The Bethe Ansatz

At high temperatues, the driving term is the equation for \(\eta_1\) becomes negligible, and the become independent of rapidity. The coupled system of equations simplifies to

\begin{equation*} \eta_1^2 = 1 + \eta_2, \hspace{5mm} \eta_n^2 = (1 + \eta_{n-1}) (1 + \eta_{n+1}) \hspace{3mm} (n > 1), \hspace{5mm} \lim_{n\rightarrow \infty} \frac{\ln \eta_n}{n} = \frac{h}{T}. \end{equation*}

This second-order difference system has the following solution:

\begin{equation*} \eta_n = \left( \frac{\sinh(\frac{h}{2T}(n+1))}{\sinh(\frac{h}{2T})} \right)^2 - 1. \end{equation*}