Theoretical study of the dynamics of the excitonic insulator order parameter of the Falicov–Kimball model excited by ultrashort laser pulse

The non-equilibrium dynamics of the excitonic insulator order parameter in the excitonic insulator phase of the Falicov–Kimball model excited by ultrashort laser pulse is studied. The Keldysh–Schwinger functional integral method with the saddle point approximation is employed. The numerical solution for the order parameter obtained from the saddle point equation is found to exhibit oscillatory behaviors, whose frequencies are consistent with the energy gap of the excitonic insulator. The fluctuations of the order parameter around the saddle point are identified with the collective modes in non-equilibrium states, and their Green’s functions are explicitly found in terms of the saddle point solution. Also, the interaction vertices responsible for the decay of the non-equilibrium order parameter is found, and the lowest order contribution for the decay process is expressed in terms of the vertices and the Green’s functions of the non-equilibrium collective modes. In the presence of a particular electron–phonon interaction, a hybrid mode of the excitonic order parameter and the phonon can be naturally defined in functional integral and its saddle point equation can be derived. It is shown that the saddle point solution with electron–phonon interaction is consistent with the experimental data and the existing theoretical result.