The Bethe Ansatz

Let us now provide the explicit monodromy operator conjugacy relations for all the different regimes of the \(XXZ\) model, building up on a.hc and a.xxz.

Our starting point is the \(L\) operator for a.r:

\begin{align*} L_j (\lambda) &= \frac{1}{\sinh(\lambda +\frac{\eta}{2})} \left( \begin{array}{cccc} \sinh (\lambda + \frac{\eta}{2}) & 0 & 0 & 0 \\ 0 & \sinh (\lambda - \frac{\eta}{2}) & \sinh \eta & 0 \\ 0 & \sinh \eta & \sinh (\lambda - \frac{\eta}{2}) & 0 \\ 0 & 0 & 0 & \sinh (\lambda + \frac{\eta}{2}) \end{array} \right)\\ &= \frac{1}{\sinh(\lambda +\frac{\eta}{2})} \left( \begin{array}{cc} \sinh\lambda \cosh\frac{\eta}{2} \,{\boldsymbol 1}_j + 2 \cosh\lambda \sinh\frac{\eta}{2} \,S^z_j & \sinh\eta \,S^-_j \\ \sinh\eta \,S^+_j & \sinh\lambda \cosh\frac{\eta}{2} \,{\boldsymbol 1}_j - 2 \cosh\lambda \sinh\frac{\eta}{2} \,S^z_j \end{array} \right). \end{align*}

with periodicity \[ L_j (\lambda + i\pi) = \sigma^z_a L_j (\lambda) \sigma^z_a \]

Axial antiferromagnet \(\Delta > 1\)

\begin{equation*} L_j (\lambda) = \frac{1}{\sin(\lambda -i\frac{\eta}{2})} \left( \begin{array}{cc} \sin\lambda \cosh\frac{\eta}{2} \,{\boldsymbol 1}_j -2i \cos\lambda \sinh\frac{\eta}{2} \,S^z_j & -i\sinh\eta \,S^-_j \\ -i\sinh\eta \,S^+_j & \sin\lambda \cosh\frac{\eta}{2} \,{\boldsymbol 1}_j + 2i \cos\lambda \sinh\frac{\eta}{2} \,S^z_j \end{array} \right). \end{equation*}

Planar antiferromagnet \(0 < \Delta < 1\)

\begin{equation*} L_j (\lambda) = \frac{1}{\sinh(\lambda -i\zeta/2)} \left( \begin{array}{cc} \sinh\lambda \cos\frac{\zeta}{2} \,{\boldsymbol 1}_j - 2i \cosh\lambda \sin\frac{\zeta}{2} \,S^z_j & -i\sin\zeta \,S^-_j \\ -i\sin\zeta \,S^+_j & \sinh\lambda \cos\frac{\zeta}{2} \,{\boldsymbol 1}_j + 2i \cosh\lambda \sin\frac{\zeta}{2} \,S^z_j \end{array} \right). \end{equation*}

Planar ferromagnet \(-1 < \Delta < 0\)

\begin{equation*} L_j (\lambda) = \frac{1}{\sinh(\lambda +i\check{\zeta}/2)} \left( \begin{array}{cc} i\cosh\lambda \sin\frac{\check{\zeta}}{2} \,{\boldsymbol 1}_j +2 \sinh\lambda \cos\frac{\check{\zeta}}{2} \,S^z_j & -i\sin\check{\zeta} \,S^-_j \\ -i\sin\check{\zeta} \,S^+_j & i\cosh\lambda \sin\frac{\check{\zeta}}{2} \,{\boldsymbol 1}_j - 2 \sinh\lambda \cos\frac{\check{\zeta}}{2} \,S^z_j \end{array} \right). \end{equation*}

Axial ferromagnet \(\Delta < -1\)

\begin{equation*} L_j (\lambda) = \frac{1}{\sin(\lambda +i\frac{\check{\eta}}{2})} \left( \begin{array}{cc} i \cos\lambda \sinh\frac{\check{\eta}}{2} \,{\boldsymbol 1}_j + 2\sin\lambda \cosh\frac{\check{\eta}}{2} \,S^z_j & -i\sinh\check{\eta} \,S^-_j \\ -i\sinh\check{\eta} \,S^+_j & i \cos\lambda \sinh\frac{\check{\eta}}{2} \,{\boldsymbol 1}_j -2\sin\lambda \cosh\frac{\check{\eta}}{2} \,S^z_j \end{array} \right). \end{equation*}

Periodicity

The periodicity of the \(L\) operator thus depends on the regime:

\begin{align*} &\text{Axial:} ~~~L_j (\lambda + \pi) = \sigma^z_a L_j(\lambda) \sigma^z_a, \\ &\text{Planar:} ~L_j (\lambda + i\pi) = \sigma^z_a L_j(\lambda) \sigma^z_a. \end{align*}

Proceeding similarly to what we did for \(XXX\) conjugacy a.hc (bearing in mind the conventions we work with for the various regimes), one can show that the monodromy matrix operators obey slightly different conjugacy relations for antiferromagnetic and ferromagnetic cases:

Antiferromagnetic conjugacy (\(\Delta > 0\))

\begin{equation*} A^\dagger(\lambda) = \frac{D(\lambda)}{d (\lambda)}, \hspace{5mm} B^\dagger(\lambda) = -\frac{C(\lambda)}{d (\lambda)}, \hspace{5mm} C^\dagger(\lambda) = -\frac{B(\lambda)}{d (\lambda)}, \hspace{5mm} D^\dagger(\lambda) = \frac{A(\lambda)}{d (\lambda)} \end{equation*}

Ferromagnetic conjugacy (\(\Delta < 0\))

\begin{equation*} A^\dagger(\lambda) = \frac{D(\lambda)}{d (\lambda)}, \hspace{5mm} B^\dagger(\lambda) = \frac{C(\lambda)}{d (\lambda)}, \hspace{5mm} C^\dagger(\lambda) = \frac{B(\lambda)}{d (\lambda)}, \hspace{5mm} D^\dagger(\lambda) = \frac{A(\lambda)}{d (\lambda)} \end{equation*}

We will refer back to these when we define the dual state basis.