Scaling limit of the staggered six-vertex model with $U_q\big(\mathfrak{sl}(2)\big)$ invariant boundary conditions

We would like to thank the referees for a careful reading of our paper and for the useful comments. Below is our reply to the issues they've raised.

The second and third referee reports give some criticisms of our discussion of the $Q$ operator in section two. Among other things, they recommend adding to the manuscript a short explanation of how the formula for the $Q$ operator (2.15) is related to the results of [28]. We have done this in the revised version, see the paragraph below eq. (2.18). It is clear from the updated paper that the relation is not particularly direct. As such, although we have checked for small lattice sizes $N=2,3,4,...$ that the commutativity condition (2.1) and $TQ$ relation (2.19) are satisfied, their proof for general $N$ as well as questions concerning uniqueness can not be literally carried over from the results of ref. [28]. We believe that such issues are beyond the scope of our study and should be addressed in a separate paper. Also, it is explained in the opening paragraphs of section 2 why the $Q$ operator is important for the construction of RG trajectories, namely, that it allows one to extract the Bethe roots for a low energy state at intermediate values of $N\sim 20$. In the 2nd referee report another approach is mentioned for finding the Bethe roots corresponding to a given state which is usually referred to as the McCoy method. It starts by computing the eigenvalues of the transfer-matrix and then recovering the eigenvalue of the $Q$ operator from the $TQ$ relation. When a numerically efficient representation for $\mathbb{Q}(\zeta)$ exists, we find that the $Q$ operator approach is more efficient than the McCoy method. In our experience, the latter leads to greater numerical errors which, in practice, limits its applicability to smaller values of intermediate $N$.

The first referee report contains a very interesting discussion regarding the problem of the identification of the boundary CFT underlying the scaling limit of the lattice system. We are grateful for this expert's perspective on the results of our study, which identifies many fruitful areas for future research. At the same time, we would prefer not to comment much on the CFT interpretation of our results in the paper. The reason is that, at most, we can only give a speculative discussion, which we would prefer to avoid in case it turns out later that we say something incorrect. The questions being raised require a separate, detailed investigation that could form the subject matter of another work. In the updated manuscript we have added an acknowledgement to Sylvain Ribault for his valuable scientific input.

Some comments to the remaining minor issues raised by the referees are:

Referee report 2: 3. The differences in the ODE/IQFT correspondence between the case with periodic Boundary Conditions (BCs) and open $U_q(\mathfrak{sl}(2))$ invariant BCs is what one would expect from the basic principles of boundary CFT. Namely, instead of there being two ODEs appearing in the scaling limit of a low energy Bethe state --- one for each chirality --- there is only one differential equation for open BCs. Also, the precise relation between the invariants labeling the RG trajectories and those entering into the ODE are different. The interested reader can compare what is written in section 3.2 in the manuscript with the discussion in, e.g., section 11 of ref.[15] for details.

1. Cardy's scheme was developed in the context of the minimal models. Its application to the CFT underlying the critical behaviour of the staggered six-vertex model, which is non-rational, would require a significant extension. This is well beyond the scope of our paper. Also, the low energy states in the scaling limit organize into different irreps of the ${\cal W}_\infty$ algebra in the case of periodic BCs as opposed to open, $U_q(\mathfrak{sl}(2))$ invariant BCs.

Referee report 1: 1. This is a good idea and we have added a contents page in the revised manuscript.

1. What is being said is correct -- all other eigenstates can be obtained from those in the sector $S^z=0$ via the $U_q(\mathfrak{sl}(2))$ raising and lowering operators. We make comments along these lines, see, e.g., the two sentences below eq.\,(2.12).

2. Yes indeed, RG trajectories are constructed via an analysis of the Bethe Ansatz equations --- as is usual in the finite size analysis of Yang-Baxter integrable lattice models. We believe that our two paragraph discussion is sufficient, and we refer the reader to the works [15] and [26] if extra clarification is required. Also, we can not remove the paragraphs entirely as is suggested. They are needed to motivate the importance of the $Q$-operator, without which an analysis of the type performed in our work would be considerably more difficult to carry out. We have tried our best to keep a balance here.

3. We agree that in order to define the density $\rho(s)$, it is enough to consider the set of values of $b(L)$ at large $L$. However, the paragraph containing formula (3.11) that involves the symbol `${\rm slim}$' concerns a different problem, namely, how to suitably define the scaling limit of an individual low energy state of the lattice model. This turns out to be rather subtle, which motivates the use of a separate symbol for the scaling limit. An additional clarifying comment has been included in the revised manuscript.

4. Section 3.1 is mainly devoted to a review of the works [22,23], where the lattice model was also studied. Formula (3.14) is an important result of ref. [23], which is why that paragraph was included. Later, in section 4.1, we give all the possible values of pure imaginary $s$ that can occur in the scaling limit of a low energy Bethe state (see (4.17)). We discuss the relation of our results to those of [23] in the paragraph containing formula (4.21).

5. We agree that the appearance of the parameter $\mu$ entering into (3.15) may be confusing and that its relation to $\zeta$ was only implicitly given later. In the revised version of the manuscript we give this relation immediately after (3.15).

6. The r.h.s. of formula (3.30) coincides with the dimension of the level subspace of an irreducible module of the $W_\infty$ algebra, which is obtained from the Verma module by removing the descendents of the null vector that occurs at level ${\tt d}-2{\cal S}-1$. This is discussed further around formulae (4.36) and (4.37) in section 4. That the algebraic system (3.27), which comes from an analysis of the ODE, correctly accounts for the case when the Verma module develops null vectors has been observed before. A detailed discussion is contained in, e.g., section 12 of the work [15].

7. By ``the above relation'' we meant formulae (3.40)-(3.42) in the original submission, which should be treated as one set. To make this clear we have changed the numbering to (3.41a)-(3.41c) and refer to this set as (3.41) later in the revised version.

8. The fact that we defined $b(L)$ to take values in the strip $-\frac{n}{2}<\Im m\big(b(L)\big)\le\frac{n}{2}$ (see eq. (3.8)) is the source of the additional complications in the conjecture on page 18. They are purely of a technical nature, included to ensure the formal correctness of the statement. It is right to say that $b_*(L)$ coming from the quantization condition approaches $b(L)$ computed from the Bethe Ansatz equations in the scaling limit and that there is essentially a one-to-one relation between the sets of values ${b_*(L)}$ and ${b(L)}$ at $L\gg1$. We believe that fig. 6 gives a good demonstration of this. In principle, for a given RG trajectory, the numerical computation of $b(L)$ is possible for any $L\gg 1$. However, obtaining $b_*(L)$ from the quantization condition requires much less programming and computing resources. The quantization condition can be analysed analytically as well, which is its main advantage. We have added a few sentences in the preamble of section 4.1 to make this clear in the revised paper.

9. The $W_\infty$ algebra indeed has the automorphism suggested which will be crucial to keep in mind for describing the CFT underlying the critical behaviour of the lattice system. As was mentioned before, this is beyond the scope of our work. We believe that our manuscript possesses some form of completion and we'd prefer not to include an additional speculative discussion that may turn out to be incorrect.

see comments above