The Bethe Ansatz

The planar Heisenberg chain (XXZ with \(-1 < \Delta < 1\))

Let us now turn to the anisotropic antiferromagnet in the gapless regime \(0 < \Delta < 1\). The anisotropy is parametrized by \(\zeta = \mbox{acos} \Delta\) with \(\zeta \in ]0, \pi/2[\). The bare momenta are here parametrized in terms of rapidities as

\begin{equation} e^{ik} = \frac{\sinh (\lambda + i\zeta/2)}{\sinh (\lambda - i\zeta/2)}, \hspace{1cm} k = \pi - 2~\mbox{atan}~ \frac{\tanh\lambda}{\tan \zeta/2} \tag{p.l}\label{p.l} \end{equation}

again such that \(k(\lambda = 0) = \pi\). Using this parametrization, the scattering phase shift again becomes function of the rapidity difference only. The Bethe equations then become

\begin{equation} \left[ \frac{\sinh (\lambda_j + i\zeta/2)}{\sinh(\lambda_j - i\zeta/2)} \right]^N = \prod_{k \neq j}^M \frac{\sinh(\lambda_j - \lambda_k + i\zeta)}{\sinh(\lambda_j - \lambda_k - i\zeta)}, \hspace{1cm} j = 1, ..., M. \tag{}\label{} \end{equation}

Taking logs and defining the fundamental kernel

\begin{equation} \phi_n (\lambda) = 2~\mbox{atan}~ \frac{\tanh \lambda}{\tan (n\zeta/2)}, \tag{p.phin}\label{p.phin} \end{equation}

the Bethe equations are rewritten as

\begin{eqnarray} \phi_1(\lambda_a) - \frac{1}{N} \sum_{b = 1}^M \phi_2(\lambda_a - \lambda_b) = 2\pi \frac{I_a}{N} \tag{p.bel}\label{p.bel} \end{eqnarray}

which are precisely of the same form as h.bel, the only difference being the definition of the \(\phi_n\) kernels. The energy of a state described by the set of rapidities \(\{ \lambda_a \}\) is then

\begin{equation} E = J \sum_{a = 1}^M \frac{-\sin^2 \zeta}{\cosh 2\lambda_a - \cos \zeta}, \tag{p.e}\label{p.e} \end{equation}

whereas the momentum again has a simple representation in terms of the quantum numbers,

\begin{equation} P = \sum_{a = 1}^M \frac{1}{i} \ln \left[\frac{\sinh(\lambda_a + i\zeta/2)}{\sinh(\lambda_a - i\zeta/2)}\right] = \pi M - \frac{2\pi}{N}\sum_{a = 1}^M I_a \hspace{0.5cm} \mbox{mod} \hspace{0.2cm}2\pi. \tag{p.p}\label{p.p} \end{equation}
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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24