# The Bethe Ansatz

### Another example: the trigonometric $$R$$-matrix and the anisotropic $$S=1/2$$ antiferromagnet ($$XXZ$$ model)a.xxz

Another solution to the Yang-Baxter equation can be found, which has the same form as the rational $$R$$-matrix but slightly different matrix elements:

\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), \hspace{1cm} b (\lambda) = \frac{\sinh \lambda}{\sinh (\lambda + \eta)}, \hspace{1cm} c (\lambda) = \frac{\sinh \eta}{\sinh (\lambda + \eta)}. \tag{a.r}\label{a.r} \end{equation}

The logic of the Algebraic Bethe Ansatz can again be applied here, following the previous example of the $$XXX$$ case. Adopting the same convention for the reference state (choosing also the homogeneous limit with $$\xi = \eta/2$$, but leaving $$\eta$$ here as a free parameter) yields

\begin{equation} a (\lambda) = 1, \hspace{1cm} d(\lambda) = b(\lambda - \eta/2)^N = \left[ \frac{\sinh (\lambda - \eta/2)}{ \sinh (\lambda + \eta/2)} \right]^N. \tag{a.ad}\label{a.ad} \end{equation}

\begin{equation*} \left[\frac{\sinh(\lambda_j + \eta/2)}{\sinh (\lambda_j - \eta/2)} \right]^N = \prod_{k (\neq j) = 1}^M \frac{\sinh (\lambda_j - \lambda_k + \eta)}{\sinh(\lambda_j - \lambda_k - \eta)}. \end{equation*}

The eigenvalue of the transfer matrix becomes

\begin{equation} \tau (\lambda | \{ \lambda_j \}) = \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda + \eta)}{\sinh(\lambda_j - \lambda)} + \left[\frac{\sinh(\lambda - \eta/2)}{\sinh(\lambda + \eta/2)}\right]^N \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda - \eta)}{\sinh(\lambda_j - \lambda)}. \tag{a.te}\label{a.te} \end{equation}

The first few conserved charges can be computed as

\begin{equation} P = -i \ln \tau (\lambda)|_{\lambda = \eta/2} \tag{a.q1}\label{a.q1} \end{equation}

giving eigenvalue

\begin{equation} P (\{ \lambda_j \}_M) = -i \sum_{j=1}^M \ln \frac{\sinh(\lambda_j + \eta/2)}{\sinh(\lambda_j - \eta/2)} . % = \pi M - \sum_{j=1}^M 2~\mbox{atan}~ \frac{\tanh \lambda_j}{\tanh \eta/2}. \tag{a.q1e}\label{a.q1e} \end{equation}

The Hamiltonian is (under an appropriate choice of the constant prefactor)

\begin{equation} H_{XXZ} = \frac{\sinh \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} + \Delta (S^z_j S^z_{j+1} - \frac{1}{4}) \right] \tag{a.q2}\label{a.q2} \end{equation}

in which $$\Delta = \cosh \eta$$. The energy eigenvalue of a state is

\begin{equation} E (\{ \lambda_j \}_M) = \sum_{j=1}^M \frac{\sinh^2 \eta}{\cosh 2\lambda_j - \cosh \eta}. \tag{a.q2e}\label{a.q2e} \end{equation}

For the gapped regime $$\Delta > 1$$ we have that $$\eta$$ is a real parameter. For the gapless regime $$0 < \Delta < 1$$ it is convenient to reparametrize the anisotropy as $$\eta= -i\zeta$$, so

\begin{equation} \Delta = \left\{ \begin{array}{cc} \cosh \eta, & 1 \leq \Delta, \nonumber \\ \cos \zeta, & -1 \leq \Delta \leq 1 \end{array} \right. \tag{da}\label{da} \end{equation} Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Created: 2023-06-07 Wed 16:02