# 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

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

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 xxz.be then become

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

Taking logs and defining the fundamental kernel

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

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

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

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

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