Finite-temperature spin dynamics and phase transitions in spin-orbital models

We study finite temperature properties of a generic spin-orbital model relevant to transition metal compounds, having coupled quantum Heisenberg-spin and Ising-orbital degrees of freedom. The model system undergoes a phase transition, consistent with that of a two-dimensional Ising model, to an orbitally ordered state at a temperature set by short-range magnetic order. At low temperatures the orbital degrees of freedom freeze out and the model maps onto a quantum Heisenberg model. The onset of orbital excitations causes a rapid scrambling of the spin spectral weight away from coherent spin waves, which leads to a sharp increase in uniform magnetic susceptibility just below the phase transition, reminiscent of the observed behavior in the Fe-pnictide materials.

Figure 1

(Color online) Schematic representation of various Hamiltonians: (a) the $J_1\text{−}{J}_2$, (b) the $J_{1a}\text{−}{J}_{1b}\text{−}{J}_2$, and (c) the spin-orbital models. At zero temperature the additional orbital degrees of freedom freeze out and the spin-orbital model reduces to the $J_{1a}\text{−}{J}_{1b}\text{−}{J}_2$ model. (d) The magnon dispersion ${\omega }_{\mathbf{k}}$ (in units of $J_1/J_{1a}$) calculated from linear spin-wave theory. Here $J_{1b}=-0.1J_{1a}$, $J_2=0.4J_{1a}$ in the $J_{1a}\text{−}{J}_{1b}\text{−}{J}_2$ model and $J_2=J_1$ in the $J_1\text{−}{J}_2$ model. The spin-wave energy forms a maximum at $\left(\pi ,\pi \right)$ in the $J_{1a}\text{−}{J}_{1b}\text{−}{J}_2$ model, which is a minimum in the $J_1\text{−}{J}_2$ model.

Figure 2

(Color online) Plots for (a) the specific heat $C_V$ and (b) the uniform magnetic susceptibility ${\chi }_m$ for the three models considered. The temperature $T$ is expressed in terms of $J_1$ (or $J_{1a}$). Compared to the other two spin-only models, there is a sharp peak in $C_V$ and a larger slope in ${\chi }_m$ in the spin-orbital model.

Figure 3

(Color online) (a) The orbital-Ising susceptibility ${\chi }_I$ and its derivative with respect to $T$ in the inset. (b) The total system $S_{\text{system}}$ and orbital $S_{\text{orb}}$ entropy in the spin-orbital model. In the vicinity of the phase transition $\sim R\text{ln}2$ entropy is lost.

Figure 4

(Color online) (a) $T=0S^{zz}\left(\mathbf{q},\omega \right)$ for the spin-orbital/$J_{1a}\text{−}{J}_{1b}\text{−}{J}_2$ models. (b) and (c) Finite temperature $S^{zz}\left(\mathbf{q},\omega \right)$ at $\left(\pi ,\pi \right)$ obtained from ED for the spin-orbital and the $J_{1a}\text{−}{J}_{1b}\text{−}{J}_2$ models, respectively. The temperature goes from $T=0.1J_1$ (the blue curve) to $T=1.0J_1$ (the red curve), with a temperature increment between each curve $\sim 0.08J_1$. A highly incoherent spin dynamics is observed in the spin-orbital model. (d) Finite temperature $\omega$ integrated form factor $S^{zz}\left(\mathbf{q}\right)$ at $\left(\pi ,0\right)$.