On the logarithmic bipartite fidelity of the open XXZ spin chain at $\Delta=-1/2$

It is known, based on initial observations of Razumov and Stroganov from about 20 years ago and much subsequent work, that the Hamiltonian of the XXZ spin chain with anisotropy parameter $\Delta=-1/2$, and various boundary conditions, possesses a special ground-state eigenvector which exhibits remarkable connections to the combinatorics of alternating sign matrices and plane partitions. The focus of the submitted paper is the XXZ spin chain with $\Delta=-1/2$ and open boundary conditions containing a single parameter. The special eigenvector for this case was recently characterized in [12], using methods involving the boundary quantum Knizhnik-Zamolodchikov equations for a multivariate generalization of the eigenvector, and several combinatorial and other properties were proved or conjectured. For example, the sum of entries of the eigenvector was conjectured to provide a generating function for totally symmetric alternating sign matrices [12, Conjecture 5.5], and an explicit expression which leads to the logarithmic bipartite fidelity (LBF) was conjectured [12, Conjecture 2.1 with $m=2$]. The LBF is a quantity which indicates the extent to which the ground-state eigenvector for a chain of length $N_1+N_2$ can be regarded as a tensor product of the ground-state eigenvectors for chains of lengths $N_1$ and $N_2$. In the submitted paper, the results and methods of [12] are used to prove the conjectured expression for the LBF in this case (Eqs. (1.6), (1.11) and (1.12)). The exact expression for finite $N_1$ and $N_2$ is then used to obtain an asymptotic expression for $N_1,N_2\to\infty$, and it is shown that this agrees with conformal field theory predictions.

I believe that this is a very interesting paper, which provides important and valuable new exact and asymptotic results for the XXZ spin chain at $\Delta=-1/2$. The exposition is very clear and comprehensive, and I have identified only a few minor matters for consideration below. Accordingly, I strongly recommend publication.

p. 7, two lines after Eq.(2.12): I think that rather than being first obtained in (44) of [33], the general solution to (2.12) for the six-vertex model was first obtained independently in (15) of https://doi.org/10.1088/0305-4470/26/12/007 and (5.12) of https://doi.org/10.1142/S0217751X94001552 The special diagonal case in (2.11) was obtained previously in Theorem 2 of https://doi.org/10.1007/BF01038545

p. 8, Eq. (2.15): Perhaps Proposition 3.1 of [12] should be cited here.

p. 8, Eqs. (2.18) and (2.21): Setting $z_2=q^{-1}z_1$ in (2.18) gives $|\Psi_2\rangle=[\beta z_1] |\downarrow\uparrow\rangle+|\uparrow\downarrow\rangle)$ rather than $[\beta z_1](|\downarrow\uparrow\rangle-|\uparrow\downarrow\rangle)$ as in (2.21). Also, the notation $|s\rangle$ might be confusing and should perhaps be changed, since $s$ was introduced previously on p. 8 for something different (i.e., a parameter with $s^2=q^3$).

p. 13, Lemma 2.7: "a most" $\rightarrow$ "at most"

p. 16, second-last line: "(the upper bound for the)'' $\rightarrow$ "(the upper bound for)''

p. 20, two lines above Corollary 3.5: Since [41] is a long paper, it may be helpful to refer to the relevant parts. In particular, the result in Corollary 3.5 follows from (3.30) of [41] (with $\mu=0$ and $\mu=1$), and references for the proof of (3.30) are given in the subsequent paragraph of [41].

p. 21, Eqs. (3.30)-(3.33): I think it should be possible to obtain an explicit expression for the determinant in (3.30) which is analogous to the expression given by (3.32) and (3.33) for the determinant in (3.31) (but it is not necessary for any of this to be done in the submitted paper). Specifically, I think that an expression for the determinant in (3.30) is
\[\prod_{i=0}^{n-2}\frac{(2i+1)\,(6i+2)!}{(6i+1)\,(2n+2i-2)!}\\
\times\sum_{0\le i\le j\le2n-2}\frac{\bigl((2n-1)(2n-i-2)+i^2\bigr)\,(2n+i-3)!\,(4n-i-4)!}{(4n-4)!\,i!\,(2n-i-1)!}\:(-1)^{i+j}\,x^j\]
which provides a counterpart to the expression
\[\prod_{i=0}^{n-2}\frac{(6i+4)!}{(2n+2i)!}\:\sum_{0\le i\le j\le2n-2}\frac{(2n+i-1)!\,(4n-i-2)!}{(4n-2)!\,i!\,(2n-i-1)!}\:(-1)^{i+j}\,x^j\]
for the determinant in (3.31).

p. 26, two lines above Eq. (4.31): "plausible assume'' $\rightarrow$ "plausible to assume''

This paper is devoted to the exact computation of the finite size bipartite fidelity of the open XXZ spin chain at the anisotropy parameter ∆ = − 12 . The asymptotic behavior of exact rigorous result is compared to the (more general) CFT prediction and perfect agreement is found.

The finite size computation is made possible thanks to the fact that the ground state of the open XXZ chain (at ∆ = − 12 ) and boundary parameters defined in eq.(1.3) can be obtained
as the homogeneous limit of the (minimal degree?) solution of the boundary qKZ equation (at q = e^2iπ/3 ). This fact is mentioned only en passant at the beginning of section 2.4 and I believe it should be more emphasized. Maybe they should state explicitly something like
|Ψ N (z 1 , . . . , z N )> =1 ∝ |ψ N >.
Working with the solution of the bqKZ equation allows the authors to defined a spectral parameter deformation of the overlaps appearing in the computation of the bipartite fidelity. These deformed overlaps are completely characterized by the symmetries and the analytical dependence on the spectral parameters. This allows to show that they can be expressed in terms of certain symplectic characters. Only then, the homogeneous limit (z i = 1) is taken.

A question came to my mind. Is it maybe possible to obtain formula (3.14) using the
Jacobi–Trudi or the Giambelli identity for symplectic characters 1 ?

In conclusion: the paper is carefully written and pleasant to read. It presents new and in my opinion interesting results, that as far as I’ve been able to check are correct. I recommend the paper for publication.

In this paper, the authors study the 'logarithmic bipartite fidelity' in the spin 1/2-XXZ spin chain. This is, roughly speaking, the overlap between the ground state of a quantum system and the tensor product of ground states of smaller systems. It can be studied using conformal field theory (CFT) methods, in the spirit of Fermi edge singularities and impurity problems often encountered in condensed matter theory. These techniques allow to make various asymptotic predictions for large systems, in particular the orthogonality catastrophe exponent, and sometimes the form of subleading corrections.

Here the focus is on exact (and even rigorous) lattice computations using quantum integrability techniques, and a precise asymptotic analysis which confirms CFT predictions. While performing this task is difficult in full generality, the authors manage to do so at the 'Razumov-Stroganov' point $\Delta=-1/2$ and for a simple family of boundary fields. The main result takes the form of two theorems, one providing an exact determinant formula for the overlaps which was previously conjectured by one of the authors, another for the asymptotic expansion using a relation to the theory of symplectic characters.


Overall the paper is clearly written and a pleasant read. The results are nontrivial, I recommend publication, provided the authors address the minor issues which are listed below.

1) Page 3, after equation (1.8), 'definite magnetisation' reads awkwardly.

2) Page 4, theorem 1.1. This is stated later, but it would be better to already mention that this was conjectured in reference [12].

3) In theorem 1.2, I think it is implied that $\xi$ is in $(0,1)$, as one could imagine other asymptotic regimes, e.g $N_2\to\infty$ first, and then $N_1\to \infty$.

4) In section 2.1, the authors make an effort to be very precise, but they might want to already comment on their convention not to take complex conjugate for the bra, which is non-standard in quantum mechanics. This would help later on for certain calculations and understanding footnote 4. For equation (2.10): even though this is well-known, add a sentence explaining what is meant by the subscripts, similar to say (2.4).

5) While reading through the introduction, it is not very clear at this stages where equation (2.15) does come from.

6) After equation (3.7). It very much looks like it is Andreev's formula.

7) Proposition 3.3. This is fine as it is, but wouldn't it be also possible to use the method of reference [35]? That is make row manipulations in the determinant, and use Taylor expansions to take the limit.

8) Around Lemma 4.2, can the authors comment on the fact that $\tau_1$ and $\bar{\tau}_1$ actually vanish? Is there some simple symmetry reason which might explain it? Two related questions: do we expect corrections of the form $N^{-3/2}, N^{-5/2},\ldots$ or should all these terms vanish? Is (4.4), (4.5) sufficient to imply $g(\xi)=0$ without relying on the results of reference [30]?

9) Before equation (4.31). Replace 'it is plausible assume' by 'it is plausible to assume'.