The Bethe Ansatz

Let us, for convenience, temporarily consider a fully anisotropic chain with Hamiltonian

\begin{equation*} H = \sum_j \left[ J_x S^x_j S^x_{j+1} + J_y S^y_j S^y_{j+1} + J_z S^z_j S^z_{j+1} - h S^z \right] \end{equation*}

We will consider the dynamical spin-spin correlations (with \(a, b = x, y, z\))

\begin{equation*} {\boldsymbol S}^{ab} (k, \omega) = \frac{1}{N} \sum_{j, j'} e^{-i k (j - j')} \int_{-\infty}^\infty dt e^{i \omega (t-t')} \frac{1}{2} \langle \left[ S^a_j (t), S^b_{j'} (t') \right] \rangle \end{equation*}

The f-sumrule fsr here takes the form

\begin{equation*} \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{ab} (k, \omega) = \sum_{j, j'} e^{-i k (j - j')} \frac{-1}{2N} \langle \left[ \left[H, S^a_j \right], S^b_{j'} \right] \rangle \end{equation*}

Defining the first frequency moments

\begin{equation*} S^{ab}_1 (k) \equiv \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{ab} (k, \omega) \end{equation*}

and the exchange operators

\begin{equation*} X^a \equiv \sum_j S^a_j S^a_{j+1} \end{equation*}

we obtain

\begin{align*} S^{xx}_1 (k) &= \frac{-1}{N} \left( (J_y - J_z \cos k) \langle X^y \rangle + (J_z - J_y \cos k) \langle X^z \rangle - \frac{h}{2} S^z_{tot} \right), \\ S^{yy}_1 (k) &= \frac{-1}{N} \left( (J_z - J_x \cos k) \langle X^z \rangle + (J_x - J_z \cos k) \langle X^x \rangle - \frac{h}{2} S^z_{tot} \right), \\ S^{zz}_1 (k) &= \frac{-1}{N} \left( (J_x - J_y \cos k) \langle X^x \rangle + (J_y - J_x \cos k) \langle X^y \rangle \right). \end{align*}

Specializing to the \(XXZ\) case with \(J_x = J_y = J\) and \(J_z = J\Delta\), we get

\begin{align*} S^{xx}_1 (k) &= S^{yy}_1 (k) = \frac{-1}{N} \left( (1 - \Delta \cos k) J \langle X^x \rangle + (\Delta - \cos k) J \langle X^z \rangle - \frac{h}{2} S^z_{tot} \right), \\ S^{zz}_1 (k) &= \frac{-2}{N} \sin^2 \frac{k}{2} J \langle X^x \rangle \end{align*}

In the special case of ground-state correlations, the expectation values of the exchange operators can be obtained from the anisotropy dependence of the energy (using the explicit form xxz.h for the Hamiltonian):

\begin{align*} \langle X^x \rangle &= \langle X^y \rangle = \frac{1}{2J} \left( E_0 - \Delta \frac{\partial E_0}{\partial \Delta} \right), \\ \langle X^z \rangle &= \frac{1}{J} \left(\frac{\partial E_0}{\partial \Delta} + \frac{N}{4} \right). \end{align*}