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Special Transition and Extraordinary Phase on the Surface of a Two-Dimensional Quantum Heisenberg Antiferromagnet
by Chengxiang Ding, Wenjing Zhu, Wenan Guo, Long Zhang
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Submission summary
| Authors (as registered SciPost users): | Chengxiang Ding |
| Submission information | |
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| Preprint Link: | scipost_202211_00001v2 (pdf) |
| Date submitted: | 2023-03-03 02:56 |
| Submitted by: | Ding, Chengxiang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmically decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, $y_{s}=0.86(4)$ and $\eta_{\parallel}=-0.33(1)$, which are distinct from the special transition of the classical O(3) model and indicate a new surface universality class of the 3D O(3) Wilson-Fisher theory.
Author comments upon resubmission
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Reports on this Submission
Report #3 by Anonymous (Referee 4) on 2023-4-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202211_00001v2, delivered 2023-04-08, doi: 10.21468/SciPost.Report.7023
Report
I thank the authors for their response to my comments. However, I still have several questions that I would like the authors to address:
1. In the definition of the Binder ratio in Eq. (7), should the numerator and denominator contain second power of `\tilde S`, instead of fourth power, since `\tilde S` already contains two powers of the order parameter? If so, is this a typo and is fig 2b) generated using the correct formula?
2. In an ordered phase, the Binder ratio should approach `1` as `L \to \infty`. Instead, in fig 2b) the Binder ratio at large `J_s` seems to be above `1.2` and, if anything, slightly increasing with the system size. Is there some explanation for this?
3. I think it would be useful to have a plot of `C_s(r)` in the paper (say, for the largest system size available). Perhaps this would shed some light on the disparity of `C_{||}(L/2)` and `{S(\pi)}/L` in Fig. 7.
4. I am confused by the discussion of `(\xi/L)^2` in the extraordinary phase at the end of section 3. In particular, I don't understand where the statement that `(\xi/L)^2` should scale as `L` for an ordered boundary is coming from. It is true that for a bulk quantum antiferromagnet in `d>1 ` spatial dimensions, `(\xi/L)^2 \propto L^{d-1}`, so d = 2 gives `(\xi/L)^2 \propto L`. But here the boundary is one-dimensional, so one might instead expect `(\xi/L)^2 =O(1)` (perhaps up to logarithmic corrections, as in the extraordinary-log phase).
Here is another comment about the analysis of `(\xi/L)^2`. The authors claim that there is a finite ordered moment in the extraordinary phase. Let me accept this claim for now. However, looking at Fig. 7, `{S(\pi)}/L` varies by some factor of `4` over the range of `L` studied, further the extrapolated value of the ordered moment is an additional factor of `3` below the value of `{S(\pi)}/L` at the largest `L` studied. So, if there is an ordered moment, the behavior of `{S(\pi)}/L` at system sizes studied is very far from the asymptotic, saturated behavior. Thus, it would be meaningless to compare `(\xi/L)^2` to predictions for asymptotic behavior in the ordered phase.
In addition, it seems that `{S(\pi)}/{S(\pi+\delta q)}` is numerically not very large for `L ` values studied (perhaps O(1)). Typically, the analysis of `\xi/L` in the ordered phase is performed assuming that this ratio is large, so that the `-1` term under the square root in Eq. (6) can be neglected.
For all these reasons, I would encourage the authors to show a plot of `S(\pi + \delta q)` as a function of `L` in the extraordinary phase by itself, without the additional algebraic manipulations that go into `\xi/L`.
Report #2 by Anonymous (Referee 5) on 2023-4-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202211_00001v2, delivered 2023-04-06, doi: 10.21468/SciPost.Report.7018
Report
The authors have positively responded to my comments, thus I think the paper is suitable for publication.
Still, concerning the fits of Sec. 3.1 and 3.2 (my comment 1. in the previous report):
On the basis of RG, one expects corrections L^(-0.759) *and* corrections L^(-1). But of course, the two are practically impossible to be distinguished.
Therefore, one should, at minimum, try omega_1=0.759 *and* try omega_1=1.
If, as it is likely, the fits results are the same, this should be briefly indicated in the paper.
If there is a significant discrepancy between fits with omega_1=0.759 fits omega_1=1, then the final uncertainty should take such discrepancy into account.
I would encourage the authors to briefly include a short discussion along these lines.
Author: Chengxiang Ding on 2023-05-02 [id 3638]
(in reply to Report 2 on 2023-04-06)
Thanks very much!
We add a short paragraph to clarify this question (two lines above Eq. (9)):
It should be noted that the finite-size scaling corrections arise from two sources, one is the leading correction proportional to $L^{-\omega_1}$, with $\omega_1 = 0.759$ for the current model, which comes from the irrelevant scaling field, another one is the background contribution analytic in $L^{-1}$. In practice, the analytic term $L^{-1}$ cannot be distinguished from $L^{-\omega_1}$ in the fitting procedure due to their close exponents. We also tried the fitting with $\omega_1=1$, the fitting quality is slightly worse and the difference of the results is very small, which falls in the range of the uncertainty of the error bars. Such a strategy has also been applied to all the other data fittings in this and the next subsections, although not explicitly stated.
Report #1 by Aleix Gimenez-Grau (Referee 1) on 2023-3-31 (Invited Report)
Report
The authors have addressed many of the points raised in the three referee reports, so I believe the paper is almost ready for publication.
However, in reading v2 I noticed the staggered magnetization Cs is no longer defined, but it still appears in some equations. Could the authors please fix this? There is also a typo "ordianry" below eq. (7). Finally, although the conclusion now discusses briefly the work by Metlitski, I strongly encourage the authors to provide a more thorough discussion, since this will definitely increase the quality of the paper.
Author: Chengxiang Ding on 2023-05-02 [id 3637]
(in reply to Report 1 by Aleix Gimenez-Grau on 2023-03-31)
Thanks very much!
-
We rewrite the definition of $C_\parallel$ as $C_{\parallel}(r)=\frac{(-1)^r}{L}\sum_{x}\langle \mbf{S}{(x,1)}\cdot\mbf{S}\rangle$ in Eq. (2).
All the $C_s$ in the text have been changed to $C_\parallel$.
-
The typo "ordianry" is fixed.
-
In Ref. [17], the extraordinary-log phase was proposed based on the perturbative RG analysis near the normal fixed point at the 1D boundary. Starting from the normal fixed point, where the spins show an infinitesimal long-range order, the spin interactions would be relevant and lead to short-range correlations at a free-standing boundary, but the coupling with the bulk critical modes reverses the RG flow direction and makes the normal fixed point stable. However, the logarithmically slow running towards this fixed point leads to the logarithmic decay of the spin correlation function instead of a long-range order, thus this is dubbed the extraordinary-log universality.
In the whole phase space, the results of our numerical work may be far from the normal surface fixed point, hence different from the extraordinary-log behavior.
This discussion is added to section 4.
Author: Chengxiang Ding on 2023-05-02 [id 3639]
(in reply to Report 3 on 2023-04-08)Thanks very much!
To question 1: It is a typo, we have fixed it.
To question 2:
We cannot thoroughly understand this result, and we think that the possible reasons may include but not limited to:
1. The surface is coupled with the bulk critical state, this is equivalent to effective long-range interactions for the surface, which may also have an impact on the value of the Binder Ratio.
In fact, in recent days, we have studied a long-range quantum Heisenberg chain by Monte Carlo simulations, it is shown the Binder Ratio in the long-range ratio is obviously larger than 1, the results will be published elsewhere.
2. Although the extraordinary phase is ordered, the value of the order parameter is very small; because of the existence of the gapless mode, the fluctuation of the AF order is still large, which makes it look like a critical state, hence the value of Binder ratio is not equal to 1.
We leave this question in future study.
To question 3:
It is a pity that we have only computed $C_\parallel(L/2)$.
P.S. We rewrite the definition of $C_\parallel$ as
$C_{\parallel}(r)=\frac{(-1)^r}{L}\sum_{x}\langle \bf{S}_{(x,1)}\cdot\bf{S}_{(x+r,1)}\rangle$ in Eq. (2); and all the $C_s$ in the text have been changed to $C_\parallel$.
To question 4:
We revisited the scaling behaviors of $S(\pi)$, $S(\pi+\delta q)$, and $(\xi_s/L)^2$ in the extraordinary phase (ordered), the paragraph after 7 lines of Eq. (27) is revised as:
Furthermore, according to the definition of Eq. (5), in an ordered phase, $S(\pi+\delta q)$ should growth logarithmically (the constant term cancels out after summing, and the integral of the $1/r$ term contribute the logarithmic term), i.e., the data of $S(\pi+\delta q)$ should satisfy the finite-size scaling form
\begin{eqnarray}
S(\pi+\delta q)=a+b\log(L);
\end{eqnarray}
combining with scaling of $S(\pi)$ in Eq. (26), we get the scaling formula of the square of the correlation ratio, which is written as
\begin{eqnarray}
(\xi_s/L)^2=a+bL/\log (L).
\end{eqnarray}
Figure 10 shows the scaling behaviors of $S(\pi+\delta q)$ and $(\xi_s/L)^2$ in the extraordinary phase, with $J_s=16$, which further demonstrates that there is a long-range AF order.