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A plane defect in the 3d O(N) model

by Abijith Krishnan, Maxim A. Metlitski

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Submission summary

Authors (as registered SciPost users): Abijith Krishnan
Submission information
Preprint Link: scipost_202304_00015v2  (pdf)
Date accepted: 2023-07-18
Date submitted: 2023-07-01 01:41
Submitted by: Krishnan, Abijith
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


{It was recently found that the classical 3d O$(N)$ model in the semi-infinite geometry can exhibit} an ``extraordinary-log" boundary universality class, where the spin-spin correlation function on the boundary falls off as $\langle \vec{S}(x) \cdot \vec{S}(0)\rangle \sim \frac{1}{(\log x)^q}$. This universality class exists for a range $2 \leq N < N_c$ {and Monte-Carlo simulations and conformal bootstrap} indicate $N_c > 3$. In this work, we extend this %analysis {result} to the 3d O$(N)$ model in an infinite geometry with a plane defect. We use renormalization group (RG) to show that in this case the extraordinary-log universality class is present for any finite $N \ge 2$. We additionally show, {in agreement with our RG analysis}, that the line of defect fixed points which is present at $N = \infty$ is lifted to the ordinary, special (no defect), and extraordinary-log universality classes by $1/N$ corrections. %in agreement with our RG analysis. {We study the ``central charge" $a$ for the $O(N)$ model in the boundary and interface geometries and provide a non-trivial detailed check of an $a$-theorem by Jensen and O'Bannon.} {Finally, we} revisit the problem of the O$(N)$ model in the semi-infinite geometry. We find evidence that at $N = N_c$ the extraordinary and special fixed points annihilate and only the ordinary fixed point is left for $N > N_c$.

Published as SciPost Phys. 15, 090 (2023)

List of changes

We would like to thank the Referee for the careful reading of our manuscript and for their comments.

1 - In page 4 below eq 1.3, it is said that "believed that the scaling dimension ∆ϵ < 2 for all finite N ≥ 1". It might be worth pointing out the existence of rigorous bootstrap bounds which show this for many values of N, including moderately large N.

Upon the Referee’s suggestion, below Eq. (1.3) we have added a citation to Ref [12] (a bootstrap study identifying the O(N) islands) and Ref [13] (a review of the large-N results).

2 - In page 16 it is said that "the interface ordinary fixed point is equivalent to two decoupled boundary ordinary fixed points for each side of the interface". Reminding the reader of eq. 1.4 could be helpful.

We have added a reference to Eq. (1.4), as suggested by the Referee.

3 - It is used in multiple occasions that the special interface transition is essentially equivalent to no defect for all N. On the other hand, the boundary special transition has non-trivial anomalous dimensions with the respect to the bulk operators. Is there a simple way to argue for the triviality of the interface with an action similar to 1.4 or 2.2 for the special boundary class?

We can start with two decoupled boundaries each at its special fixed point (assuming N < N_c ~ 5). However, the dimension of the boundary O(N) vector at the boundary special fixed point is quite small (please see table I in Ref. [1]). Even for N=1, this dimension is ~0.36 and the dimension decreases with N. Thus, the analog of the coupling u in Eq. (1.4) is highly relevant. We cannot track the RG flow induced by u, however, the natural guess is that it is towards the trivial interface, which, importantly, has only a single relevant O(N) singlet perturbation.

4 - The folding construction is used a few times in the text. Could one have anticipated eq. 5.37 from 3.34 + the folding argument?

We were very intrigued by the apparent doubling of the shift of the g^3 coefficient in the beta-function from the single boundary to the interface. One way to uncover the origin of this doubling would be to pursue the strategy explained on page 24:

``From the form of the action (5.1) a direct computation of the coefficient b in beta(g) requires the knowledge of the four-point function of the tilt operator t_i at the normal fixed point. (This should be compared to the computation of the coefficient alpha_{bound}, which relies only on the two-point function of t_i and the knowledge of the coefficient s. In addition, a number of higher order counter-terms in the action, omitted in
Eq. (5.1), such as e.g. delta L_{bound} ~ pi^2 pi_i t_i, would have to be fixed by the requirement of O(N) invariance. We do not pursue this route to computing b here.”

Instead of pursuing this plan, we chose to extract the g^3 coefficient in the beta-function from the boundary order parameter dimension at the special transition in d=3+epsilon. However, it would be interesting to pursue the calculation sketched above in future work.

5 - Reference computes conformal data for the O(N) conformal interface in the epsilon expansion (for the ordinary and special transitions ). Maybe these results could be mentioned. Perhaps some of the them are useful to compare with?

We thank the Referee for pointing out this paper. Since we do not use the OPE coefficients at the ordinary/special transition anywhere in our paper, we don’t see a natural place to cite this reference (we also don’t cite older papers on the epsilon expansion for the ordinary/special transition, since we don’t use the corresponding results.)

Reports on this Submission

Anonymous Report 1 on 2023-7-10 (Invited Report)


I am happy with the changes and recommend the paper for publication.

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