A strong-weak duality for the 1d long-range Ising model

The authors have studied the long-range Ising model (LRI) in one dimension with interaction decaying as $ \frac{1}{r^{1+s}} $. This model shows nontrivial critical behaviour for $ \frac{1}{2} \leq s \leq 1 $. Near s close to $ \frac{1}{2} $, it has a perturbative description around a mean-field theory. On the other hand, near s close to $ 1 $, there was no weakly coupled description available before the work by these authors.

In higher dimensions, it has been proposed that the LRI interpolates between the short-range Ising model (SRI) and mean-field theory. Therefore, the LRI admits a perturbative description in terms of the short-range Ising model coupled to a Gaussian free field. The puzzle in one dimension is that SRI is a trivial TQFT, and it is not clear how to generate an RG flow. The authors propose a field-theoretic description of the 1D LRI in this limit, near s close to $ 1 $.

The paper is well written and clearly explained. The authors give a good introduction to 1D CFT and then introduce their model. The weakly coupled model turns out to be a compact GFF perturbed by a vertex operator, which they explain in detail. Then they carry out a detailed study of this dual description and validate it in two independent ways: (1) by performing a standard perturbative renormalisation of the dual field-theoretic model near $ s = 1 $, and (2) by applying the analytic conformal bootstrap (in 1D) to compute CFT data and show agreement between the two methods. I am convinced by their proposal and would recommend it for publication after addressing a few minor concerns that I have mentioned below.

1. In Section 5, the authors have shown that correlators involving $\sigma$, $\chi$, $O_+$, and $O_-$ are fixed up to some high order in $\delta = (1 - s)$, essentially up to the order where an infinite set of primary operators first appears. For the first few orders, the crossing equation can be satisfied with only a finite number of primary operators, which is a unique feature of 1D CFT. This analysis is impressive, as the CFT data are uniquely fixed using crossing symmetry and an additional constraint that demands symmetry under the flip $\sqrt{\delta} \to -\sqrt{\delta}$. This ambiguity appears at order $O(\delta^0)$ and, as such, it is not a symmetry of the full finite-$\delta$ theory. Nevertheless, in perturbation theory, this feature enables the authors to isolate the LRI model in the space of theories. I wonder if there is any physical meaning to the invariance of the data under this flip, since if so, imposing it could also help numerically isolate the theory non-perturbatively. It is interesting that once they impose this symmetry, the data also automatically satisfy the expected OPE relations.

2. Another question is related to analytic functionals. One could have chosen to work with a Regge-bounded basis from the beginning, in which case one would not have to worry about its action on unitary CFT correlators. However, that approach would introduce additional subtractions, and I wonder whether that would lead to additional ambiguities. Therefore, it was important that the authors chose to work with a single-zero functional with a fixed kernel, whose action is not guaranteed to be well-defined on unitary CFT correlators in general. A related issue is that, on the field-theory side, the perturbing operators do not have well-defined UV dimensions. My concern is whether this flow could actually be an irrelevant flow. In my opinion, any such feature would manifest in the correlator when nontrivial infinite sums appear in the expansion. The authors have addressed this in the conclusion, but I wonder if they have further insight on this point.

3. The vanishing of equation (5.70) does not seem obvious. In particular, the blocks contain unknown parameters, but perhaps the blocks simplify here, and the unknowns are just factored out nicely so that the integration over z can be performed. I think an explanation of this fact will be helpful for the reader.

4. In equation (5.79), there appears to be a conformal block corresponding to dimension 1. Hence, in principle, there is an ambiguity, and a choice has been made. Could the authors clarify this to understand what assumptions go into fixing the data uniquely?

5. Below equation (5.61), $y=z/z-1$ is not defined.

6. Given the data at order $O(\delta^0)$, can one predict that operators with $\Delta = n$ will appear in the $\sigma \times \sigma$ OPE at $O(\delta^2)$, or is this taken as an input?

7. This is more of a curiosity. The proposed model possesses a $U(1)$ global symmetry, and the $U(1)$ singlets are identified with the operators of the LRI model. However, these singlets must also satisfy crossing equations involving charged operators, even though such charged operators are not visible within the LRI model itself. Does this imply that, in a numerical bootstrap study, including these additional crossing equations could lead to stronger constraints for the singlet sector? Or should one disregard them altogether when isolating the LRI sector?

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This paper studies the phase transition the 1D long-range model Ising model - the 1D Ising model with powerlike interactions among spins. When the power decay of interactions is not too strong, the phase transition happens at a nonzero temperature and is described by a nontrivial 1D CFT with powerlike correlations. When the power decay is too strong, there is no phase transition, like for the short-range 1D Ising model. The question is how the transition happens between the two regimes. A similar question in 2D and 3D was addressed by Behan et al (completing old work by Fisher-Nickel and Sak) who found a field theory description, coupling the short-range Ising CFT to a generalized free field (GFF). The paper under review makes a compelling proposal for a solution in 1D, completing old work by Yuval-Anderson and Kosterlitz. Their construction couples the "1D Ising TQFT" to a compactified GFF.

This long paper explains in detail the construction and performs many checks. I am convinced that their construction is correct at a technical level. I do however have some questions about what this construction means and how it should be presented.

I will give a list of questions in the order they occurred to me while reading the paper.

1. p.4 "Thus, while the generalized..." They have in mind here some construction which does not work. Which is it?

2. p.5, before Section 1.1 - What's the prediction for what will happen beyond the crossover, for $s>1$? Behan et al had a prediction in this regime as well.

3. p.8 - Earlier work by S. Ferrara, R. Gatto, and A. F. Grillo, “Properties of Partial Wave Amplitudes in Conformal Invariant Field Theories,” Nuovo Cim. A26 (1975) 226 (see Eq. (3.4)) as well as Belavin, Polyakov and Zamolodchikov (see App.A) should, I believe, be included along with [41].

4. p.13 - Section 3.1. This central section introduces a crucial field theoretic ingredient - compactified 1D GFF with negative dimension. As this is the first time that this theory is discussed in detail in the physics literature, it would be great to have it marked with the usual QFT labels. Is it a fixed point? Is it a CFT? Is it a gapped or gapless theory? What's the relation of theories with different values of $b_0$? (As of now, the reader has to ask and answer these basic questions themselves.) It would be also great to develop already here an interesting hint currently found in the conclusions, where the parameter $b_0$ is compared to the mass term of the free scalar theory - it may help the reader greatly.

5. Concerning the mathematical literature, it seems to me the theories are related but not quite identical, since there is no compactification there. The class of observables that the mathematicians consider is more narrow. They deal with the random continuous field $\phi$, but are not so interested in other operators like derivatives of $\phi$ and the vertex operators. There is probably no theorem in those mathematics paper proving the existence of the QFT that the authors use, with all needed properties. So ultimately the existence of that mathematics work does not seem sufficient for tranquilizing the reader that the author's construction makes sense.

6. p.13 "In fact, the resulting random distribution is almost surely a continuous function (modulo an additive constant). In other words, such GFF preserves shift symmetry." What does "In other words" refer to, to the additive constant or to continuity? In fact, is continuity used anywhere? Is there a simple physics way to see that paths are continuous?

7. App.A, p.51 "can be defined without introducing any action functional" - is this an important point? Can it also be introduced with an action functional, and if it can, why should we not use the standard physics way rather than the mathematics construction? Is this a specific lesson about 1D GFF, or a general lesson urging us to change the way we should think about Gaussian theories?

8. Third paragraph on p.51 - the authors cite many mathematics papers, but do any of those refer to their work as "conformal GFF"? If not it would be good to stress that the point of view taken here is a bit different.

9. What's the physical meaning of the last but one paragraph on p.52?

10. p.53 "the integral is regular at $x_1\sim x_2$" Is this significant? Of course if $\phi$ is differentiable the integral is convergent, but if it's not, then not necessarily. For random $\phi$'s distributed according to $\exp(-S)$ the integral will be finite almost surely, but this is a tautology.

11. Section A.2 - I got a bit lost in the discussion of observables. I understood that $<\phi[f]>$ is defined only for $f$'s of zero mean. But what's the meaning of "well-defined quantity" in the phrase "$<(\phi(x)-\phi(y))^2>$ is a well-defined quantity" below (A.14), in the rigorous language of random distributions being used here. Whatever it is, how does this fact follow from the above discussion?

12. Section 3.2, 1st line "1d continuum model". The word "model" is used in modern QFT for discrete models like the Ising model, apart from a few historically important theories like the Schwinger model. Otherwise continuous theories are referred to as QFTs. Is there a reason the authors don't call (3.9) a QFT? Is there a message that the considered theory somehow falls short of being a QFT?

13. The coupling to Pauli matrices in (3.9) through the path ordered exponential appears without much ado. In the introduction, the authors describe nicely their construction as coupling a 1D TQFT to a 1D GFF. Is (3.9) the ONLY coupling of these two theories which exists, or is it the only one which works? There seems to be an interesting general lesson here about what it means to couple a 1D TQFT to a 1D QFT, but the authors avoid stating it.

14. p.16 3rd paragraph "The first justification" - are any others and where in the paper do they occur?

15. p.16 4th paragraph - the formulation in terms of bosonic spinors - can the authors comment about advantages and disadvantages of different formulations?

16. It would be great if this section culminated in classifying theory (3.9) as an RG flow originating from some UV theory through some relevant deformation. That's what we do with any QFT, and I hope it's a QFT.

17. Another question is how their theory differs from the original theory (2.20), and where it lives on the diagram of Fig. 1(b). If it's precisely the same theory, then it would be great to state early on what's the advantage of such a reformulation. If it's a different theory which shares the same fixed point then it's interesting and would be great to know it early on.

18. p.18 below (3.26), "Interestingly, $D_x O$ can be a descendant..." - I believe what the authors what to say here is that $D_x O$ can be nonvanishing. It's a descendant by definition.

19. General question/comment for Section 3.2.2: That $O_h$ can be compensated by a shift of $b_0$ - could this be interpreted by saying that some operator of the theory is redundant? But what is the operator which implements the shift of $b_0$? Is there such an operator?

20. p.19 below (3.32) "the only relevant or marginal operators" Here the authors are treating $O_g$ and $O_h$ as if they were scaling operators, but at the beginning of Section 4 and elsewhere they stress they are not. The way to reduce the reader's confusion would be to develop the already mentioned analogy between $b_0$ and the mass of the free massive scalar, earlier than in the conclusions. Could it be that their flow is then similar to the $\phi^4$ flow in massive free scalar theory in 2d (except of course $\phi^4$ is strongly relevant)?

21. Where is the relevant perturbation which needs to be tuned to reach the phase transition? In the beta-function calculations it probably drops out because power-law divergences are dropped, but who is it? Is it by any chance the operator which deforms the two-state Quantum Mechanics to weakly gapped (from being topological), while preserving $\mathbb{Z}_2$ invariance? Is this coupling forbidden or does it have to be tuned? In the 2D and 3D work of Behan et al, the thermal perturbation of the 2D Ising CFT had to be tuned to reach the fixed point. It would be great to know if here the situation is similar or different.

22. p.20 What would the theory flow to if the couplings were turned off?

23. p.20 "it should be said that the UV completeness of the model is not needed for our purposes, as we are only interested in its IR behavior." I sense this comment is off the mark. UV completeness is not essential, but perturbative renormalizability is. Has this work solved the problem of finding a field theoretic description capable to compute critical exponents to all orders near the long-short crossover point? (If the model is not perturbatively renormalizable, then in what aspects is it better than Kosterlitz's model (2.20)? Can the computations they performed be performed from (2.20)?) Perhaps the authors don't have a rigorous proof of perturbative renormalizability yet, but can they build on their insight about $b_0$ and $m^2$ analogy and give at least intuitive arguments that it is?

This also concerns the discussion in the second paragraph on p.49 in the conclusions where the authors also seem to conflate UV completeness with perturbative renormalizability.

1. p.24 1st line - where was $\chi=i/\sqrt{2}\partial\phi$ identification made?

2. 2nd line - which of the two marginal operators is exactly marginal?

3. Section 5 - can the authors state advantages and disadvantages of the analytic bootstrap approach vs perturbative RG? Could Section 5 be done completely independently from the previous sections? Given Section 5, is perturbative RG obsolete for this model? Or will Section 5 treatment fail at higher powers of perturbing parameter?

I would be grateful to the authors if they could shed light on these questions and incorporate the answers whenever possible into the revised version of the paper, for the benefit of the readers.

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In this work, the authors develop a powerful weakly coupled description of the 1d long-range Ising model (LRI) in the region of parameter space where it looks almost short-ranged. In so doing, they resolve a problem raised 8 years ago, concerning a duality for the dimension $d \geq 2$ LRI and how it can be extended to one dimension. This problem is highly non-trivial because the "CFT" governing the critical short-range Ising model (SRI) in $d = 1$ is really the simplest TQFT with two ground states. Coupling the spin field of this SRI to a generalized free theory, which is the naive expectation, fails to generate a sensible flow.

The improved proposal, explained in detail here after appearing in a shorter version of this paper last year, draws inspiration from work by Anderson, Yuval and Kosterlitz (AYK) in the 70s. Their statistical model, which is valid at leading order in the expansion parameter $\delta$, rewrites the LRI as a gas of alternating kinks and anti-kinks. The gas is dilute since the critical point of the LRI at small $\delta$ is close to the ordered phase of the SRI. The same physics can be reproduced in a field theory which involves exponentials of a bosonic field with a shift symmetry. Although this is most often seen for free fields in 2d, the authors consider generalized free fields in 1d and carefully define normal ordering so that the propagator becomes logarithmic at $\delta = 0$. Using Pauli matrices and path ordering to enforce the alternation of kinks and anti-kinks leads to a novel theory which has the symmetries of the LRI plus an extra $U(1)$. A fully predictive framework follows by restricting to the $U(1)$ singlet sector and this is shown to pass many checks.

While some parts of the paper are more fast-paced than others, it is very self-contained. Some aspects which are likely to make it a useful reference for many years are the modern review of the AYK model and the careful discussion of test functions which require a certain number of vanishing derivatives. It is also nice to see that this theory occupies a sort of "sweet spot" for implementing perturbation theory with the analytic bootstrap, being more complicated than the most commonly studied deformations of generalized free fields but more controlled than the LRI near the crossover in $d = 2$. All in all, I have mostly minor changes to suggest.

1. It looks strange to jump from $\frac{ax + b}{cx + d}$ to another convention in equation (2.5).

2. The Figure 1 caption says $s = 0$ instead of $s = 1$.

3. While most lines in Figure 1 are easy to reproduce using the beta functions (2.21), it would be nice to state where the equation of the dashed blue line can be found. In particular, the notation in equation (B.6) is not consistent with the rest of the paper.

4. Section 2.4 says that references 9 and 10 are about the $d > 1$ LRI but it would be more correct to say $d \geq 2$.

5. Under (4.9), the dots seem to be terms that vanish as $a \to 0$. Terms that are independent of $a$ and therefore regular are already shown.

6. Under (4.20), I am not sure why $b^2$ is a more physical parameter than $h$. But it seems that something interesting is going on here. Even though it is easy to check that (4.21) follows from (4.20), the fixed points $(g_, h_)$ can only obtained by using (4.21) and then translating back to the $h$ variable. Getting this directly from (4.20) would fail because when $g_*^2$ and $h_*$ are both $O(\delta)$, the included term $g^2 h$ and the excluded term $g^4$ are equally important. It would be nice to comment on this.

7. The presentation of equation (4.32) seems a bit strange. This quantity is being considered in order to compute the scaling dimension $\Delta_\sigma$ so it should not be necessary to already know $\Delta_\sigma$ in order to define it.

8. Given the other novel aspects (matrix degrees of freedom and non-scaling operators for instance) a reader who is new to conformal perturbation theory might come away from Appendix F with the impression that mixing with the identity is also a peculiar feature of this theory. It should be stated that this is not the case. Indeed, other theories also need this to ensure that perturbative corrections to one-point functions vanish. This type of additive renormalization is equivalent to subtracting power-law divergences. Most papers use the latter terminology exclusively but this paper uses both.

9. Both limits are the same in (5.22) but one should probably be $+i\infty$.

10. Under (5.61), it seems that $y$ is being used as a shorthand for $\frac{z}{z - 1}$.

11. There is a typo "whihc" on page 65 and the indentation after equation (G.2) looks like a typo as well.

12. In section G.1, $s$ is normally defined as $p/2 + \epsilon/2$ and the expressions in equation (G.1) are wrong because they are not defect shadows. Same with (G.4).

13. When the codimensions $q$ and $q'$ are worked out in section G.1, there is another mistake. We should have $q = \frac{4 - \epsilon - p}{2}$.

14. In the last paragraph of the paper, it sounds like the implication goes the other way. The vanishing of $c_{11T}$ and $c_{22T}$ is what we know a priori. The fact that the dimension of an odd-spin operator cannot leave the pole (G.16) is then a consequence.

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