Magnetization dynamics fingerprints of an excitonic condensate $t_{2g}^4$ magnet

The competition between spin-orbit coupling (SOC) $\lambda$ and electron-electron interaction $U$ leads to a plethora of novel states of matter, extensively studied in the context of $t_{2g}^4$ and $t_{2g}^5$ materials, such as ruthenates and iridates. Excitonic magnets—the antiferromagnetic state of bounded electron-hole pairs-are prominent examples of phenomena driven by those competing energy scales. Interestingly, recent theoretical studies predicted that excitonic magnets can be found in the ground state of SOC $t_{2g}^4$ Hubbard models. Here we present a detailed computational study of the magnetic excitations in that excitonic magnet, employing one-dimensional chains (via density matrix renormalization group) and small two-dimensional clusters (via Lanczos). Specifically, first we show that the low-energy spectrum is dominated by a dispersive (acoustic) magnonic mode, with extra features arising from the $\lambda =0$ state in the phase diagram. Second, and more importantly, we found a novel magnetic excitation forming a high-energy optical mode with the highest intensity at wave-vector $q\to 0$. In the excitonic condensation regime at large $U$, we also have found a novel high-energy $\pi$ mode composed solely of orbital excitations. These features do not appear all together in any of the neighboring states in the phase diagram and thus constitute unique fingerprints of the $t_{2g}^4$ excitonic magnet, of importance in the analysis of neutron and resonant inelastic x-ray scattering experiments.

Figure 1

To guide readers, the $\lambda$ vs $\mathrm{U}$ phase diagram is shown in panel (a), reproduced from Ref. [23]. Panels (b) and (c) show the DMRG dynamical magnetic struture factor $M_z(q,\omega )$ for the parameter points $(U/t,\lambda /t)=(40.0,0.45)$ and $(8.0,1.15)$, respectively, depicted as golden stars in (a). Frequency resolution $\mathrm{\Delta }\omega /t=0.005$ and $0.02$ is used for (b) and (c), respectively, while fixing the broadening to $\eta /t=0.05$. All above results are calculated using a 16-site three-orbital Hubbard model chain. The red arrow in (a) shows the $\lambda$ range where the large-$U$ effective model was used.

Figure 2

Spin (a) and orbital (b) static structure factors for the three-orbital Hubbard model (triangles) and the effective spin-orbital $SL$ model (blue filled curves), for various values of $\lambda$. The evolution with $\lambda$ of $S(q=0)$, $S(q=\pi )$, and $L(q=\pi )$ is shown in panel (c) using the $SL$ model. The vertical arrows depict values of $\lambda$ for which dynamical spin and orbital structure factors are presented in Fig. 4. The excitonic order parameter for the three-orbital Hubbard model is shown in panel (d). For all panels we have used $U/t=40$.

Figure 3

(a) Dynamical magnetic structure factor $M_z(q,\omega )$, obtained with DMRG, $L=48$ sites, and the $SL$ model. (b) A cartoon illustrating the allowed $\mathrm{\Delta }J=1$ excitations on the ground state of the BEC phase. Panel (c) describes visually the optical magnetic excitation using $J=1$ and $J=2$ spin-orbit excitons in momentum space. (d–f) Dynamical structure factors calculated for the specific excitation channels shown in panel (b) using $L=36$ sites chain. For all the results above, $\lambda /t=0.45$, $U/t=40$, $\mathrm{\Delta }\omega /t=0.01$, and $\eta /t=0.02$ were used.

Figure 4

Panels on the left- and right-hand side show the evolution of the dynamical spin structure factor $S_z(q,\omega )$ and dynamical orbital structure factor $L_z(q,\omega )$ of the $SL$ model, respectively, with increasing $\lambda /t$. An $L=36$ chain was used for all results at fixed $U/t=40.0$, $\mathrm{\Delta }\omega /t=0.01$, and $\eta /t=0.02$.

Figure 5

Panel (a) shows the $\lambda /t\text{−}U/t$ phase diagram calculated via the Lanczos method on an $L=6$ chain and using the $SL$ model. Colors inside the BEC phase are chosen following the optical excitation energy ${\omega }_{op}^{q=0}$. Panels (b, c) show the evolution of the long wavelength dynamical spin structure factor, $S(q=0,\omega )$, for the $P_{1\left(2\right)}$ paths shown in panel (a). The small discontinuities visible in the plot are caused by changes in the total $J$ of the ground state by one unit. Panels (d, e) depict Lanczos results for an $\sqrt{8}\times \sqrt{8}$ cluster at fixed $U/t=40$ and $J_H=U/4$. In (d), $S\left(\mathbf{q}\right)$ at the dominant momenta $\mathbf{q}=(0,0)$ and $(\pi ,\pi )$, and $L(\pi ,\pi )$ are shown for various $\lambda /t$'s. The cluster used in Lanczos analysis appears as inset. The small fluctuations present in $S(\pi ,\pi )$ and $S(0,0)$ near $\lambda =0.73$ depicts the small uncertainty in the numerical results, near the critical point. (e) Dynamical magnetic structure factor $M(\mathbf{q},\omega )$ at $\lambda /t=0.76$, with magnon and optical modes similar to the one dimensional case.