SciPost Submission Page
Entanglement parity effects in the KaneFisher problem
by Chunyu Tan, Yuxiao Hang, Stephan Haas, Hubert Saleur
Submission summary
Authors (as registered SciPost users):  Chunyu Tan 
Submission information  

Preprint Link:  https://arxiv.org/abs/2405.09046v1 (pdf) 
Date submitted:  20240610 05:12 
Submitted by:  Tan, Chunyu 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We study the entanglement of a segment of length $\ell$ in an XXZ chain with one free extremity and the other connected to the rest of the system with a weak bond. We find that the vonNeumann entropy exhibits terms of order $O(1)$ with strong parity effects, that probe the physics associated with the weakened bond and its behavior under the RG (Kane Fisher problem). In contrast with the XX case studied previously the entropy difference $\delta S\equiv S^eS^o$ gives rise now to a "resonance" curve which depends on the product $\ell T_B$, with $1/T_B$ a characteristic length scale akin to the Kondo length in Kondo problems. The problem is studied both numerically using DMRG and analytically near the healed and split fixed points. Interestingly  and in contrast with what happens in other impurity problems $\delta S$ can, at least at lowest order, be tackled by conformal perturbation theory.
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 multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Reports on this Submission
Strengths
The analytical findings in this article are sound and in a reasonable agreement with the numerical data.
Weaknesses
Broader context of the problem is missing.
Results are not placed into context or perspective either.
Numerical details are not provided.
The manuscript is poorly structured with many errors and inconsistencies.
Figures a poorly explained.
Report
In the article ”Entanglement parity effects in the KaneFisher problem” the authors investigate the order one terms of the entanglement in a segment in an XXZ chain with a conformal impurity in the middle in the form of a weak bond. The authors use analytical methods based on conformal perturbation theory and check their predictions numerically with DMRG. This was done for two different scenario’s in which the chain is either healed or split in two under renormalization. A key finding is that the difference in order one terms of the entanglement entropy between even and odd sites, called $\delta S $, in XXZ chains, in accordance with the XX chain, shows resonance curves and universal scaling with the length of the segment $l$ and the Kondo temperature $T_B$. Their analytical findings are well performed, sound and agree well with the numerical data.
Although the key findings of the paper scientifically sound and could make a relevant contribution to the field of impurity problems in spin chains, akin Kondo problems, the manuscript is hardly accessible for nonexperts. Also, focusing on the selected narrow problem that authors provide neither a broadscope motivation behind the problem nor a wider perspective of their results. Presented results seems to me incremental compare to the previous works by authors.
To summarize, I cannot recommend the publication of this manuscript until the level of presentation is significantly improved (detailed comments are listed below). I also believe that the manuscript lacks originality (and undoubtedly it lacks a broad perspective) to justify its publication in SciPost Physics, though it might be suitable for SciPost PhysicsCore.
Requested changes
Below are my detailed comments:
1. Broader context of the results is missing. In large detail the results are derived and compared with numerical data. But there is no indepth discussion, for instance, when numerical data demonstrate some deviation – what is the source of it, how to improve, etc? There is also no discussion on the broader impact of the results.
2. Introduction simply does not serve its role. No motivation is given, no context has been provided. When the authors write in the introduction “Think for instance of the Hamiltonian (1)” defined in the main body of the text it is extremely confusing. I understand that the authors worked on similar problems a lot and for them it is intuitive indeed to think about this Hamiltonian, but it is not as intuitive for the reader who have to see this Hamiltonian for the first time. In short, the manuscript requires reformatting and a better structured and clearer introduction to set the problem and a much more elaborate conclusion to place the findings into context, what they contribute to the problems mentioned above and proposals for further research.
3. What is the impact on specifically the KaneFisher problem never mentioned in the paper except in the introduction.
4. Numerical details are missing, making all numerical results nonreproduceable. This includes: Type of DMRG algorithm, bond dimension, convergence criteria, truncation error, number of sweeps, system sizes etc.
5. Lots of formulas are not formally introduced. The quantity of interest in this paper is the difference in the entanglement entropy between even and odd sectors $\delta S$. Besides the abstract it is nowhere introduced. Furthermore, this also holds for others important one. Take for example the definition of the constant $g$ appearing in the AffleckLudwig boundary entropy, shown in equation (2). This term includes boundary effects and is thus rather important for this study. Another example is the definition of the entropy in equation (34), which is not the well known formula for the VonNeumann entropy and definitely requires the reference.
6. Details of the system is missing. An important part of the XXZ chain is the Luttinger Liquid that is realized for $J_z<1$. For someone that is known with either quantum spin chains or impurity problems it is obvious that the Luttinger Liquid is studied due to its free bosonic degrees of freedom and its corresponding logarithmic contributions to the entanglement entropy. But this is not clear for others. Furthermore, it puzzled me for a while whether the authors consider finite or semiinfinite segment length $l$. This should be clearly stated.
7. Some parts of the findings are not elaborately touched on. Take for example equation 67. When $\Delta \rightarrow 1$ the system approaches the phase boundary of the Luttinger Liquid with an exponential diverging Luttinger Liquid exponent $K$. This is reflected in this equation but is never described anywhere in the text. But this also hold for some behaviour in the numerical data. In figure 5 for example, there is a quite a bit of a discrepancy between the two curves. This has to be properly explained.
8. Important concepts and terms (though standard in the field) are not introduced, examples of which are healed and split, UV and IR, etc.
9. In section 3.3 the authors start to refer to equations in one of the references. It would simplify readers experience a lot if the authors provide these equations essential to understand part of the derivations.
10. Caption are far to insufficient in describing the figures. The simply do not give any idea on what is presented.
11. Sometimes the authors mention that results are checked numerically but not shown in the paper. Why are these not shown in an appendix?
12. In the healed case the Hamiltonian $H^B$ contains an extra term $J_z \sigma^z_l \sigma^z_{l+1}$ in comparison to the split case. There is no explanation, why it is so.
13. Some results of Table 1 are not mentioned in the text.
14. Variables and constants appearing in formulas are quite often not described.
And there a few minor comments:
15. Quite a few typo’s in the text. Dots at the end of a sentence are missing.
16. Inconsistencies and errors in the formulas such as capitals $P$ in equation 32 , $(0)+\rangle$ instead of $(0)\rangle$ in equation 38, same equation dash instead of double minus sign, etc.
17. As mentioned by another referee, all equations appear in bold.
18. Missing DOI in quite a few references.
Recommendation
Ask for major revision
Strengths
New effects and results on a timely problem
Weaknesses
Very minimalistic bibliography
Report
The manuscript "Entanglement parity effects in the KaneFisher problem" investigates the entanglement properties of a segment within an XXZ chain, with one entangling site connected to the rest of the system with a weak bond which is the well known KaneFisher problem. The other entangling site is left free. The authors identify a significant order 1 parity effects in the entropy, which they analyse through both numerical (DMRG) and analytical methods. A key finding is the resonance behaviour of the entropy difference between even and odd sites (which they call $\delta S$). such difference is a universal scaling function of $\ell T_B$, where the length scale $1/T_B$ is analogous to the Kondo length.
The paper addresses an important problem in the field of quantum manybody systems and entanglement theory. The interplay between entanglement entropy and impurities in XXZ chains is a topic of considerable interest in recent times. The introduction of a resonance curve in the entropy difference $\delta S$ and the application of conformal perturbation theory to analyse this quantity represent novel and fundamental contributions to the field. These results are likely to stimulate further research into similar impurity problems and the role of parity effects in quantum spin chains.
The numerical results are robust and provide clear evidence of the predicted parity effects. Near the healed and split fixed points, the authors use conformal field theory and conformal perturbation theory to derive analytical expressions for $\delta S$. The perfect agreement between numerical and analytical results enhances the credibility of the findings.
The paper is wellwritten and organised. The introduction provides adequate background information. It could be published also in the present form.
I have only three minor optional suggestions for improvement:
1. Some readers might benefit from a more detailed explanation of the terminology "healed" and "split" fixed points/chains in the introduction. As written now it is understandable to the specialists and it is a very easy concept that every reader of Scipost could understand.
2. The bibliography is extremely minimalistic with only the most relevant manuscripts cited. It could be integrated by adding some of the many references on entanglement for impurity problems both for interacting and noninteracting systems, including, but not limited to, some of the large literature on entanglement in Kondo problems.
3. Strangely enough the formulas appears all in bold in my PDF viewer. If this is a problem of the source file, it should be fixed.
Requested changes
See above
Recommendation
Ask for minor revision