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

by Christian Hagendorf, Gilles Parez

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Submission summary

Authors (as registered SciPost users): Christian Hagendorf · Gilles Parez
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-06-09
Date submitted: 2022-05-27 15:26
Submitted by: Hagendorf, Christian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


The open XXZ spin chain with the anisotropy $\Delta=-\frac12$ and a one-parameter family of diagonal boundary fields is studied at finite length. A determinant formula for an overlap involving the spin chain's ground-state vectors for different lengths is found. The overlap allows one to obtain an exact finite-size formula for the ground state's logarithmic bipartite fidelity. The leading terms of its asymptotic series for large chain lengths are evaluated. Their expressions confirm the predictions of conformal field theory for the fidelity.

Author comments upon resubmission

Dear Editor,

We enclose our revised manuscript. Moreover, we provide here below a detailed response to the referee reports and a complete list of changes.

Sincerely yours, Gilles Parez and Christian Hagendorf.

Reply to referee 1:

We thank you for your positive review, interesting comments and questions. Here are our answers to your comments, questions and requested changes:

1) Below (1.8) (and below (2.43)), we replaced "definite magnetisation" with "magnetisation".

2) We now explicitly mention in the paragraph above (1.9) that the expressions for the overlaps of Theorem 1.1 were conjectured in [12].

3) We slightly modified the statement of Theorem 1.2 to exclude the cases where $\xi$ converges to 0 or 1.

4) Below (2.3), we now point out that our definition of the overlap differs from the standard Hermitian scalar product of quantum mechanics.

As for the meaning of the indices in (2.10), we explain that they correspond to the standard tensor-leg notation in the paragraph above (2.4). To stress that (2.4) is merely an example of this notation, we added "For example, ..." above the equation.

5) Below (2.13), we now mention that the components of $|\Psi_N\rangle$ have multiple contour-integral representations. Moreover, below (2.15), we added a sentence explaining that the simple factorised expression follows from the contour-integral representation of this special component.

6) Fixed.

7) The methods of reference [37] (previously [35]) can indeed be used. We checked explicitly that they lead to determinant formulas for the specialised symplectic characters that differ from ours (and turn out to be slightly more complicated).

However, one can prove the equality of these determinant formulas and ours' by establishing that the corresponding matrices are related one to another through conjugation with simple (and explicitly computable) triangular matrices.

We decided not to include this (technical) comment in the revised version of the text. Nonetheless, we thank the referee for their question.

8) The differential equations of Proposition 4.1 are linear with coefficients that depend polynomially on $n$. It is this polynomial dependence that forbids terms of the type $N^{-1/2}$ in the asymptotic expansion of the ratio (4.1) as $N\to \infty$ and, hence, leads to $\tau_1(z)=\bar \tau_1(z)=0$, as stated in Lemma 4.2. By recurrence, it also leads to the absence of terms of the type $N^{-(k+1/2)}$ for any integer $k\geqslant 1$.

Moreover, we believe that the polynomial dependence of the coefficients on $n$ also forbids the appearance of terms involving $\ln N$ in the asymptotic series of the ratio (4.1). Terms of this type would certainly be necessary to obtain a non-zero $g(\zeta)$. Hence, one should be able to conclude that $g(\xi)=0$ on the sole basis of the differential equations.

More generally, a detailed analysis of the differential equations should lead to the asymptotic series (4.2) and even determine all terms beyond the orders presented in (4.2). We have, however, not undertaken this analysis because the results of [30] are sufficient for our purposes.

As the reason for the different points raised by the referee is an analytic feature and not a "symmetry reason", we have refrained from modifying or expanding our manuscript with regard to this point.

9) Fixed.

Reply to referee 2:

Thank you for your review. We are pleased that you found our paper interesting and pleasant to read. As for your comments:

1) The fact that the ground-state vector of the spin chain is obtained as the homogeneous limit of the bqKZ equation is clearly stated at the beginning of section 3.2 (with a reference to [12] where it is proven). The relation between the two vectors is explicitly written in (3.27).

2) We have not investigated the possibility of deriving (3.14) with the help of Jacobi–Trudi or the Giambelli identity for the symplectic characters.

Nonetheless, we point out that there there is an alternative method for deriving (3.14), relying on the methods of reference [37] (previously [35]), as discussed above in our response to point 7 of referee 1.

Reply to referee 3:

We thank you for your positive review. Your interesting and constructive comments helped us improve our manuscript. Here are our answers to your comments, questions and requested changes:

1) Page 7: Thank you for bringing the references to several articles on the boundary Yang-Baxter equation to our attention. We added them as references [33]-[35] to the bibliography (and removed our original reference [33]).

2) Page 8: We added a sentence below (2.15). It explains that the simple factorised expressions follow from the components' multiple contour-integral expressions and cites [12]. See our response to point 5 of referee 1.

3) Page 8, (2.18): Thank you for your comment! It made us notice a missing minus sign in the expression for component $(\Psi_2)_{\uparrow\downarrow}$, which we added in the revised version of the manuscript.

Moreover, we replaced $|s\rangle$ with $|\zeta>$ throughout the text to avoid any confusion with the parameter $s$ obeying $s^2=q^3$.

4) Page 14 (previously page 13): Fixed.

5) Page 17 (previously page 16): Fixed.

6) Page 20, above Corollary 3.5: We added an explicit reference to Theorem 37 with $\mu=0$ and $\mu=1$ of [43] (previously [41]).

7) Page 21, (3.30)-(3.33): We thank the referee for their interesting observation.

We have compared for small values of $n$ the referee's explicit expression for the determinant in the case of even $N_1, N_2$ and our determinant in (3.30). The two expressions do indeed coincide, provided that one replaces $n$ with $n+1$ in the referee's expression.

Nonetheless, as suggested by the referee, we decided not to elaborate on this point in the revised version of the manuscript.

8) Page 26: Fixed.

List of changes

- Page 3, below (1.8): Changed "definite magnetisation" to "magnetisation" (R1 point 1). The same replacement was made at the top of page 12.

- Page 3, above (1.9): Added a sentence explaining that the expressions of Theorem 1 were conjectured in [12] (R1 point 2).

- Page 4, Theorem 1.2: Modified statement of the theorem to exclude the cases where $\xi$ converges to $0$ or $1$ (R1 point 3).

- Page 6, below (2.3): Modified the paragraph to point out that the overlap is different from the standard Hermitian scalar product of quantum mechanics (R1 point 4).

- Page 6, above (2.4): Replaced "For $M=1$" with "For example, if $M=1$" (R1 point 4).

- Page 7, below (2.10): Replaced "the following $K$-matrix" with "a diagonal $K$-matrix".

- Page 7, below (2.12): Added references suggested by referee 3 (R3 point 1).

- Page 7, below (2.13): Added a comment explaining that the components of $|\Psi_N\rangle$ that do not vanish trivially have explicit expressions in terms of multiple contour integrals (related to R1 point 5 and R3 point 2).

- Page 8, below (2.15): Added a sentence explaining that the factorised expression follows from the multiple contour-integral formula for the component and cited [12] (R1 point 5, R3 point 2).

- Page 8, (2.18): Added missing minus sign in the expression of $(\Psi_2)_{\uparrow\downarrow}$ (R3 point 3).

- Page 9, just above (2.21): Replaced $|s\rangle$ with $|\zeta\rangle$ (R3 point 3). The same replacement was made on page 8, (2.21); page 9, first paragraph and (2.22); and page 10, first paragraph.

- Page 14, Lemma 2.7: Corrected "is a most" to "is at most" (R3 point 4).

- Page 17, above (2.76): Replaced "(the upper bound for the) the degree width" with "(the upper bound for) the degree width" (R3 point 5).

- Page 18, below (3.7): Replaced "following the proof on Andreev's formula" with "using Andreev's formula" (R1 point 6).

- Page 20, above Corollary 3.5: Replaced "with the help of Krattenthaler's formula [41]" with the detailed reference "[43, Theorem 37 with $\mu=0$ and $\mu=1$]" (R3 point 6).

- Page 24, below (4.21): Replaced "parametrisation" with "parameterisation" to be consistent throughout the manuscript.

- Page 26, above (4.31): Corrected "is plausible assume" to "is plausible to assume" (R1 point 9, R3 point 8).

- Page 29: Added to [12] a reference to a published erratum to the original article.

- Page 30: Replaced original reference [33] with new references [33]-[35] suggested by referee 3 (R3 point 1).

Published as SciPost Phys. 12, 199 (2022)

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