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Scaling of disorder operator at deconfined quantum criticality
by Yan-Cheng Wang, Nvsen Ma, Meng Cheng, Zi Yang Meng
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We study scaling behavior of the disorder parameter, defined as the expectation value of a symmetry transformation applied to a finite region, at the deconfined quantum critical point in (2+1)$d$ in the $J$-$Q_3$ model via large-scale quantum Monte Carlo simulations. We show that the disorder parameter for U(1) spin rotation symmetry exhibits perimeter scaling with a logarithmic correction associated with sharp corners of the region, as generally expected for a conformally-invariant critical point. However, for large rotation angle the universal coefficient of the logarithmic corner correction becomes negative, which is not allowed in any unitary conformal field theory. We also extract the current central charge from the small rotation angle scaling, whose value is much smaller than that of the free theory.
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- Cite as: Anonymous, Report on arXiv:2106.01380v2, delivered 2022-03-02, doi: 10.21468/SciPost.Report.4602
This paper applies the technology of symmetry disorder operators to deconfined quantum criticality. The paper is extremely interesting and worthwhile and apart from the specific points below it is clear. In addition to obtaining some scaling functions for the O(3) universality class, one main claim is that the line operator shows a violation of conformal invariance for deconfined quantum critical point (DCP) in negativity of quantity s(pi), a striking result.
However, I have a concern about the procedure for defining the line operators in a way that avoids strong bonds and the extent to which finite size effects are controlled. I hope that the authors can clarify these issues to solidify what will then be a nice paper.
1. Various definitions of the line operator are compared and Figure S3 (supplemental material) shows that for some of these choices of definition the expected behavior is not found. The authors' conclusion is that these choices of definition suffer from severe finite size effects.
What I do not see presented is evidence that similar finite size effects are absent for the preferred definition, in the DCP case where there is no independent result to check against. This is important because the authors explain in S3 that the finite size effects can be of the same magnitude as the phenomenon of interest and can artificially give a negative s(pi).
Therefore, the severity of finite size effects and the error bars in the large L extrapolation should be quantified directly from the data. This might allow the authors to directly support the claim that their chosen definitions are free of the severe finite size effects seen in Figure S3. Alternatively, it is possible that the size of differences between curves in S5 is reflective of the error bar in the final result.
The authors suggest in the supplemental material that the problem with finite size effects could be related to having to fit a nonuniversal leading term and also a universal subleading one. If it is possible with available data, an alternative may be to subtract off the leading term by taking appropriate differences of f(l)=ln |<X>| for different shapes/sizes instead of simply fitting it. For example f(l)-b f(l/b) for some b.
It would also be useful to show in the supplemental material some intermediate plots that give a sense of the error bars involved in extracting the logarithmic coefficient.
2. The procedure of adjusting the path of the defect operator in response to the VBS configuration is justified on the grounds that it reduces the contribution from strong bonds. If I understand correctly, the adjustment is a simple translation of the operator.
It would seem that this procedure cannot change the coefficient of the leading perimeter law term. This term is set by microscopic correlations. Since the critical VBS order parameter is small at large L, a translation only changes microscopic conditions, on average, by an amount that is small at large L. Therefore, the change in the definition will only affect subleading terms.
For this reason, the argument for this protocol based on the J1-J2 model example does not seem fully conclusive on its own to me. In the J1-J2 model the differences in the protocols affect the leading term. The meaning of a strong bond is also different in the explicitly dimerized model and DCP model.
If point 1 was addressed with evidence quantifying finite size and fitting error then this could avoid needing to rely on this argument.
“is arguably the enigma of…”. I did not understand the exact meaning: this may be my ignorance of a use of the word, otherwise this could be clarified.
Why is the J-Q3 model chosen (rather than J-Q)?
“Previous studies of the (2+1)d Ising and … suggest”: is it a conjecture or has it been derived using RG?
“measurements of local observables in the H_J-Q3 model appear to exhibit conformal invariance.” Is this correct? For a different DCP model in ref 14 the anomalous dimensions from the correlators become negative, violating unitarity bound, at scales roughly comparable with these.
What does it mean to calculate the VBS order parameter for a “spin configuration”?
Though there is an emergent U(1) for the VBS there is no corresponding microscopic onsite symmetry, so there is an obstacle to defining a similar disorder operator. Is there any way around this?
- Cite as: Anonymous, Report on arXiv:2106.01380v2, delivered 2022-03-01, doi: 10.21468/SciPost.Report.4592
1. Timely study of a new aspect of the DQC scenario.
2. Marked differences in the behavior of the disorder operator mean value scaling between conventional QCP and anticipate DQC point are clearly exposed.
3. A careful numerical analysis is provided.
4. Details of the measurement scheme in the presence of translational symmetry breaking are provided.
1. A few details need to be better explained, as detailed below.
2. The paper formatting needs to be adapted.
3. Several language flaws need to be corrected (see list below).
The authors present a detailed numerical study using QMC methods of the scaling behavior of appropriate disorder operators in several quantum spin models at a quantum phase transition out of an AF ordered phase. In all cases the authors fit their numerical data, obtained on varying finite-size systems and square subregions, to a scaling form that contains a logarithmic corner contribution. They then carefully analyze the dependence of the corner contribution on the U(1) rotation angle entering the disorder parameter. For the bilayer and columnar dimer model (referred to as the J1-J2 model in the manuscript) the observed behavior is in accord with earlier results by the same authors on related models and the low-angle scaling is in agreement with universal predictions from conformal symmetry and current conservation.
Interestingly, for the anticipated DQC in the J-Q3 model, the behavior differs in two important aspects: (i) the extrapolated current central charge falls significantly below the free boson value, indicative of strong coupling, (ii) the large-angle value becomes negative, and the authors argue that this is not possible in any unitary CFT. This finding in certainly very interesting and adds an important new piece of information to the DQC scenario, in particular in view of several recent developments, as detailed by the authors.
In addition to the physical results, the authors provide detailed explanations of how to measure the disorder parameter mean value in the presence of (explicit or spontaneous) translational symmetry breaking, and such technical details are certainly very useful.
I find that all the general criteria for publication in SciPost Phys. are already met by the manuscript or will be met after my requests for changes are implemented, and also the expectation No. 3 is fulfilled, since this work opens a new perspective and approach to study the DQC scenario, calling for follow-up work from both computational and field-theoretical perspectives, e.g., on related models and transitions.
In summary, I suggest to publish this work after the following changes and clarifications are performed on the current manuscript:
1. All the SM should be turned into regular appendices or integrated into the main text.
2. The authors should provide details regarding the quality of the fits of their data to Eq. (4) in the form of chi^2 values that add information beyond the statement in .
3. The definition of the perimeter l=4R-R should be justified (why not l=4R)?
4. The authors should explain how they arrived at the estimated scaling dimension for the spin operator in . Furthermore, it should be made more explicit, how the last sentence in  relates to that result, i.e., the smallness of the imaginary part (I suppose).
5. In the current SM subsection on the choice of M, the authors claim that for the Even-A subregion the negative corner contribution comes about because of a large perimeter contribution that affects the fitting prediction. If this was indeed the case, then why does a similar problem not affect/explain the observed negative value of s in the case of the J-Q3 model? Would the large-angel values of s turn out to become positive for the J-Q3 model if better data would be available?
6. The authors should show ( e.g. included in the current Fig. S5) the results of a naive measurement of <X_M> at the anticipated DQC point in order for the reader to compare to the results from their correction procedure.
7. Overall, the manuscript contains a large number of language flaws, which must be corrected. Below is an incomplete list focusing on the first page only:
-title: “Scaling of disorder operator…“ -> “Scaling of the disorder operator…“
-page 1: “exciciting“ -> “exciting“
-page 1: “which is defined“ -> “which are defined“
-page 1: “U(1) disorder operator“ -> “The U(1) …“
-page 1: “one side of DQC exhibits the Valence Bond Solid“ -> “one site of the DQC exhibits valence bond solid“
-page 1: “behavior of disorder operator“ -> “behavior of the disorder operator“
-page 1: “on square lattice, using the ubiased Stochastic…“ -> “on the square lattice, using unbiased stochastic …“