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Critical properties of a comb lattice

by Natalia Chepiga, Steven R. White

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Submission summary

Authors (as registered SciPost users): Natalia Chepiga
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-07-23
Date submitted: 2020-06-09 02:00
Submitted by: Chepiga, Natalia
Submitted to: SciPost Physics
Ontological classification
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 $2L$.

Author comments upon resubmission

We would like to thank to all referees for their careful reading of the manuscript and for their relevant comments that allows us to improve the manuscript. Below we address questions raised by referees 2 and 3.

  1. Referee 2: 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. The way we observe this on a comb lattice is really a non-equally separated degrees of freedom, as sketched in Fig.15. However, an effective interaction between these degrees of freedom depends on an effective distance between them. Very intuitively, we can expect it to be proportional to the spin-spin correlations, which is in critical 1D systems know to decay with the distance between spins as a power-low. So the referee is right, the system can effectively be described by a chain with the non-uniform coupling constant J. A corresponding sentence has been added to the manuscript.

  2. Referee 2: 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? Exact solutions are always useful, of course. However, I do not see how Jordan-Wigner transformation can be applied here when both N and L are macroscopic? The coupling along the backbone causes the same problem as Jordan-Wigner transformation in 2D, the only difference is that there is only L problematic terms for a comb instead of LN as for 2D lattice. If I miss the point, further comments are welcome.

Referee 3: 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?

3.We have significantly extended the discussion of log-corrections. The notion of an apparent scaling dimension has been properly introduced, and its deviation from the CFT value caused by the log-corrections has been discussed. A quantitative study of the log corrections goes beyond the scope of this manuscript and would require more complicated model with longer range interactions along the teeth and the backbone as pointed by the referee. Indeed, longer-range couplings, for example a J1-J2 model along the teeth at J2/J1=0.2411, can tune the marginal operator to zero and suppress the log-corrections for weak backbones and L-even. It is less clear though how to remove the corrections associated with the weakening of the boundary conditions that appears upon increasing the backbone interactions (see for example the curve for Jbb=0.5 in Fig.8(a)) without affecting the backbone too much. More puzzling in this respect is the absence of log-corrections for L-odd and Jbb<<1 and almost excellent agreement with linear scaling with 1/(NL) (see Fig.6). Either the corrections along the tooth and along the backbone have opposite signs and compensate each other or there is an effective interaction between the delocalized spins; we do not know yet. The exploration continues.

All the remaining comments from referees were helpful and the manuscript has been modified accordingly.

List of changes

1. The definition of the entanglement entropy has been included along with a sketch of the bi-partition of a comb.
2. The discussion on CFT predictions for Friedel oscillations has been extended and relevant references have been included.
3. Eq.4 has been re-written and all parameters entering the equation have been defined.
4. We have added an extended discussion about the CFT prediction for the structure of the excitation spectrum.
5. We have significantly extended the discussion of log-corrections.

Published as SciPost Phys. 9, 013 (2020)

Reports on this Submission

Anonymous Report 2 on 2020-7-17 (Invited Report)


The authors replied carefully to all three reports and the changes they have made in their resubmitted manuscript are convincing. I therefore recommend the publication of this manuscript in SciPost Physics as it is. Only a small remark: In Figs. 4b) and 6c) the indices i and j of the local magnetization on the backbone still seemed to be interchanged.

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Anonymous Report 1 on 2020-6-25 (Invited Report)


I would like to recommend the publication of the revised manuscript. Regarding my questions about the possibility of applying a Jordan-Wigner transformation, there is some literature about JW transformation in tree graphs, see e.g. S. Backens et al.Scientific Reports 9, 2598 (2019).

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