Thermal Hall conductivity from semiclassical spin dynamics simulations: implementation and applications to chiral ferromagnets and Kitaev magnets

Abstract

We investigate thermal Hall transport in magnetic systems, using semiclassical spin dynamics simulations. Building on a linear response framework, we discuss the intricacies of computing the thermal Hall conductivity from real-time energy current correlations and the energy magnetization. We then apply this methodology to two models: a square-lattice chiral magnet with in-plane Dzyaloshinskii-Moriya interaction, and the antiferromagnetic Kitaev model in a field. Our results demonstrate the efficiency of semiclassical spin dynamics to study thermal Hall transport capturing quantitative effects beyond the simple intrinsic non-interacting approximation. They can serve as a benchmark for comparison with experiments in regimes where non-linearities from magnon-magnon interactions and strong thermal fluctuations play a crucial role.

Figures (10)

• Figure 1: Schematic representation of local equilibrium (i.e. magnetization) thermal Hall currents in a honeycomb lattice with a temperature gradient $\nabla T$. The lattice bonds are colored according to the local temperature. On each bond, an arrow represents the local energy current, whose width encodes the current's magnitude. In the systems we consider, we find that at local equilibrium, only chiral edge energy currents are sizable, however bulk currents remain crucial to determine the response out of equilibrium.
• Figure 2: Models studied in this paper and notational conventions. (a) Chiral ferromagnetic model studied in Sec. \ref{['sec:chiral_fm']}. The ground state is polarized out of plane and the dark arrows indicate the Dzyaloshinskii-Moriya vectors $\bm{D}_{ij}$ for the direction indicated by the bond vectors $\bm{r}_{ij}$. (b) Antiferromagnetic Kitaev model studied in Sec. \ref{['sec:kitaev_afm']}.
• Figure 3: (a-c) Current-current correlation function $C^{xy}(t)$ (in arbitrary units) as a function of time (in s), for different values of the dimensionless Gilbert damping coefficient $\alpha_{\rm G}$ and for $B/J\approx 0.17$. For each curve we indicate the extracted Kubo component $\tilde{\kappa}_0^{xy} = \kappa^{xy} \mu_{\rm B} / \gamma k_{\rm B}$ of the thermal Hall conductivity (in units of $J$). Note the different time scales on the horizontal axes. (d) Kubo component of the Hall conductivity $\tilde{\kappa}_0^{xy}$ as a function of $\alpha_{\mathrm{G}}$, showing deviations from the plateau value for $\alpha_{\mathrm{G}} \gtrsim 10^{-2}$, indicating a breakdown of the nearly-microcanonical regime required for a reliable extraction of the conductivity.
• Figure 4: Decomposition of the normalized thermal Hall conductivity $\tilde{\kappa}^{xy} = \kappa^{xy} \mu_{\rm B} / \gamma k_{\rm B}$ of the model Sec.\ref{['sec:phys-mod-A']} as a function of the magnetic field $B/J$. Shown are the Kubo contribution $\tilde{\kappa}^{xy}_{\rm Kubo}$ (blue), the energy magnetization contribution $\tilde{\kappa}^{xy}_{\rm EM}$ (red), and the total quantity $\tilde{\kappa}^{xy}_{\rm tot} = \tilde{\kappa}^{xy}_{\rm Kubo}+\tilde{\kappa}^{xy}_{\rm EM}$ (purple). Note that the point at $B/J=0$ was computed, not postulated on physical grounds.
• Figure 5: Cross-correlation analysis of the energy currents in the Kitaev model at three representative temperatures: (a–c) $k_{\rm B} T/J \approx 0.026$, (d–f) $k_{\rm B} T/J \approx 0.13$, and (g–i) $k_{\rm B} T/J \approx 0.35$. Left column: raw transverse current correlators $C^{xy}(t)$ and $C^{yx}(t)$. Middle column: fitted correlators $C^{xy}_{\text{fit}}(t)$ using damped sine functions, with $R^2$ values indicating fit quality. Right column: Fourier transforms $\tilde{C}^{xy}(\omega)$, where dominant frequencies are marked by green circles. With increasing temperature, the oscillatory signal is damped more rapidly and spectral peaks broaden, reflecting enhanced magnon loss of coherence due to thermal fluctuations.