Theoretical study of third harmonic generation in excitonic insulators

Recent progress in experimental techniques has made it possible to observe the dynamical properties of quantum many-body systems with laser light. Thus, the investigation of nonequilibrium states far away from equilibrium has become one of the central issues of condensed matter physics. In addition, nonlinear optical responses in quantum materials, e.g., high harmonic generation in strongly correlated electron systems, have actively been studied in recent years from both basic and applied aspects.
Motivated by such developments in the field, we study the nonlinear optical response of excitonic insulator (EI) theoretically. Specifically, using the extended Falicov-Kimball model (spinless two-band model), we investigate third harmonic generation (THG) in the EI. We use the time-dependent mean-field theory to incorporate the dynamical behavior of the order parameter of the EI and carry out analytical calculations using the perturbation expansion in terms of the nonequilibrium Green's function to investigate the behavior of the photocurrent.
In the EI with the inversion symmetry, the collective motion of the order parameter is activated at second order of the external electric field and its effects appear in the THG spectrum. We find three peaks in the THG susceptibility at energies $\hbar\Omega=\Delta_g/3,\Delta_g/2,$ and $\Delta_g$, where $\Delta_g$ is the bandgap in equilibrium. While the THG response at $\Delta_g/3$ is originated from three-photon excitation of the independent particle, the latter two peaks are caused by the collective order-parameter motion activated at second order. The resonant peaks caused by the collective modes are prominent in the BCS (weak-coupling) regime but they become less significant in the BEC (strong-coupling) regime.
We have demonstrated that the resonant peaks caused by the collective order-parameter motion is observable in the temperature dependence of the THG intensity. Our study suggests that the THG measurement is useful for detecting the collective nature in the EIs.