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Critical properties of a comb lattice

by Natalia Chepiga, Steven R. White

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Submission summary

As Contributors: Natalia Chepiga
Arxiv Link: (pdf)
Date submitted: 2020-02-27 01:00
Submitted by: Chepiga, Natalia
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


In this paper we study the critical properties of the Heisenberg spin-1/2 model on a comb lattice --- a 1D backbone decorated with finite 1D chains -- the teeth. We address the problem numerically by a comb tensor network that duplicates the geometry of a lattice. We observe a fundamental difference between the states on a comb with even and odd number of sites per tooth, which resembles an even-odd effect in spin-1/2 ladders. The comb with odd teeth is always critical, not only along the teeth, but also along the backbone, which leads to a competition between two critical regimes in orthogonal directions. In addition, we show that in a weak-backbone limit the excitation energy scales as $1/(NL)$, and not as $1/N$ or $1/L$ typical for 1D systems. For even teeth in the weak backbone limit the system corresponds to a collection of decoupled critical chains of length $L$, while in the strong backbone limit, one spin from each tooth forms the backbone, so the effective length of a critical tooth is one site shorter, $L-1$. Surprisingly, these two regimes are connected via a state where a critical chain spans over two nearest neighbor teeth, with an effective length $2L$.

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Heisenberg spin chains
Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2020-5-11 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2002.11405v1, delivered 2020-05-11, doi: 10.21468/SciPost.Report.1678


1) High quality of data for all relevant regimes in parameter space

2) Testing of new method

2) Profound analysis of data combined with intuitive pictures

3) Very clearly written


1) No discussion of possible experimental realizations

2) Deviations from conformal limit not quantitative analyzed


In this paper the authors study the critical properties of the Heisenberg spin-1/2 model on a comb lattice using a newly developed comb tensor network method which they have introduced themselves in a previous publication. The model is well adapted to the strength of the method and large data sets for all regimes of interest are presented. Concurrently, it provides insights into the crossover of single impurity models such as y-junctions and non-trivial boundaries to the Kondo necklace model. The authors investigate the scaling of excited energy levels and the central charge with system size as well as the dimerization and magnetization along the backbone of the comb and the teeth. A comparison is made to CFT predictions. Particularly, there is an interesting effen-odd effect depending on the number of teeth sites and backbone sites. Besides the thorough analysis of the data this manuscript provides very convincing intuitive pictures of the different regimes and related scaling depending on the strength of the backbone coupling. My only criticism relates to the relatively short discussion of deviations from these simple pictures. In particular, the introduction of the effective exponent in the scaling as a result of the logarithmic corrections usually present in the Heisenberg chain is rather abrupt. Would it be possible to tune the disturbing marginal operator away such as in the Heisenberg chain with longer-range couplings? If the discussion of the corrections to the underlying CFT would be more substantiated I would highly recommend this manuscript for publication.

Requested changes

1) Include a more detailed discussion of the deviations of the CFT predictions, in particular give more details on the introduction and derivation of the effective exponent used in the fits.

2) In some figure legends for the magnetization profiles the indices i and j seem to be interchanged.

3) There are several typos.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: perfect

Anonymous Report 2 on 2020-4-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2002.11405v1, delivered 2020-04-22, doi: 10.21468/SciPost.Report.1636


1- A very detailed numerical study of a model with a geometry that allows for precise DMRG calculations.
2- Provided insight into a model that interpolates between a 1D and 2D problem.
3- The authors provide a clear understanding of the different regimes of the model, including a comparison with the conformal theory predictions.
4- An interesting crossover region was identified.


1- The model itself is a kind of artificial.


It is a nicely prepared manuscript about a numerical study of a spin-1/2 model with a particular geometry. While probably difficult to realize in experiments, the geometry allows for a precise numerical study of the model using DMRG. Different regimes have been identified, and the overall picture follows what we expect from the physics of 1D spin chains once the numerical results are properly interpreted. The critical correlations have been critically compared to expressions from conformal theory.
I would like to recommend the publication of the paper.

In addition to the report, a remark: it appears to me that geometry of the model allows a solution by Wigner-Jordan transformation for the XY anisotropic case - is this so, and would it be useful?

Requested changes

Fig. 17: It is not clear to me what does the "chain with a non-uniform lattice spacing" refers to. Does it mean that the couplings are different on the different bonds.

In the paragraph starting "According to Affleck et al. [25], the structure...", what is the "n=1,4,9..." ? Please explain.

  • validity: high
  • significance: good
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2020-4-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2002.11405v1, delivered 2020-04-01, doi: 10.21468/SciPost.Report.1605


1-Interesting model interpolating between various critical limits
2-Very thorough study containing a large number of reliable numerical data
3-Good analysis by comparing with predictions in the conformal limit


1-Explanations on how certain quantities are precisely defined and calculated and how some of the symbols in the equations are defined are sometimes missing
2-Some of the CFT results appear without references.
3-Some of the panels are very hard to read because the symbols are very small and the legends are crammed into a corner


The authors study a comb lattice of spin-1/2 coupled by
nearest-neighbor Heisenberg interactions. This model can be seen as a
generalization of spin systems studied previously in the literature
such as Y-junctions and spin chains with modified bonds. The results
are based on numerical data obtained using a tensor network adapted to
this geometry. This tensor network has been recently introduced by the
same authors (Ref. [17] in the manuscript). The study is very
thorough, presents a wealth of data, and analyzes them based on
theoretical predictions in the conformal limit. The results are quite
interesting: a number of critical regimes are identified. Before the
manuscript can be published, the following issues should be addressed:

Requested changes

1) Definitions of the entanglement entropy and, in particular, on how the lattice is cut exactly to obtain the entanglement entropies in Fig. 2 should be given.
2) Fig. 3 (c,e,g) and similar figures are difficult to read because the symbols are extremely small. Perhaps these figures can be improved?
3) It seems that no references are given with regard to the Friedel oscillations, e.g. around Eq. (2). Such references should be added.
4) Above Eq. (3) the authors say 'Here we expect ...'. It is not clear to which case the authors are refering here.
5) Eq. (4): The quantities in the equation should be explained, i.e., what is 'n', 'd(n)', 'g'?
6) Fig. 9: Instead of panel (i) - which does not exist - it should read panel (h) [twice].
7) There are a several typos which should be corrected.

  • validity: top
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: excellent
  • grammar: excellent

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