The Bethe Ansatz

We start by rewriting h.tba1 as

\begin{equation*} \ln (1 + e^{\varepsilon_n (\lambda)/T}) = hn - \pi J a_n (\lambda) + \sum_{m=1}^\infty \left( \delta_{nm} + A_{nm}\right) * \ln (1 + e^{-\varepsilon_m (\lambda)/T}). \end{equation*}

As can be seen from the equilibrium conditions (most easily from h.tbaf), all functions \(\varepsilon_n\) with \(n > 1\) are positive-valued for any \(\lambda\), leaving \(\varepsilon_1\) as the only function which suffers a sign change. Considering the \(T \rightarrow 0^+\) limit and writing the positive(resp. negative)-valued parts of \(\varepsilon_1\) as \(\varepsilon_1^{(+)}\) (resp. \(\varepsilon_1^{(-)}\)), we obtain

\begin{equation*} \varepsilon_n (\lambda) = hn - \pi J a_n (\lambda) - A_{n1} * \varepsilon_1^{(-)} (\lambda), \hspace{5mm} n > 1 \end{equation*}

for all \(n > 1\) functions, which are fully determined once we solve

\begin{equation*} \varepsilon_1^{(+)} (\lambda) + (1 + A_{11}) * \varepsilon_1^{(-)} (\lambda) = h - \pi J a_1 (\lambda). \end{equation*}

Performing a Fourier transformation, [TBC]