The Bethe Ansatz

Parametrization in terms of rapiditiesc.h.r

Let us now specialize to the most important case of the isotropic antiferromagnet ($$\Delta = 1$$).

The bare momenta $$k_a$$ will be parametrized in terms of rapidities $$\lambda$$ according to

$$\lambda = \frac{1}{2} \cot \frac{k}{2}, \hspace{1cm} k = \pi - 2~\mbox{atan} (2\lambda) \tag{h.l}\label{h.l}$$

such that $$k(\lambda = 0) = \pi$$. Such a parametrization is purposefully chosen to make the scattering phase shift become a new function depending only on the rapidity difference, namely $$\phi(k_1, k_2) \equiv \phi (\lambda_1 - \lambda_2)$$ where

\begin{equation*} \phi (\lambda) = 2~\mbox{atan} \lambda. \end{equation*}

The Bethe equations for the $$XXX$$ model are written in terms of rapidities as

$$\left[ \frac{ \lambda_a + i/2}{\lambda_a - i/2} \right]^N = \prod_{b \neq a}^M \frac{\lambda_a - \lambda_b + i}{\lambda_a - \lambda_b - i}, \hspace{1cm} a = 1, ..., M \tag{h.be}\label{h.be}$$

or, in logarithmic form,

$$%2\mbox{atan} \left( 2\lambda_a \right) - \frac{1}{N} \sum_{b = 1}^M 2\mbox{atan} \left(\lambda_a - \lambda_b\right) = 2\pi \frac{I_a}{N} \phi_1(\lambda_a) - \frac{1}{N} \sum_{b = 1}^M \phi_2(\lambda_a - \lambda_b) = 2\pi \frac{I_a}{N} \tag{h.bel}\label{h.bel}$$

where the quantum numbers $$I_j$$ are half-odd integers for $$N - M$$ even, integers for $$N - M$$ odd (with $$I_j$$ defined mod$$(N)$$). For convenience, we have introduced the functions

$$\phi_n (\lambda) = 2~\mbox{atan}~ \frac{2\lambda}{n}. \tag{h.phin}\label{h.phin}$$

The energy of a state is given as a function of the rapidities by

$$E = J \sum_{a = 1}^M \frac{-2}{4\lambda_a^2 + 1}, \tag{h.e}\label{h.e}$$

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

$$P = \sum_{a = 1}^M \frac{1}{i} \ln \left[\frac{\lambda_a + i/2}{\lambda_a - i/2}\right] = \pi M - \frac{2\pi}{N}\sum_{a = 1}^M I_a \hspace{5mm} \mbox{mod} \hspace{1mm}2\pi. \tag{h.p}\label{h.p}$$