CFT and Lattice Correlators Near an RG Domain Wall between Minimal Models

Dear Editor,

We would like to firstly thank the referee for the detailed report regarding our draft!


Here are our reply to the questions and requests from the referees:

Referee 3:

From the report:

a) For the novelty of our work, we agree with the referee that we should emphasize it properly in our paper. Hence we added a paragraph (the second last paragraph) in the introduction to emphasize the novelty of our work.

b) We implemented PBC by choosing a Hamiltonian that is invariant under translations around the circle, so the true ground state will necessarily have periodic boundary conditions. The only numeric concern with DMRG and PBCs is whether or not the DMRG procedure has in fact found (a very accurate approximation to) the true ground state. This in fact had been an issue for us that we resolved by increasing the number of sweeps much higher than might typically be taken in DMRG analyses, but was not a major obstacle. We have added fig (Cameron will add and work into the text) demonstrating that all two-point correlators in the ground state do indeed only depend on |i-j mod N|, where N is the number of lattice sites, so that PBCs have indeed been obtained numerically.

c) The referee asks about the imperfect agreement in the UV and the IR in figure 3, especially at large |i-j|. We thank the referee for pointing this out which we believe was simply due to not obtaining sufficiently high convergence with our DMRG calculation. We have redone the calculation to higher accuracy and the result is shown in the new figure 3, which agrees much more accurately with the CFT calculation at large |i-j|. The issue in the UV is due to the presence of higher dimension operators in the expansion of lattice operators in terms of CFT operators, as we noted in footnote 13 and around equation (4.8).

Requested Changes:

1) We explained after Equ. (2.2) the meaning of a and b: The \mu matrices sit on the bonds of the \sigma-lattice and \mu_{i,a} and \mu_{i,b} denote the \mu matrices sitting on the bonds on the left and right of the ith site of the \sigma-lattice.

2)We thank the referee for this suggestion. We have added a figure with our setup shown on a cylinder, in Fig. 3.

3)We added footnote 7 explaining the folded and unfolded picture.

4) We disagree that using the two-point correlators to identify the leading CFT primary operators for the lattice operators is naive or somehow incorrect. However it is true that better accuracy for this expansion could be obtained by using the state-operator correspondence, so we have added such an analysis, where we calculate <vac|mu^z_i|epsilon> in DMRG to extract the coefficient of the operator epsilon(x) in the expansion of the lattice operator mu^z_i, as well as the coefficients of the first few subleading operators, where the CFT operators are identified by their energy on the circle in DMRG.

5)See our response to point c above, we have improved the numerics to resolve the discrepancy.

6)We added Equ. (4.4) to explain the coordinates we use. We changed x to u in section 4.1 for consistency of the notations.

7)We agree with the referee that our discussion of the experimental setup is just a brief review of existing ideas in the literature. We tried to make this explicit in the intro to section 5 but we have revised the language to make it hopefully more clear. We felt that this literature was nevertheless relevant and hopefully interesting to the reader and included it to make the paper more self-contained. There have already been papers in the literature demonstrating the realization of Ising model and Tricritical Ising model using Rydberg arrays so we just borrowed the results from https://arxiv.org/abs/2108.09309 and commented on how the RG domain wall studied in our paper can be easily engineered in their setup in a controlled manner.

Referee 2: From the report:

a) The CFT coordinate systems we are using is discussed in detail in Equ. (3.15) and the discussion around it. We further changed the notation in section 4.1 from x to u to avoid confusions. We emphasize that the “x->-x” symmetry is an emergent symmetry in the continuum limit which is not present in the lattice model, and we have added an explicit demonstration of this fact. Our lattice results in Fig.4 showed that the “x->-x” symmetry manifests away from the interface and is violated slightly close to the interface.

b) We appreciate the observation from the referee about the relatively large deviation of the lattice results from the analytical computation in the Ising case compared to the Tricritical Ising case. This has been noticed and articulated in our footnote 14 where we point out this fact and provide an explanation of it. The basic reason is that the leading deviation is controlled by the conformal dimension of the energy operator whose dimension is five times larger in the Ising case than the Tricritical Ising case. Hence the deviation of the lattice result in the Ising case from the analytical computation is expected to be larger than the Tricritical Ising case. Additionally, we have added a new calculation of the contribution from the subleading operators to operator expansion of \mu^z_i, providing an independent check that the contribution from the subleading operator is the size we claim. This last results are shown in Equ. (4.10) in our revised draft. (SAY EXACTLY WHAT EQUATION WE ADDED THAT THEY SHOULD LOOK AT)

We also thank the referee for noticing a typo in the legends of Figure 4, we have corrected this.

Referee 1:

Requested Changes:

1) The referee has requested an explanation of why the holomorphic/antiholomorphic factorization used in equation (3.6) and similar formulas is justified, since it is not true at the level of operators. Indeed it is true that this factorization is a property of specific correlators rather than the operators in general. It is a trick often used in the 2d boundary CFT literature referred to as the `mirroring trick’ and explained for example in DiFrancesco et al.s book, chapter 11. We have added a footnote below equation (3.6) explaining that it follows from the structure of the differential equations satisfied by the correlator. That is, the n-point correlators in the presence of the boundary obey differential equations that take the same form as if one were working with a 2n-point correlator where the 2n holomorphic and antiholomorphic variables of the n-point correlator with boundary get mapped to the 2n purely holomorphic variables of the 2n-point correlator without boundary. The basic reason for this equivalence is that the diagonal subgroup of the chiral algebra that is preserved by the boundary is isomorphic to the holomorphic half of the full chiral algebra.

2) We have added a fuller explanation of (3.9), and (3.5) which is its general form. Below (3.5) we give an intuitive explanation of the decomposition, which hopefully will make the basic structure at least plausible to a casual reader. On the page before (3.9), we have added a long footnote explaining how the argument in Crnkovic et al. works; a full rederivation of their result would be fairly lengthy but we hope this footnote is sufficient for a more careful reader to understand what the key steps are and have a clearer sense of how the detailed result on the form of the OPE in the B theory in the presence of the RG wall arises.

3) We have added further comments to the draft to address this point, see our response to point 7 of referee 3.

4) We’ve revised the caption indicating that the solid curves are theoretical calculations and the dots are from numerical lattice calculations.

5) We thank the referee for bringing the paper to our attention. However, part of the point of Gaiotto’s construction and part of what makes it more interesting is exactly that it does not appear to rely on integrability of the RG flow itself (though of course the CFT fixed points are integrable). So in keeping with this motivation we think it is philosophically cleaner not to use results that explicitly depend on the integrability of the flow.

1) We explained after Equ. (2.2) the meaning of a and b: The \mu matrices sit on the bonds of the \sigma-lattice and \mu_{i,a} and \mu_{i,b} denote the \mu matrices sitting on the bonds on the left and right of the ith site of the \sigma-lattice.

2) We have added a figure with our setup shown on a cylinder, in Fig. 3.

3)We added footnote 7 explaining the folded and unfolded picture.

4) We disagree that using the two-point correlators to identify the leading CFT primary operators for the lattice operators is naive or somehow incorrect. However it is true that better accuracy for this expansion could be obtained by using the state-operator correspondence, so we have added such an analysis, where we calculate <vac|mu^z_i|epsilon> in DMRG to extract the coefficient of the operator epsilon(x) in the expansion of the lattice operator mu^z_i, as well as the coefficients of the first few subleading operators, where the CFT operators are identified by their energy on the circle in DMRG.

5) The referee 1 asks about the imperfect agreement in the UV and the IR in figure 3, especially at large |i-j|. We have redone the calculation to higher accuracy and the result is shown in the new figure 3, which agrees much more accurately with the CFT calculation at large |i-j|. The issue in the UV is due to the presence of higher dimension operators in the expansion of lattice operators in terms of CFT operators, as we noted in footnote 13 and around equation (4.8).

6)We added Equ. (4.4) to explain the coordinates we use. We changed x to u in section 4.1 for consistency of the notations.

7) The referee 3 has requested an explanation of why the holomorphic/antiholomorphic factorization used in equation (3.6) and similar formulas is justified, since it is not true at the level of operators. We have added a footnote below equation (3.6) explaining this point in detail.

8) We have added a fuller explanation of (3.9), and (3.5) which is its general form. Below (3.5) we give an intuitive explanation of the decomposition, which hopefully will make the basic structure at least plausible to a casual reader. On the page before (3.9), we have added a long footnote explaining how the argument in Crnkovic et al. works; a full rederivation of their result would be fairly lengthy but we hope this footnote is sufficient for a more careful reader to understand what the key steps are and have a clearer sense of how the detailed result on the form of the OPE in the B theory in the presence of the RG wall arises.

9) We’ve revised the caption in Fig.6 indicating that the solid curves are theoretical calculations and the dots are from numerical lattice calculations.