## Selective amplification of collective excitations in excitonic insulator

We have established a microscopic model to understand the dynamical behavior of excitonic insulator Ta2NiSe5 and a reversal of excitonic was successfully predicted and later demonstrated by coherent phonon spectroscopy. We adopted this model and tuned the electronic damping such that an coherent oscillation of the amplitude and phase of the order parameter emerges, which correspond to the celebrated Higgs and Goldstone mode. Due to the electron phonon coupling, Goldstone mode gains mass. By tuning the pump wavelength, a strong amplification of both the Higgs and the Goldstone modes was identified when pumping at half of the Higgs mode frequency, accompanied by a concomitant drop of the fluence where the order reverses. This matches the second-order Raman-active symmetry of the Higgs mode. The simultaneous amplification of the Goldstone mode arises from the possible decay from Higgs modes to two Goldstones modes at opposite momenta with energies identical to half of the Higgs frequency.

## A comprehensive model considering the interplay of different degrees of freedom in transition metal oxides

The low-energy physics governing the electronic properties of complex transition metal oxides are very complex. Entangled charge, orbital, spin, and lattice degrees of freedom in the *d* shell all play a role with nonnegligible interaction, engendering a complex phase diagram with a plethora of exotic orders. We consider a microscopic model counting for (1) the Kanamori-type electronic interaction parameterized by onsite (*U*) and interorbital (*U’*) electro-electron interaction and Hund’s coupling (*JH*), (2) Jahn-Teller coupling (*g*) which captures the orbital-lattice interaction, (3) spin-orbital coupling (λ), (4) lattice geometric splitting (Δ) and elastic energy (*B*). Surprisingly, the eigenvalue of the Hamiltonian can be calculated analytically, and the potential energy surface can be deduced with respect to the lattice order parameters. By tuning the aforementioned various parameters, we are able to explore the diverse ground states and phase transitions. By further considering either an impulsive or displacive excitation, the dynamics of the system can be fully simulated.

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