Dynamical Spin Structure Factor for the Anisotropic Spin-$1/2$ Heisenberg Chain

The longitudinal spin structure factor for the $XXZ$-chain at small wave vector $q$ is obtained using Bethe ansatz, field theory methods, and the density matrix renormalization group. It consists of a peak with a peculiar, non-Lorentzian shape and a high-frequency tail. We show that the width of the peak is proportional to $q^2$ for finite magnetic field compared to $q^3$ for a zero field. For the tail we derive an analytic formula without any adjustable parameters and demonstrate that the integrability of the model directly affects the line shape.

Figure 1

Form factors squared for the $2p$ excitations and different $N$ at $\Delta =0.25$, $s=-0.1$, and $q=2\pi /25$. The inset shows the scaling of the width ${\gamma }_q$. The points are obtained by an extrapolation $N\to \infty$ of the numerical data. The solid line is the prediction (4), ${\gamma }_q=0.356q^2$.

Figure 2

Form factors (dots) obtained by BA compared to formula (8) (line) for $\Delta =0.75$, $s=-0.1$, $N=600$, and $q=2\pi /50$. The form factors for the exact eigenstates at $\omega \approx 2\pi vl/N$ are added and represented as a single point. The inset shows each form factor separately. The number of states at each level agrees with a simple counting based on multiple particle-hole excitations created around the Fermi points.

Figure 3

Sum of form factors at $\omega \approx 2\pi v(l+4K)/N$ (dots) and at $\omega \approx 2\pi vl/N$ (squares) obtained by BA for $\Delta =0.25$, $s=0$, $N=600$, and $q=2\pi /50$. The solid line corresponds to (11), the dotted line to (11) with finite size corrections included [22], and the dashed line to (10).