The Bethe Ansatz

We consider here first the limit of infinitely strong interactions, \(c \rightarrow \infty\), which is also known as the Tonks-Girardeau limit. We have that our Cauchy kernel l.ck trivializes to

\begin{equation*} \lim_{c \rightarrow \infty} {\cal C} (\lambda) = 0 \end{equation*}

so

\begin{equation*} \epsilon(\lambda) = \lambda^2 - \mu. \end{equation*}

The free energy becomes

\begin{equation*} g (T, \mu) = -T \int_{-\infty}^{\infty} \frac{d\lambda}{2\pi} \ln \left[ 1 + e^{-(\lambda^2 - \mu)/T}\right] \end{equation*}

and the density

\begin{equation*} \rho(\lambda) + \rho_h(\lambda) = \frac{1}{2\pi} = \rho(\lambda) [1 + e^{(\lambda^2 -\mu)/T}], \end{equation*}

so

\begin{equation*} \rho(\lambda) = \frac{1}{2\pi} \frac{1}{e^{(\lambda^2 - \mu)/T} + 1} \end{equation*}

which are the same formulas as for free spinless fermions.