The Bethe Ansatz

The ground state is characterized by a fully-packed distribution of quantum numbers, symmetric with respect to \(\lambda = 0\). The limits \(\lambda_{\pm}\) are thus equal to a magnetization-dependent constant \(\lambda_F\). For \(|\lambda| < \lambda_F\), the hole density identically vanishes, while the particle density vanishes for \(|\lambda| > \lambda_F\). The continuum Bethe equation l.bec therefore becomes an equation for the single root distribution \(\rho_g(\lambda)\),

\begin{equation} \rho(\lambda) + \int_{-\lambda_F}^{\lambda_F} d\lambda' a_2 (\lambda - \lambda') \rho(\lambda') = a_1 (\lambda), \tag{h.rge}\label{h.rge} \end{equation}

which is the \(XXX\) version of the Lieb equation l.le.

For nonzero magnetization, the boundaries \(\lambda_F\) are finite. However, as the magnetic field goes to zero, \(h \rightarrow 0^+\), we have that \(\lambda_F \rightarrow \infty\) as can be seen from the discussion on limiting quantum numbers in c_h_e_rr. In this zero field case (and this case only), h.rge can be solved using the convolution theorem. We define Fourier transforms as

\begin{equation} \rho(\lambda) = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} e^{-i\omega \lambda} \rho(\omega), \hspace{1cm} \rho(\omega) = \int_{-\infty}^{\infty} d\lambda e^{i\omega \lambda} \rho(\lambda). \tag{h.f}\label{h.f} \end{equation}

The kernels \(a_n (\lambda)\) then have the Fourier transforms

\begin{equation} a_n (\omega) = \int_{-\infty}^{\infty} \frac{d\lambda}{2\pi} e^{i\omega \lambda} \frac{n}{\lambda^2 + n^2/4} = e^{-|\omega|n/2}. \tag{h.anf}\label{h.anf} \end{equation}

The convolution theorem used in h.rge thus allows to immediately solve for the zero-field ground state root distribution \(\rho_g\) \[ \rho_g (\omega) = \frac{a_1(\omega)}{1 + a_2(\omega)} = \frac{1}{2\cosh \omega/2}, \]

Derivation

\begin{equation*} \rho (\lambda) = \int_{-\infty}^\infty \frac{d\omega}{2\pi} e^{-i\omega \lambda} \rho(\omega) \end{equation*}

Poles at \(\omega_n = 2\pi i (n + 1/2)\), \(n = 0, 1, ...\) in UHP (close there for \(\lambda < 0\)). Residues:

\begin{equation*} \frac{d}{d\omega} \cosh \frac{\omega}{2} = \frac{1}{2} \sinh \omega \end{equation*}

whose inverse Fourier transform gives the distribution in rapidity space:

\begin{equation} \rho_g(\lambda) = \frac{1}{2\cosh \pi \lambda}. \tag{h.rg}\label{h.rg} \end{equation}

The ground-state energy 1938.Hulthen.AMAF.26A and momentum can thus be computed exactly by substituting h.rg in h.ep,

\begin{equation} E_g = -NJ\ln 2, \hspace{1cm} P_g = \pi \frac{N}{2} ~~\mbox{mod}~ 2\pi. \tag{h.epg}\label{h.epg} \end{equation}