The Bethe Ansatz

Parametrization in terms of rapidities c.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

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

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

\begin{equation} \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{}\label{} \end{equation}

or, in logarithmic form,

\begin{equation} %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} \end{equation}

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

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

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

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

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

\begin{equation} 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} \end{equation}

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24