Finite-temperature dynamics and thermal intraband magnon scattering in Haldane spin-one chains

The antiferromagnetic spin-one chain is considerably one of the most fundamental quantum many-body systems, with symmetry-protected topological order in the ground state. Here, we present results for its dynamical spin structure factor at finite temperatures, based on a combination of exact numerical diagonalization, matrix-product-state calculations, and quantum Monte Carlo simulations. Open finite chains exhibit a subgap band in the thermal spectral functions, indicative of localized edge states. Moreover, we observe the thermal activation of a distinct low-energy continuum contribution to the spin spectral function with an enhanced spectral weight at low momenta and its upper threshold. This emerging thermal spectral feature of the Haldane spin-one chain is shown to result from intraband magnon scattering due to the thermal population of the single-magnon branch, which features a large bandwidth-to-gap ratio. These findings are discussed with respect to possible future studies on spin-one chain compounds based on inelastic neutron scattering.

Figure 1

Sketch of the low-energy excitations of the Haldane spin-one chain. The black line shows the gapped single-magnon dispersion, and the upper shaded regions denote the two- and three-magnon continua. The lower shaded region encloses the intraband-magnon-scattering contribution to the dynamical spin structure factor that emerges from the thermal population of the single-magnon branch. For open chains, an additional subgap edge-state mode extends from $q=\pi$ toward smaller momenta, indicated by the dashed line.

Figure 2

Dynamical spin structure factor $S(q,\omega )$ for the Haldane spin-one chain from DMRG with OBC (left panels), and QMC with PBC (right panels) for various temperatures $T$ (a,b) $T/J=1/24$, (c,d) $T/J=0.2$, (e,f) $T/J=0.4$, and (g,h) $T/J=1.0$.

Figure 3

Comparison of the DMRG spectral function $S(q,\omega )$ (left panel) for $L=32$ (OBC) with $S^{IB}(q,\omega )$ calculated for the IBMS model (right panel) at $T/J=0.3$. A Gaussian broadening with ${\sigma }_{\omega }=0.1J$, similar to the DMRG spectra, was applied to the IBMS model spectral function.

Figure 4

Comparison of the ED spectral functions $S(q,\omega )$ with the IBMS model for $S^{IB}(q,\omega )$ at $T/J=0.3$. The main panel shows results for $q=\pi /2$, and the inset those for $q=0.4\pi$. For comparison, ED results for $T=0$ are also included. A Gaussian broadening with ${\sigma }_{\omega }=0.05J$ was applied to all spectral functions in this figure.