general audience.
8. The manuscript could benefit from numerical simulations, which could verify the theoretical predictions, validate theoretical claims and provide visual insights into the spin liquid behavior of the proposed models. The insights on extensive ground-state degeneracy and spin liquidity may inspire for further theoretical investigations and experimental realizations of spin-liquid quantum materials.
9. The section Summary and outlook is too broad and contains several speculative statements without any rigorous proofs. The main take-home message of the present manuscript is therefore lost. This section should be substantially shortened when leaving out comments on only marginally related research topics such quantum chaos, SYK models, higher-form symmetries, field-theoretic gauge theories, quantum gravity, etc. Instead, the key findings should be reported within a clear and concise summary.
To conclude, the manuscript addresses extensive ground state entropy and spin liquidity of bond-dependent quantum spin models. The manuscript has several serious insufficiencies and weak points and hence, the major revisions along the lines of this report are needed before it can be accepted for publication in the reputable journal such as SciPost Physics Core.
Requested changes
See the numbered list in the report
Recommendation
Ask for major revision
Author: Sumiran Pujari on 2025-01-29 [id 5156]
(in reply to Report 1 on 2025-01-15)
The Referee writes:
1. First of all, the introduced quantum spin models with a rather unconventional setting of bond-dependent interactions represent, at least in my opinion, rather unrealistic toy models. While the theoretical constructs are quite robust, the manuscript lacks explicit discussion on experimental validations of the studied models suitable for experimental testing of the relevant theoretical predictions. The relevance of the studied models to real world systems remains uncertain and hence, the manuscript should include a dedicated section on potential experimental realizations that could manifest these intriguing spin liquid ground states.
Reply: I agree with the Referee that the toy models appear unrealistic. In terms of experimental testing or realizations, the only thing I can think of are artificial quantum systems. I am not an expert on quantum chemistry enough to guess what sort of materials can potentially host such couplings (given the Jackeli-Khaillulin mechanism for Kitaev bond-dependent couplings). In the revision, I can point to the possibility of realizing the spin liquid physics described here in artificial quantum platforms and possibly at play in some quantum spin ice materials if the quantum Ising couplings' quantization axes somehow gets "staggered" across the lattice.
The Referee writes:
2. Although the manuscript brings a rather broad discussion of research even from a relatively unrelated or just marginally related research area (see for instance the point 9.), the manuscript only marginally discusses the connection of the studied models with some closely related quantum spin models such as quantum compass models, Kitaev honeycomb model, branched and comb-like quantum spin chains, etc. The manuscript is thus missing connection to closely related literature, some of the studied models such as the ladder model shown in Fig. 2 was even exactly solved in previous studies (see e.g. Brzezicki, W., Oleś, A.M. Physical Review B 80 (2009) 014405). From this perspective, the paper should be put more carefully into a more proper context of existing literature on this topic.
Reply: I thank the Referee for pointing out the relevant reference above. It will be cited appropriately in the revised paper when discussing the ladder model in Fig. 2. The relation of $2d$ models to quantum compass models and the Kitaev honeycomb model has been discussed a fair deal in the paper in Sec. II I believe. I will look into the branched and comb-like quantum spin chain literature and look for relevant connections.
The Referee writes:
3. The main paper’s outcome is the theorem: "The resultant eigenspectrum is still the same as that of the quantum Ising model with h/J = J_z/J_x." and "The degeneracy of the spectrum is 2N∂i where N∂I is the number of the auxiliary spins." While the proof of the second statement is rather trivial, the proof of the first statement concerning the resultant eigenspectrum is quite speculative and ambiguous. The idea of this proof should be substantially expanded to prove the equivalence between both resultant energy spectra. The diagonalized form of the Hamiltonians (in fermionic representation) should be also presented.
Reply: The equivalence of the spectra is unambiguously seen from Eq. (17) from Sec. I C when $J'_x=0$. The paper will be revised to bring this argument to Sec. I A where the degeneracy is first discussed.
The Referee writes:
4. The physical interpretation for some terms is completely missing. For instance, the manuscript does not explain the physical meaning of the operator M^z mod 2 entering the commutator in Eq. (7).
Reply: I am not sure what the Referee has in mind here. The interpretation I know is that this conservation is a $Z_2$ superselection sector which becomes the fermion parity conservation in the fermionic language. This has been mentioned. I would be happy to learn what the Referee has in mind. Thank you.
The Referee writes:
5. The abstract is highly imbalanced. Three-quarters of the abstract relate to the findings for the model (a) though the manuscript addresses other nine models and the content related to the model (a) does not exceed 20% of the whole manuscript.
Reply: This point is well taken. The abstract will be rewritten to better reflect the contents of the paper.
The Referee writes:
6. The manuscript predicts spin liquids for a rather wide class of one-dimensional and two-dimensional quantum spin models with bond-dependent interactions. How about the nature of spin liquids? Is the nature of spin liquid ground states of one-dimensional and two-dimensional quantum spin models identical? Provide convincing evidence and characterization of the relevant quantum spin liquids.
Reply: Unfortunately, at the present moment, I do not have more insights or more detailed understanding into the nature of spin liquids. This is certainly a lack of understanding on my part, but it also highlight the reach of the anticommutation structure to guarantee spin liquidity without knowing the details of the ground state manifold. I believe the paper stands on its own without a characterization of the spin liquids. I hope to make progress on this front in the future.
Regarding the relation between the $1d$ and $2d$ cases, the only thing I can say at the present moment is that the $1d$ situation hosts a classical spin liquid coexisting with standard Ising order, whereas the $2d$ constructions are fully quantum mechanical and that they do not possess any Ising order.
The Referee writes:
7. The paper's notation and structure may be challenging for readers outside the given field. Simplified explanations or a brief summary of key results could enhance readability and accessibility for a general audience.
Reply: Thank you for this suggestion. The paper will be revised to have a paragraph on the organization and summary of results of the paper.
The Referee writes:
8. The manuscript could benefit from numerical simulations, which could verify the theoretical predictions, validate theoretical claims and provide visual insights into the spin liquid behavior of the proposed models. The insights on extensive ground-state degeneracy and spin liquidity may inspire for further theoretical investigations and experimental realizations of spin-liquid quantum materials.
Reply: Thank you for this suggestion. A numerical verification of the degeneracy and (classical) spin liquidity of a $2d$ model related to Fig. 1a was done in arXiv:2406.17034 (Ref.[17] of submitted version). This numerical observation is what put me on track to understand the mechanism and find the fully quantum constructions discussed here. As mentioned in the reply to point 6 raised by the Referee, once a theoretical prior or lens in which to view the $2d$ quantum spin liquids is available or developed, numerical simulations will certainly be warranted to confirm them (apart from the OTOC computations motivated by quantum chaos speculations mentioned in the outlook section).
The Referee writes:
9. The section Summary and outlook is too broad and contains several speculative statements without any rigorous proofs. The main take-home message of the present manuscript is therefore lost. This section should be substantially shortened when leaving out comments on only marginally related research topics such quantum chaos, SYK models, higher-form symmetries, field-theoretic gauge theories, quantum gravity, etc. Instead, the key findings should be reported within a clear and concise summary.
Reply: In the revised paper, the final section will be sub-sectioned into a dedicated summary section where the main results are recapitulated again. The outlook section can be renamed as "Outlook and Speculations" perhaps to make explicit that the goal is to "think out aloud" about possible connections. As such, this does not seem amiss in a research paper. Along with point 7 and the corresponding revision, the sub-sectioning mentioned above should hopefully take care of the Referee's concerns.
Sumiran Pujari on 2025-01-03 [id 5080]
Few remarks:
1) I recently realized that the argument for spin liquidity in the ground state manifold can be extended to dynamical situations as well which is worth pointing out. Let us take the example of the 2-point correlator $\langle \sigma^\mu_{i_\times}(t) \sigma^\nu_{j_\times}(0) \rangle$. In the public version, the proof of the static case $t=0$ was given based on the $\ll$anticommutation$\gg$ structure of the conserved quantities of the (time-independent) Hamiltonian. Briefly, as shown in Table I, this structure naturally subdivides the ground state manifold into disjoint sets and thus also the partition sum $\langle O \rangle(T \rightarrow 0) = \sum_{|\psi\rangle \in {|\psi_{\text{gs}}\rangle }} \langle \psi | O | \psi \rangle$ (Eq. 26 in the paper). The sum over the ground states in each disjoint set cancel out to zero.
In the dynamical situation, we have $\langle \sigma^\mu_{i_\times}(t) \sigma^\nu_{j_\times}(0) \rangle = \langle e^{-i H t} \sigma^\mu_{i_\times} e^{i H t}\sigma^\nu_{j_\times} \rangle $. Since 1) the subdivision of the ground state manifold into disjoint sets is based on the conserved quantities of the Hamiltonian, and 2) the conserved quantities will commute across the $e^{-i H t}$ and $e^{i H t}$ factors by definition, the cancellation argument will also apply to this dynamical case as well. I.e, Table I will essentially be reproduced for finite $t$ after replacing "$\sigma^\mu_{i_\times} \sigma^\nu_{j_\times}$" with "$\sigma^\mu_{i_\times}(t) \sigma^\nu_{j_\times}(0)$" everywhere. Thus the sum over the ground states in each disjoint set contributing towards $\langle \sigma^\mu_{i_\times}(t) \sigma^\nu_{j_\times}(0) \rangle$ will again cancel out to zero. Similarly, all the arguments in the paper for the static correlators extend to the corresponding dynamical correlators as well.
The implication of the above result is that correlations are hyperlocal not only in space but also in time. This is a natural consequence of the local $\ll$anticommutation$\gg$ structure and the resulting spectral degeneracies. This is true throughout the spectrum and thus for all temperatures. We may recall for comparison that the Kitaev spin liquid also has this hyperlocal property [29] (Baskaran et al, PRL 2007) for its 2-point spin correlators where it is a consequence of fractionalization of the spins into Majorana and $Z_2$ gauge degrees of freedom.
2) From the spin liquid argument section on page 7, right column, 2nd para:
In the present public version, it reads: "To make explicit what we mean by "appear", take the case of $\sigma^\mu_i \sigma^\nu_j \sigma^\gamma_k$. If the bonds ${\langle i,j\rangle,\langle k,l\rangle}$ host ${\sigma^x_i \sigma^x_j, \sigma^x_k \sigma^x_l}$, then $\sigma^\mu_i \sigma^\nu_j \sigma^\gamma_k$ must equal $\sigma^x_i \sigma^x_j \sigma^x_k$ (or $\sigma^x_i \sigma^x_j \sigma^x_j \sigma^x_k = \sigma^x_i \sigma^0_j \sigma^x_k = \sigma^x_i \sigma^x_k$ which is actually a 2-spin operator) for any correlator involving this 3-site operator to be non-zero. If the bonds ${\langle i,j\rangle,\langle k,l\rangle}$ host ${\sigma^x_i \sigma^x_j, \sigma^z_k \sigma^z_l}$, then $\sigma^\mu_i \sigma^\nu_j \sigma^\gamma_k$ must equal $\sigma^x_i \sigma^x_j \sigma^z_j \sigma^x_k = - i \sigma^x_i \sigma^y_j \sigma^x_k $ for any correlator involving this 3-site operator to be non-zero."
It should read as: "To make explicit what we mean by "appear", take the case of $\sigma^\mu_i \sigma^\nu_j \sigma^\gamma_k$. If the bonds ${\langle i,j\rangle,\langle j,k\rangle}$ host ${\sigma^x_i \sigma^x_j, \sigma^z_j \sigma^z_k}$, then $\sigma^\mu_i \sigma^\nu_j \sigma^\gamma_k$ must equal $\sigma^x_i \sigma^x_j \sigma^z_j \sigma^z_k = - i \sigma^x_i \sigma^y_j \sigma^z_k $ for any correlator involving this 3-site operator to be non-zero. If the bonds ${\langle i,j\rangle,\langle j,k\rangle}$ host ${\sigma^x_i \sigma^x_j, \sigma^x_j \sigma^x_k}$, then $\sigma^\mu_i \sigma^\nu_j \sigma^\gamma_k$ must equal $\sigma^x_i \sigma^x_j \sigma^x_j \sigma^x_k = \sigma^x_i \sigma^0_j \sigma^x_k = \sigma^x_i \sigma^x_k$ which is actually a 2-spin operator for any correlator involving this "3-site" operator to be non-zero. "
3) A minor rephrasing from the introductory section on page 2, right column towards the bottom:
In the present public version, it reads: "Before proceeding, it must be remarked here that the lemma above has been used previously in other contexts [18] as well as on ground state degeneracy [19], however with zero ground state entropy density which is not the focus of this work."
It may be rephrased as: "Before proceeding, it must be remarked here that the lemma above and its variants have been used previously in other contexts [18] as well as on ground state degeneracy [19], however with focus on the physics of sub-extensive degeneracies which is not the focus of this work."