A Gaudin-like determinant for overlaps of Néel and XXZ Bethe states

In order to prove overlap formula (9) we start with an expression for the overlap of the Néel state with an unnormalized off-shell state $|\lbrace \tilde{\lambda }_j\rbrace _{j=1}^{M}\rangle$ of the form (2), which was proven in [11, 12] (see equations (2.26)–(2.27) in [11] with ξ ≕ −iη/2). Introducing the short-hand notation s_{α, β} = sin (α + iβ) these equations from [11] can be written as (note that N = 2M)

$$\begin{equation} \fl \big\langle \Psi _0|\lbrace \tilde{\lambda }_j\rbrace _{j=1}^M\big\rangle = \sqrt{2} \left[\prod _{j=1}^M\frac{s_{\tilde{\lambda }_j,+\eta /2}}{s_{2\tilde{\lambda }_j,0}}\;\frac{s_{\tilde{\lambda }_j,-\eta /2}^{M}}{ s_{\tilde{\lambda }_j,+\eta /2}^{M}} \right] \left[\prod _{j>k=1}^{M}\frac{s_{\tilde{\lambda }_j+\tilde{\lambda }_k,\eta }}{s_{\tilde{\lambda }_j+\tilde{\lambda }_k,0}}\right]\det {}_{\!M}(\delta _{jk}+U_{jk}), \end{equation} \tag{ 10a }$$

$$\begin{equation} \fl U_{jk} = \frac{s_{2\tilde{\lambda }_k,\eta }s_{2\tilde{\lambda }_k,0}}{s_{\tilde{\lambda }_j+\tilde{\lambda }_k,0}s_{\tilde{\lambda }_j-\tilde{\lambda }_k,\eta }}\Bigg[\prod _{{{\scriptstyle {\begin{array}{@{}c@{}}\scriptstyle l=1\\ \scriptstyle l\ne k\end{array}}}}}^{M}\frac{s_{\tilde{\lambda }_k+\tilde{\lambda }_l,0}}{s_{\tilde{\lambda }_k-\tilde{\lambda }_l,0}}\Bigg]\Bigg[\prod _{l=1}^M\frac{s_{\tilde{\lambda }_k-\tilde{\lambda }_l,-\eta }}{s_{\tilde{\lambda }_k+\tilde{\lambda }_l,+\eta }}\Bigg]\bigg(\frac{s_{\tilde{\lambda }_k,+\eta /2}}{s_{\tilde{\lambda }_k,-\eta /2}}\bigg)^{2M}. \end{equation} \tag{ 10b }$$

This expression is unhandy to perform the thermodynamic limit as well as for parity-invariant states $|\lbrace \pm \lambda _j\rbrace _{j=1}^{M/2}\rangle$ due to zeroes of the determinant and singularities in the prefactor. To perform the limit to parity-invariant states (not necessarily on-shell Bethe states) we set $\tilde{\lambda }_j = \lambda _j + \epsilon _j$ for j = 1, ..., M/2 and $\tilde{\lambda }_j = -\lambda _{j-M/2} + \epsilon _{j-M/2}$ for j = M/2 + 1, ..., M. Here, the parameters λ_j, j = 1, ..., M/2, are arbitrary complex numbers. We shall see that the main ingredients to the derivation of formula (9) are the limits _j → 0, j = 1, ..., M/2, and the pseudo parity invariance of the set $\lbrace \tilde{\lambda }_j\rbrace _{j=1}^{M} = \lbrace \lambda _j+\epsilon _j\rbrace _{j=1}^{M/2}\cup \lbrace -\lambda _j+\epsilon _j\rbrace _{j=1}^{M/2}$. We derive then an off-shell version of equation (9), which has the same form up to corrections which are zero when the rapidities satisfy the Bethe equations (4).

If we multiply the prefactor in (10a) with $\alpha _{\rm reg}=\prod _{j=1}^{M/2}\left(\frac{s_{2\epsilon _j,0}}{s_{0,\eta }}\right)$ and the determinant by its inverse $\alpha _{\rm reg}^{-1}$ we get regular expressions with well-defined limits _j → 0, j = 1, ..., M/2, as well as a well-defined XXX scaling limit λ → ηλ and η → 0 afterwards. Assuming an appropriate order of rapidities the lowest order in $\lbrace \epsilon _j\rbrace _{j=1}^{M/2}$ of the regularized prefactor and of the determinant read

$$\begin{equation} \gamma = \sqrt{2}\left[\prod _{j=1}^{M/2}\frac{s_{\lambda _j,+\eta /2}s_{\lambda _j,-\eta /2}}{s_{2\lambda _j,0}^2}\right] \left[\prod _{{{\scriptstyle {\begin{array}{@{}c@{}}\scriptstyle j>k=1\\ \scriptstyle \sigma =\pm \end{array}}}}}^{M/2}\frac{s_{\lambda _j+\sigma \lambda _k,+\eta }s_{\lambda _j+\sigma \lambda _k,-\eta }}{s_{\lambda _j+\sigma \lambda _k,0}^2}\right], \end{equation} \tag{ 11a }$$

$$\begin{equation} \det {}_{\rm reg} = \lim \nolimits _{\lbrace \epsilon _j\rightarrow 0\rbrace _{j=1}^{M/2}} \left\lbrace \prod _{j=1}^{M/2}\frac{s_{0,\eta }}{s_{2\epsilon _j,0}}\det {}_{\!M}(\delta _{jk}+U_{jk})\right\rbrace . \end{equation} \tag{ 11b }$$

The task is now to calculate the limits _j → 0, j = 1, ..., M/2, in (11b). For this purpose we reorder the rows and columns of the matrix $\mathbb{1}+U$ under the determinant in such a way that the pair (λ_k + _k, −λ_k + _k) belongs to the two rows and two columns with indices 2k − 1 and 2k.

We consider the resulting M × M matrix as a M/2 × M/2 block matrix built of 2 × 2 blocks. Collecting all terms up to first order in all _j, j = 1, ..., M/2, we obtain on the diagonal the 2 × 2 blocks

$$\begin{equation*} 1+U_{2k-1,2k-1} = 1+\delta _k\frac{s_{2\lambda _k,+\eta }}{s_{2\lambda _k,0}}\mathfrak {a}_k,\quad U_{2k-1,2k} = -\frac{s_{2\lambda _k,-\eta }}{s_{2\lambda _k,+\eta }}\mathfrak {a}_k^{-1} + \delta _k\mathfrak {b}_k^{-} ,\nonumber \end{equation*}$$

$$\begin{equation} U_{2k, 2k-1} = -\frac{s_{2\lambda _k,+\eta }}{s_{2\lambda _k,-\eta }}\mathfrak {a}_k + \delta _k\mathfrak {b}_k^{+} ,\quad 1+U_{2k, 2k} = 1+\delta _k\frac{s_{2\lambda _k,-\eta }}{s_{2\lambda _k,0}}\mathfrak {a}_k^{-1}, \end{equation} \tag{ 12a }$$

and for the off-diagonal blocks (1 ⩽ j, k ⩽ M/2, j ≠ k)

$$\begin{eqnarray*} &&U_{2j-1,2k-1} = \delta _k\frac{s_{2\lambda _k,+\eta }s_{0,\eta }}{s_{\lambda _j+\lambda _k,0}s_{\lambda _j-\lambda _k,+\eta }}\mathfrak {a}_k,\quad U_{2j-1,2k} = \delta _k\frac{s_{2\lambda _k,-\eta }s_{0,\eta }}{s_{\lambda _k-\lambda _j,0}s_{\lambda _j+\lambda _k,+\eta }}\mathfrak {a}_k^{-1} ,\nonumber \end{eqnarray*}$$

$$\begin{equation} U_{2j, 2k-1} = \delta _k\frac{s_{2\lambda _k,+\eta }s_{0,\eta }}{s_{\lambda _j-\lambda _k,0}s_{\lambda _j+\lambda _k,-\eta }}\mathfrak {a}_k ,\quad U_{2j, 2k} = \delta _k\frac{s_{2\lambda _k,-\eta }s_{0,\eta }}{s_{\lambda _j+\lambda _k,0}s_{\lambda _k-\lambda _j,+\eta }}\mathfrak {a}_k^{-1}, \end{equation} \tag{ 12b }$$

where we defined the abbreviations $\delta _k=s_{2\epsilon _k,0}/s_{0,\eta }$ for k = 1, ..., M/2 and

$$\begin{equation} \mathfrak {a}_k = \tilde{\mathfrak {a}}(\lambda _k)= \left[\prod _{{{\scriptstyle {\begin{array}{@{}c@{}}\scriptstyle l=1\\ \scriptstyle \sigma =\pm \end{array}}}}}^{M/2}\frac{s_{\lambda _k-\sigma \lambda _l,-\eta }}{s_{\lambda _k-\sigma \lambda _l,+\eta }}\right]\left(\frac{s_{\lambda _k,+\eta /2}}{s_{\lambda _k,-\eta /2}}\right)^{2M}. \end{equation} \tag{ 13 }$$

Note that the function $\tilde{\mathfrak {a}}$ is different from the function $\mathfrak {a}$ of equation (5) because in the definition of $\mathfrak {a}$ the parameters $\lbrace \lambda _k\rbrace _{k=1}^{M}$ are Bethe roots whereas here in equation (13) they are arbitrary. The symbols $\mathfrak {b}_k^+$ and $\mathfrak {b}_k^-$ in equation (12a) denote the first order corrections of the elements U_{2k, 2k − 1} and U_{2k − 1, 2k}, respectively. After a short calculation we obtain up to zeroth order in δ_k

$$\begin{eqnarray} &&\fl \frac{s_{2\lambda _k,-\eta }}{s_{2\lambda _k,+\eta }}\mathfrak {a}_k^{-1}\mathfrak {b}_k^{+} + \frac{s_{2\lambda _k,+\eta }}{s_{2\lambda _k,-\eta }}\mathfrak {a}_k\mathfrak {b}_k^{-} = 2\cosh (\eta )-s_{0,\eta }\partial _{\lambda _k}\log \left \lbrace \frac{s_{\lambda _k,+\eta /2}^{2M}}{s_{\lambda _k,-\eta /2}^{2M}}\prod _{{{\scriptstyle {\begin{array}{@{}c@{}}\scriptstyle l=1 \\ \scriptstyle l\ne j\end{array}}}}}^{M/2}\prod _{\sigma =\pm }\frac{s_{\lambda _k+\sigma \lambda _l,-\eta }}{s_{\lambda _k+\sigma \lambda _l,+\eta }}\right \rbrace . \nonumber\\ \end{eqnarray} \tag{ 14 }$$

We further define $\alpha _k = \sqrt{-\frac{s_{2\lambda _k,+\eta }}{s_{2\lambda _k,-\eta }}\mathfrak {a}_k}$ and multiply the M × M matrix $\mathbb{1}+U$ from the left and from the right respectively with the matrices

$$\begin{equation} {\rm diag}_{M}\left(\alpha _1,\alpha _1^{-1},\ldots ,\alpha _M,\alpha _M^{-1}\right),\quad {\rm diag}_{M} (\alpha _1^{-1},\alpha _1,\ldots ,\alpha _M^{-1},\alpha _M). \end{equation} \tag{ 15 }$$

Since the determinants of these diagonal matrices are equal to one this transformation does not change the value of the determinant $\det _M(\mathbb{1}+U)$. We see that the structure of the matrix becomes

$$\begin{equation} \fl \left(\begin{array}{@{}ccc@{}} \left[\begin{array}{@{}cc@{}}1-\delta _1\frac{s_{2\lambda _1,-\eta }}{s_{2\lambda _1,0}}\alpha _1^2 & 1+\delta _1\mathfrak {b}_1^{-}\alpha _1^2 \\ 1+\delta _1\mathfrak {b}_1^{+}\alpha _1^{-2} & 1-\delta _1\frac{s_{2\lambda _1,+\eta }}{s_{2\lambda _1,0}}\alpha _1^{-2}\end{array} \right]&\delta _2\left[\begin{array}{@{}cc@{}}a_{12}&b_{12}\\ c_{12}&d_{12}\end{array}\right] & \dots\\ \delta _1\left[\begin{array}{@{}cc@{}}a_{21}&b_{21}\\ c_{21}&d_{21}\end{array}\right]&\left[\begin{array}{@{}ll@{}}1-\delta _2\frac{s_{2\lambda _2,-\eta }}{s_{2\lambda _2,0}}\alpha _2^2 & 1+\delta _2\mathfrak {b}_2^{-}\alpha _2^2 \\ 1+\delta _2\mathfrak {b}_2^{+}\alpha _2^{-2} & 1-\delta _2\frac{s_{2\lambda _2,+\eta }}{s_{2\lambda _2,0}}\alpha _2^{-2}\end{array}\right] &\\ \vdots & & \ddots \end{array}\right), \end{equation} \tag{ 16 }$$

where the elements a_jk, b_jk, c_jk, and d_jk, j, k = 1, ...M/2, of the off-diagonal blocks can be calculated by multiplying the 2 × 2 block (12b) with ${\rm diag}_2(\alpha _j,\alpha _j^{-1})$ from the left and with ${\rm diag}_2(\alpha _k^{-1},\alpha _k)$ from the right.

Under the determinant the matrix (16) can be further simplified by replacing column 2k − 1 by the difference of columns 2k − 1 and 2k for all k = 1, ..., M/2 and afterwards by replacing row 2j − 1 by the difference of rows 2j − 1 and 2j for all j = 1, ..., M/2. Up to first order in each δ₁, δ₂, ..., δ_M/2 the determinant of $\mathbb{1}+U$ becomes

$$\begin{eqnarray} \fl \det {}_{\!M}\left (\begin{array}{@{}c@{\ \ }c@{\ \ }c@{\ \ }c@{}} \left[\begin{array}{@{}c@{\ \ }c@{}}\delta _1 D_{1} & 0 \\ 0 & 1\end{array}\right] & \left[\begin{array}{@{}cc@{}}\delta _2 e_{12} & 0\\ 0 & 0\end{array}\right] & \left[\begin{array}{@{}cc@{}}\delta _3 e_{13} & 0 \\ 0 & 0\end{array}\right] & \dots\\ \ms \left[\begin{array}{@{}c@{\ \ }c@{}}\delta _1 e_{21} & 0 \\ 0 & 0\end{array}\right] & \left[\begin{array}{@{}cc@{}}\delta _2 D_{2} & 0 \\ 0 & 1\end{array}\right] & \left[\begin{array}{@{}cc@{}}\delta _3 e_{23} & 0 \\ 0 & 0\end{array}\right] &\\ \ms \left[\begin{array}{@{}c@{\ \ }c@{}}\delta _1 e_{31} & 0 \\ 0 & 0\end{array}\right] & \left[\begin{array}{@{}cc@{}}\delta _2 e_{32} & 0 \\ 0 & 0\end{array}\right] & \left[\begin{array}{@{}cc@{}}\delta _3 D_{3} & 0 \\ 0 & 1\end{array}\right] & \\ \vdots & & & \ddots \end{array}\right)= \left[\prod _{j=k}^{M/2}\delta _k\right]\det {}_{\!M/2} \left[\begin{array}{@{}cccc@{}}D_{1} & e_{12} & e_{13} & \dots\\ e_{21} & D_{2} & e_{23} & \\ e_{31} & e_{32} & D_{3} & \\ [-0.7ex] \vdots & & & \ddots \end{array}\right] \nonumber\\ \end{eqnarray} \tag{ 17 }$$

where e_jk = a_jk − b_jk − c_jk + d_jk. The diagonal elements D_k, k = 1, ..., M/2, are given by

$$\begin{eqnarray} &&\fl D_k = \lim {}_{\lbrace \delta _k\rightarrow 0\rbrace _{k=1}^{M/2}}\left(-\frac{s_{2\lambda _k,-\eta }}{s_{2\lambda _k,0}}\alpha _k^{2} - \frac{s_{2\lambda _k,+\eta }}{s_{2\lambda _k,0}}\alpha _k^{-2} - \mathfrak {b}_k^{+}\alpha _k^{-2} - \mathfrak {b}_k^{-}\alpha _k^2 \right)\nonumber \\ &&\quad \,\,\fl = \frac{s_{2\lambda _k,+\eta }}{s_{2\lambda _k,0}}\mathfrak {a}_k + \frac{s_{2\lambda _k,-\eta }}{s_{2\lambda _k,0}}\mathfrak {a}_k^{-1}+2\cosh (\eta )-s_{0,\eta }\partial _{\lambda _k}\log \left \lbrace \frac{s_{\lambda _k,+\eta /2}^{2M}}{s_{\lambda _k,-\eta /2}^{2M}}\prod _{{{\scriptstyle {\begin{array}{@{}c@{}}\scriptstyle l=1 \\ \scriptstyle l\ne k\end{array}}}}}^{M/2}\prod _{\sigma =\pm }\frac{s_{\lambda _k+\sigma \lambda _l,-\eta }}{s_{\lambda _k+\sigma \lambda _l,+\eta }}\right \rbrace \nonumber \\ &&\quad \,\,\fl = \frac{s_{2\lambda _k,+\eta }}{s_{2\lambda _k,0}}\mathfrak {A}_k + \frac{s_{2\lambda _k,-\eta }}{s_{2\lambda _k,0}}\bar{\mathfrak {A}}_k +2Ms_{0,\eta }K_{\eta /2}(\lambda _k)-\sum _{{{\scriptstyle {\begin{array}{@{}c@{}}\scriptstyle l=1\\ \scriptstyle l\ne k\end{array}}}}}^{M/2} s_{0,\eta }K_\eta ^{+}(\lambda _k,\lambda _l), \end{eqnarray} \tag{ 18 }$$

where we defined $K_\eta ^{+}(\lambda ,\mu )=K_\eta (\lambda -\mu )+K_\eta (\lambda +\mu )$, $K_\eta (\lambda )=\frac{s_{0,2\eta }}{s_{\lambda ,+\eta }s_{\lambda ,-\eta }}$ and $\mathfrak {A}_k=1+\mathfrak {a}_k$, $\bar{\mathfrak {A}}_k=1+\mathfrak {a}_k^{-1}$. Using these definitions the off-diagonal elements can be simplified to

$$\begin{equation} e_{jk} = \sqrt{\frac{s_{2\lambda _k,+\eta }s_{2\lambda _k,-\eta }\mathfrak {a}_j}{s_{2\lambda _j,+\eta }s_{2\lambda _j,-\eta }\mathfrak {a}_k}}(s_{0,\eta }K_{\eta }^{+}(\lambda _j,\lambda _k)+ f_{jk}), \end{equation} \tag{ 19 }$$

where the symbols f_jk are specified below (see equation (20c)). We can forget about the square roots as they cancel each other under the determinant. The factor $\prod _{k=1}^{M/2}\delta _k$ cancels exactly the factor $\alpha _{\rm reg}^{-1}=\prod _{j=1}^{M/2}\frac{s_{2\epsilon _j,0}}{s_{0,\eta }}=\prod _{k=1}^{M/2}\delta _k^{-1}$ in equation (11b).

Hence, we reduced the M dimensional determinant of overlaps with deviated parity-invariant states in the limit of vanishing deviations to an M/2 dimensional determinant which depends on M/2 independent half of the parameters. The result can be eventually summed up as follows:

$$\begin{eqnarray} \fl \big\langle \Psi _0 |\lbrace \pm \lambda _j\big\rbrace _{j=1}^{M/2}\rangle &= \left.\big\langle \Psi _0 |\lbrace \lambda _j+\epsilon _j\rbrace _{j=1}^{M/2}\cup \lbrace -\lambda _j+\epsilon _j\rbrace _{j=1}^{M/2}\big\rangle \right|_{\lbrace \epsilon _j\rightarrow 0\rbrace _{j=1}^{M/2}} \nonumber \\ &=\big[\gamma \det {}_{\!M/2}(G_{jk}^{+}) + \mathcal {O}\big(\lbrace \epsilon _j\rbrace _{j=1}^{M/2}\big)\big]_{\lbrace \epsilon _j\rightarrow 0\rbrace _{j=1}^{M/2}} = \gamma \det {}_{\!M/2}(G_{jk}^{+}), \end{eqnarray} \tag{ 20a }$$

where the prefactor γ is given by equation (11a) and the matrix $G_{jk}^+$ reads

$$\begin{eqnarray} &&\fl G_{jk}^{+} = \delta _{jk}\bigg(Ns_{0,\eta }K_{\eta /2}(\lambda _j)-\sum _{l=1}^{M/2}s_{0,\eta }K_\eta ^{+}(\lambda _j,\lambda _l)\bigg) + s_{0,\eta }K_\eta ^{+}(\lambda _j,\lambda _k)\nonumber \\ &&+ \delta _{jk}\frac{s_{2\lambda _j,+\eta } \mathfrak {A}_j+s_{2\lambda _j,-\eta } \bar{\mathfrak {A}}_j}{s_{2\lambda _j,0}} + (1-\delta _{jk})f_{jk}, \quad j,k=1,\ldots ,M/2 \end{eqnarray} \tag{ 20b }$$

$$\begin{eqnarray} &&\fl f_{jk} = \mathfrak {A}_k\bigg( \frac{s_{2\lambda _j,+\eta } s_{0,\eta }}{s_{\lambda _j+\lambda _k,0}s_{\lambda _j-\lambda _k,+\eta }} - \frac{s_{2\lambda _j,-\eta }s_{0,\eta }}{s_{\lambda _j-\lambda _k,0}s_{\lambda _j+\lambda _k,-\eta }} \bigg) + \mathfrak {A}_k\bar{\mathfrak {A}}_j\frac{s_{2\lambda _j,-\eta }s_{0,\eta }}{s_{\lambda _j-\lambda _k,0}s_{\lambda _j+\lambda _k,-\eta }} \nonumber \\ &&- \bar{\mathfrak {A}}_j\bigg(\frac{s_{2\lambda _j,-\eta }s_{0,\eta }}{s_{\lambda _j-\lambda _k,0}s_{\lambda _j+\lambda _k,-\eta }} + \frac{s_{2\lambda _j,-\eta }s_{0,\eta }}{s_{\lambda _j+\lambda _k,0}s_{\lambda _j-\lambda _k,-\eta }}\bigg). \end{eqnarray} \tag{ 20c }$$

Note that here the matrix elements $G_{jk}^{+}$ contain an additional factor s_{0, η} = isinh (η) compared to the earlier definition of $G_{jk}^{+}$ in equation (9b), which is convenient for taking the XXX limit η → 0 which we shall do in the next section, section 3.2. Formula (20) holds for arbitrary complex numbers λ_j, j = 1, ..., M/2.

If we are on-shell in equations (20a-c), i. e. if the set $\lbrace \pm \lambda _j\rbrace _{j=1}^{M/2}$ satisfies the Bethe equations (5), and if we are not in the XXX limit with rapidities at infinity, $\mathfrak {A}_j$ and $\bar{\mathfrak {A}}_j$ just vanish. Together with the norm of Bethe states and using the symmetry of the Gaudin matrix (6c) as well as the relation

$$\begin{equation} \det {}_{\!M}\left(\begin{array}{@{}cc@{}}A & B\\ B & A\end{array}\right)= \det {}_{\!M/2}(A+B)\det {}_{\!M/2}(A-B) \end{equation} \tag{ 21 }$$

for block matrices we finally gain the overlap formula (9), where the additional factors s_{0, η} in $G_{jk}^{+}$, 1 ⩽ j, k ⩽ M/2, cancel against the prefactor $\sqrt{s_{0,\eta }^M}$ in the norm formula (6).

At the isotropic point Δ = 1, there are Bethe states with rapidities at infinity that need to be specially treated. In order to obtain an appropriate overlap formula for the XXX case including such rapidities at infinity we use the off-shell formula (20a-c). As long as we are off-shell we can scale all rapidities by η, send η to zero and can afterwards send some of the rapidities to infinity. Within this scaling equations (20) reduce to

$$\begin{equation} \big\langle \Psi _0 |\lbrace \pm \lambda _j\rbrace _{j=1}^{M/2} \big\rangle = \tilde{\gamma }\det {}_{\!M/2}(\tilde{G}_{jk}^{+}), \end{equation} \tag{ 22a }$$

where now

$$\begin{eqnarray} &&\fl \tilde{\gamma }= \sqrt{2}\left[\prod _{j=1}^{M/2}\frac{\lambda _j^2+\frac{1}{4}}{4\lambda _j^2}\right] \left[\prod _{j>k=1}^{M/2}\prod _{\sigma =\pm }\frac{(\lambda _j+\sigma \lambda _k)^2+1}{(\lambda _j+\sigma \lambda _k)^2}\right] \end{eqnarray} \tag{ 22b }$$

$$\begin{eqnarray} &&\fl \tilde{G}_{jk}^{+} = \delta _{jk}\left(2M\tilde{K}_{1/2}(\lambda _j)-\sum _{l=1}^{M/2}\tilde{K}_1^{+}(\lambda _j,\lambda _l)\right) + \tilde{K}_1^{+}(\lambda _j,\lambda _k) \nonumber\\ &&+\, \delta _{jk}\frac{(2\lambda _j+i) \mathfrak {A}_j+(2\lambda _j-i) \bar{\mathfrak {A}}_j}{2\lambda _j} + (1-\delta _{jk})\tilde{f}_{jk},\quad j,k=1,\ldots ,M/2 \end{eqnarray} \tag{ 22c }$$

$\tilde{K}_\alpha ^{+}(\lambda ,\mu )=\tilde{K}_\alpha (\lambda -\mu )+\tilde{K}_\alpha (\lambda +\mu )$, and $\tilde{K}_\alpha (\lambda )=\frac{2\alpha }{\lambda ^2+\alpha ^2}$. The explicit form of $\tilde{f}_{jk}$ is not interesting since either $\tilde{f}_{jk}$ vanishes due to Bethe equations or it gives subleading corrections if one of the parameters λ_j goes to infinity. The term $\mathfrak {a}_j$ reads now

$$\begin{equation} \mathfrak {a}_j = \left[\prod _{k=1}^{M/2}\prod _{\sigma =\pm }\frac{\lambda _j-\sigma \lambda _k-i}{\lambda _j-\sigma \lambda _k+i}\right]\left(\frac{\lambda _j+i/2}{\lambda _j-i/2}\right)^{2M}, \end{equation} \tag{ 23 }$$

and we have again $\mathfrak {A}_j=1+\mathfrak {a}_j=0$ if the set $\lbrace \pm \lambda _j\rbrace _{j=1}^{M/2}$ satisfies the Bethe equations of the XXX model.

Let us consider the case when n pairs of Bethe roots are at ±∞ in such a way that the difference and the sum of two Bethe roots which do not belong to the same pair is also infinity. Let us denote the corresponding parameters by μ_j, j = 1, ..., n. We have m = M/2 − n finite pairs (λ_j, −λ_j) that fulfil the Bethe equations

$$\begin{equation} \mathfrak {a}_j = \left[\prod _{k=1}^{m}\prod _{\sigma =\pm }\frac{\lambda _j-\sigma \lambda _k-i}{\lambda _j-\sigma \lambda _k+i}\right]\left(\frac{\lambda _j+i/2}{\lambda _j-i/2}\right)^{2M} = -1, \quad j=1,\ldots ,m. \end{equation} \tag{ 24 }$$

We used the fact that all factors including one of the parameters μ_k, k = 1, ..., n, are equal to one. The Bethe equations for all infinite Bethe roots are trivial. The prefactor $\tilde{\gamma }$ has no divergencies in μ_j and just reads

$$\begin{equation} \tilde{\gamma }= \frac{\sqrt{2}}{4^n}\left[\prod _{j=1}^{m}\frac{\lambda _j^2+1/4}{4\lambda _j^2}\right]\left[\prod _{j>k=1}^{m}\prod _{\sigma =\pm }\frac{(\lambda _j+\sigma \lambda _k)^2+1}{(\lambda _j+\sigma \lambda _k)^2}\right]. \end{equation} \tag{ 25 }$$

Now, we take the limits μ_j → ∞, j = 1, ..., n, in equation (22c) and treat all terms containing $\mathfrak {A}_j$ or $\bar{\mathfrak {A}}_j$ carefully. The determinant simplifies to

$$\begin{equation} \lim \nolimits _{\lbrace \mu _j\rightarrow \infty \rbrace _{j=1}^n}\left(\frac{\det {}_{\!M/2}(\tilde{G}_{jk}^{+})}{\mathcal {N}(\lbrace \mu _j\rbrace _{j=1}^{n})}\right) = 4^n(2n)!\det {}_{\!m}(\hat{G}_{jk}^{+}) \end{equation} \tag{ 26 }$$

where $\hat{G}_{jk}^{+}$ is a reduced m × m version of the M/2 × M/2 matrix $\tilde{G}_{jk}^{+}$ in equation (22c), and where we used a factor $\mathcal {N}(\lbrace \mu _j\rbrace _{j=1}^{n})=\prod _{j=1}^n\mu _j^{-2}$ to get a non-vanishing result.

We need the same factor $\mathcal {N}(\lbrace \mu _j\rbrace _{j=1}^{n})$ to correctly renormalize the norm formula (6). We introduce the abbreviations $\Lambda _{\pm }^{\!2m} = \lbrace \pm \lambda _j\rbrace _{j=1}^{m}$, $\mathcal {M}_{\pm }^{2n}=\lbrace \pm \mu _j\rbrace _{j=1}^{n}$, and $\Lambda _{\pm }^{M} = \Lambda _{\pm }^{\!2m} \cup \mathcal {M}_{\pm }^{2n}$ where the subscripts ± and 2m, 2n are reminiscent to the parity invariance of the state and the 2m flipped spins, 2n pairs at infinity, respectively. The square of the norm of the state can be written as

$$\begin{equation} \fl \left.\frac{\langle \Lambda _{\pm }^{M}|\Lambda _{\pm }^{M}\rangle }{\mathcal {N}^2(\lbrace \mu _j\rbrace _{j=1}^{n})} \right |_{\lbrace \mu _j\rightarrow \infty \rbrace _{j=1}^n} = \langle \Lambda _{\pm }^{\!2m}|\left(S^+\right)^{2n}\left(S^-\right)^{2n}|\Lambda _{\pm }^{\!2m}\rangle = (4n)!\: \langle \Lambda _{\pm }^{\!2m} |\Lambda _{\pm }^{\!2m}\rangle . \end{equation} \tag{ 27 }$$

The norm square of the Bethe state $|\Lambda _{\pm }^{\!2m}\rangle$ is given by the XXX limit of the norm formula (6b) and reads

$$\begin{equation} \fl \langle \Lambda _{\pm }^{\!2m} |\Lambda _{\pm }^{\!2m}\rangle \ = \left[\prod _{j=1}^m \frac{\lambda ^2_j+1/4}{\lambda ^2_j}\right] \left[\prod _{j>k=1}^m\prod _{\sigma =\pm }\frac{((\lambda _j+\sigma \lambda _k)^2+1)^2}{(\lambda _j+\sigma \lambda _k)^4}\right] \det {}_{\!2m}(\hat{G}_{jk}), \end{equation} \tag{ 28 }$$

where $\hat{G}_{jk}$ is a reduced 2m × 2m version of the M × M Gaudin matrix $\tilde{G}_{jk}$.

All together we obtain for the overlap of the Néel state (7) with a normalized, parity-invariant XXX Bethe state $|\Lambda _{\pm }^{(m,n)}\rangle$, where m pairs (λ_j, −λ_j) are finite and all other rapidities ±μ_j, j = 1, ..., n, are at ±∞ (N_∞ = 2n, 2m + 2n = M = N/2):

$$\begin{equation} \fl \langle \Psi _0 |\Lambda _{\pm }^{(m,n)}\rangle = \left.\frac{\langle \Psi _0 |\Lambda _{\pm }^{M}\rangle }{ \Vert \Lambda _{\pm }^{M}\Vert }\right|_{\lbrace \mu _j\rightarrow \infty \rbrace _{j=1}^{n}} = \frac{\sqrt{2} N_{\infty }!}{\sqrt{(2N_{\infty })!}} \left[\prod _{j=1}^m \frac{\sqrt{\lambda ^2_j+1/4}}{4\lambda _j}\right] \sqrt{\frac{\det {}_{\!m}\hat{G}_{jk}^{+}}{\det {}_{\!m}\hat{G}_{jk}^{-}}}, \end{equation} \tag{ 29a }$$

$$\begin{eqnarray} &&\fl \hat{G}_{jk}^{\pm } = \delta _{jk}\bigg(2M\tilde{K}_{1/2}(\lambda _j)-\sum _{l=1}^{m}\tilde{K}_1^{+}(\lambda _j,\lambda _l)\bigg) + \tilde{K}_1^{\pm }(\lambda _j,\lambda _k), \quad j,k=1,\ldots ,m \end{eqnarray} \tag{ 29b }$$

with $\tilde{K}_\alpha ^{\pm }(\lambda ,\mu )=\tilde{K}_\alpha (\lambda -\mu )\pm \tilde{K}_\alpha (\lambda +\mu )$ and $\tilde{K}_\alpha (\lambda )=\frac{2\alpha }{\lambda ^2+\alpha ^2}$. Note again that in equations (29) Bethe roots can be complex numbers (string solutions).