The Bethe Ansatz

R-matrix

\begin{equation*} R (\lambda) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & b (\lambda) & c (\lambda) & 0 \\ 0 & c (\lambda) & b (\lambda) & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \end{equation*}

With \(\varphi\) an odd function, we take

\begin{align*} a(\lambda) &= 1, & b(\lambda) &= \frac{\varphi(\lambda)}{\varphi(\lambda + \eta)}, \\ c(\lambda) &= \frac{\varphi(\eta)}{\varphi(\lambda + \eta)}, & d(\lambda) &= b(\lambda - \eta/2)^N = \left(\frac{\varphi (\lambda -\eta/2)}{\varphi(\lambda + \eta/2)}\right)^N \end{align*}

Transfer matrix

\begin{equation*} L_j (\lambda) = R_{aj} (\lambda - \eta/2) \end{equation*}

Transfer matrix eigenvalue

\begin{align*} \tau (\lambda | \{ \lambda \}) &= \frac{a(\lambda)}{b(\bar{\lambda}, \lambda)} + \frac{d(\lambda)}{b(\lambda, \bar{\lambda})} \\ &= \frac{\varphi(\bar{\lambda} - \lambda + \eta)}{\varphi (\bar{\lambda} - \lambda)} + \left[ \frac{\varphi(\lambda - \eta/2)}{\varphi (\lambda + \eta/2)} \right]^N \frac{\varphi(\bar{\lambda} - \lambda - \eta)}{\varphi (\bar{\lambda} - \lambda)} \\ &= \frac{\varphi (\bar{\lambda} - \lambda + \eta)}{\varphi(\bar{\lambda} - \lambda)} \left[ 1 + \left( \frac{\varphi(\lambda - \eta/2)}{\varphi(\lambda + \eta/2)} \right)^N \frac{\varphi(\bar{\lambda} - \lambda - \eta)}{\varphi(\bar{\lambda} - \lambda + \eta)} \right] \end{align*}

or equivalently

\begin{align*} \tau (\lambda | \{ \lambda \}) &= \frac{a(\lambda)}{b(\bar{\lambda}, \lambda)} \bigl( 1 + {\cal E}^{-1} (\lambda | \{ \lambda \}) \bigr) = \frac{d(\lambda)}{b(\lambda, \bar{\lambda})} \bigl( 1 + {\cal E} (\lambda | \{ \lambda \}) \bigr), \\ {\cal E} (\lambda | \{ \lambda \}) &\equiv \frac{a(\lambda)}{d(\lambda)} \frac{b(\lambda, \bar{\lambda})}{b(\bar{\lambda}, \lambda)} = \left(\frac{\varphi (\lambda + \eta/2)}{\varphi (\lambda - \eta/2)}\right)^N \frac{\varphi (\lambda - \bar{\lambda} - \eta)}{\varphi(\lambda - \bar{\lambda} + \eta)} \end{align*}

Hamiltonian

\begin{equation*} H_{XXZ} = \frac{\varphi (\eta)}{2} \frac{d}{d\lambda} \ln \tau (\lambda) |_{\lambda = \eta/2} = \sum_{j=1}^N \left[S^x_j S^x_{j+1} + S^y_j S^y_{j+1} + \cosh \eta ~(S^z_j S^z_{j+1} - \frac{1}{4}) \right] \end{equation*}

Eigenstates

\begin{equation*} | \{ \lambda \} \rangle = B(\bar{\lambda}) | 0 \rangle \end{equation*}

Bethe equations

\[ \frac{a(\lambda_j)}{d(\lambda_j)} \frac{b(\lambda_j, \bar{\lambda}_j)}{b(\bar{\lambda}_j, \lambda_j)} = \left(\frac{\varphi (\lambda_j + \eta/2)}{\varphi (\lambda_j - \eta/2)}\right)^N \frac{\varphi (\lambda_j - \bar{\lambda}_j - \eta)}{\varphi(\lambda_j - \bar{\lambda}_j + \eta)} = 1 \]

or equivalently (using \(\varphi (-\lambda) = -\varphi (\lambda)\) and the fact that the term \(j\) is now included in the product)

\begin{align*} &r (\lambda_j) + s(\lambda_j | \{ \lambda \}) = 0, \\ &r (\lambda_j) \equiv \frac{a(\lambda_j)}{d(\lambda_j)} = \left(\frac{\varphi (\lambda_j + \eta/2)}{\varphi (\lambda_j - \eta/2)}\right)^N, \\ &s (\lambda_j | \{ \lambda \}) \equiv \frac{b(\bar{\lambda}, \lambda_j)}{b(\lambda_j, \bar{\lambda})} = \frac{\varphi (\lambda_j - \bar{\lambda} + \eta)}{\varphi(\lambda_j - \bar{\lambda} - \eta)} \end{align*}

or equivalently

\[ {\cal E} (\lambda_j | \{ \lambda \}) + 1 = 0 \]

Energy

\[ E(\{ \lambda \}) = \frac{\varphi(\eta)}{2} \left. \frac{d}{d\lambda} \ln \frac{\varphi(\bar{\lambda} + \eta/2)}{\varphi(\bar{\lambda} - \eta/2)} \right|_{\lambda = \eta/2} = \sum_j \left(\cos k(\lambda_j) - \Delta\right) \]

Momentum

\[ e^{i k(\lambda_j)} = \frac{\varphi(\lambda_j -\eta/2)}{\varphi(\lambda_j + \eta/2)} \]

\[ P(\{ \lambda \}) = i \ln \tau (\eta/2 | \{ \lambda \}) = \frac{i}{N} \ln \bar{r} = i \ln \frac{\varphi(\bar{\lambda} +\eta/2)}{\varphi(\bar{\lambda} - \eta/2)} = \sum_j k(\lambda_j) \]

\[ e^{-i \hat{P}} {\cal O}_j e^{i \hat{P}} = {\cal O}_{j+1} \]

Scalar products

\[ S_M (\{ \mu \}, \{ \lambda \}) \equiv \langle 0 | C(\bar{\mu}) B(\bar{\lambda}) | 0 \rangle \]

\begin{align*} &S_M (\{ \mu \}, \{ \lambda \}) = \frac{\mbox{det}~T(\{ \mu \}, \{ \lambda \})}{\mbox{det}~V(\{ \mu \}, \{ \lambda \})}, \\ &T_{jk} = \frac{\partial}{\partial \lambda_k} \tau (\mu_j | \{ \lambda \}), \hspace{5mm} V_{jk} = \frac{1}{\varphi (\mu_k - \lambda_j)} \end{align*}

\[ \mathbb{B}(\lambda) \equiv \frac{B(\lambda)}{d(\lambda)}, \hspace{5mm} \mathbb{C}(\lambda) \equiv \frac{C(\lambda)}{d(\lambda)} \]

Gaudin matrix and state norm

\begin{equation*} {\mathbb N}_M = \langle 0 | C (\bar{\lambda}) B (\bar{\lambda}) | 0 \rangle = \varphi^M(\eta) %\prod_{j \neq k} \frac{\varphi(\lambda_j - \lambda_k + \eta)}{\varphi(\lambda_j - \lambda_k)} \frac{\varphi(\bar{\lambda\!\lambda}_= + \eta)}{\varphi(\bar{\lambda\!\lambda}_=)} \det {\cal G} (\{ \lambda \}) \end{equation*} \begin{align*} {\cal G}_{jk} (\{ \lambda \}) &= -\frac{\partial}{\partial \lambda_k} \ln \left[ \frac{a(\lambda_j)}{d(\lambda_j)} \frac{b(\lambda_j, \bar{\lambda}_j)}{b(\bar{\lambda}_j, \lambda_j)}\right] = -\frac{\partial}{\partial \lambda_k} \ln \bigl(-{\cal E} (\lambda_j | \{ \lambda \})\bigr) \\ &= -\frac{\partial}{\partial \lambda_k} \ln r(\lambda_j) + \frac{\partial}{\partial \lambda_k} \ln s (\lambda_j | \{ \lambda \}) \\ &= -\delta_{jk} ~N ~\partial_\lambda \ln \left.\frac{\varphi(\lambda + \eta/2)}{\varphi(\lambda - \eta/2)} \right|_{\lambda_j} + \partial_{\lambda_k} \ln \frac{\varphi(\lambda_j - \bar{\lambda} + \eta)}{\varphi(\lambda_j - \bar{\lambda} - \eta)} \end{align*}

Matrix element for \(\sigma^-\)

\begin{equation*} F_j^- (\{ \mu \}_{M+1}, \{ \lambda \}_M) = \langle 0 | C(\bar{\mu}) ~\sigma^-_j ~B(\bar{\lambda}) | 0 \rangle \end{equation*} \begin{align*} F_j^- (\{ \mu \}_{M+1}, \{ \lambda \}_M) &= \frac{\phi_{j-1} (\{ \mu \})}{\phi_{j-1} (\{ \lambda \})} \frac{\varphi(\bar{\mu} + \eta/2)}{\varphi(\bar{\lambda} + \eta/2)} %\frac{\det_{M+1} H^- (\{ \mu \}, \{ \lambda \})}{\prod_{M+1 \geq l > m \geq 1} \varphi(\mu_l - \mu_m) \prod_{1 \leq l < m \leq M} \varphi(\lambda_l - \lambda_m)} \frac{\det_{M+1} H^- (\{ \mu \}, \{ \lambda \})}{\varphi(\bar{\mu\!\mu}_{\scriptscriptstyle \leq}) \varphi(\bar{\lambda\!\lambda}_{\scriptscriptstyle \geq})} \end{align*}

\[ \phi_j (\{ \lambda \}) = \left(b(\bar{\lambda} - \eta/2)\right)^{-j} = e^{iP(\{ \lambda \}) j} \]

\begin{align*} \left. H^{-}_{jk} \right|_{k < M+1} &= \frac{\varphi(\eta)}{\varphi(\mu_j - \lambda_k)} \left( a(\lambda_k) \varphi(\bar{\mu}_j - \lambda_k + \eta) - d(\lambda_k) \varphi (\bar{\mu}_j - \lambda_k -\eta) \right), \nonumber \\ H^{-}_{j M+1} &= \frac{\varphi(\eta)}{\varphi(\mu_j + \eta/2) \varphi(\mu_j - \eta/2)}. \end{align*}

Matrix element for \(\sigma^z\)

\begin{equation*} F_j^z (\{ \mu \}_{M}, \{ \lambda \}_M) = \langle 0 | C(\bar{\mu}) ~\sigma^z_j ~B(\bar{\lambda}) | 0 \rangle \end{equation*} \begin{align*} F_j^z (\{ \mu \}_{M}, \{ \lambda \}_M) = \frac{\phi_{j-1} (\{ \mu \})}{\phi_{j-1} (\{ \lambda \})} \frac{\varphi(\bar{\mu} + \eta/2)}{\varphi(\bar{\lambda} + \eta/2)} %\frac{\det_{M} \bigl(H (\{ \mu \}, \{ \lambda \}) - 2P (\{ \mu \}, \{ \lambda \}) \bigr)}{\prod_{M \geq l > m \geq 1} \varphi(\mu_l - \mu_m) \prod_{1 \leq l < m \leq M} \varphi(\lambda_l - \lambda_m)} \frac{\det_{M} H^z (\{ \mu \}, \{ \lambda \})}{\varphi(\bar{\mu\!\mu}_{\scriptscriptstyle \leq}) \varphi(\bar{\lambda\!\lambda}_{\scriptscriptstyle \geq})} \end{align*} \begin{align*} H^z (\{ \mu \}, \{ \lambda \}) &\equiv H^z (\{ \mu \}, \{ \lambda \}) - 2P (\{ \mu \}, \{ \lambda \}),\\ H_{jk} &= \frac{\varphi(\eta)}{\varphi(\mu_j - \lambda_k)} \left( a(\lambda_k) \varphi(\bar{\mu}_j - \lambda_k + \eta) - d(\lambda_k) \varphi (\bar{\mu}_j - \lambda_k -\eta) \right), \\ P_{jk} &= \varphi(\eta) \frac{\varphi(\bar{\lambda} - \lambda_k + \eta)}{\varphi(\mu_j - \eta/2) \varphi(\mu_j + \eta/2)} \end{align*}