Finite-temperature dynamic structure factor of the spin-1 XXZ chain with single-ion anisotropy

Improving matrix-product state techniques based on the purification of the density matrix, we are able to accurately calculate the finite-temperature dynamic response of the infinite spin-1 $\text{XXZ}$ chain with single-ion anisotropy in the Haldane, large-$D$, and antiferromagnetic phases. Distinct thermally activated scattering processes make a significant contribution to the spectral weight in all cases. In the Haldane phase, intraband magnon scattering is prominent, and the on-site anisotropy causes the magnon to split into singlet and doublet branches. In the large-$D$ phase response, the intraband signal is separated from an exciton-antiexciton continuum. In the antiferromagnetic phase, holons are the lowest-lying excitations, with a gap that closes at the transition to the Haldane state. At finite temperatures, scattering between domain-wall excitations becomes especially important and strongly enhances the spectral weight for momentum transfer $\pi$.

Figure 1

Graphical representation of Eq. (4). The blue symbols represent the tensors in the finite window that distinguishes between $|{\psi }_A\rangle$ and $|{\psi }_B\rangle$ while the gray tensors represent the iMPS unit cell.

Figure 2

Finite-temperature dynamic structure factor in the Haldane phase ($J_z=J$) in units of $J^{-1}$. The temperature $T/J=0.4$; the on-site anisotropy $D/J=-0.04$ in (a), (b) and $D/J=0.2$ in (c), (d). All spectral functions are convoluted with a Gaussian of width $0.1J$.

Figure 3

Dynamic spin structure factor in the large-$D$ phase for $D/J=2$. Again, $J_z/J=1$. (a), (b) Zero-temperature data obtained by pure-state MPS techniques are contrasted with the results for (c), (d) $T/J=0.4$ and (e), (f) 1.

Figure 4

Dynamic spin structure factor in the antiferromagnetic phase. Model parameters are $D/J=0.2$ and $J_z/J=2$. We use $T/J=0$, 0.4, and 1 as in Fig. 3.