Figure 1
Illustration of the application of the protocols to the Hubbard dimer. This results in four possible approximations to obtain out-of-equilibrium dynamics and related thermodynamic quantities for the Hubbard dimer. These approximations can be used depending on the desired precision and the regime of interest. The worse-case scenario is to use a standard zero-order non-interacting Hamiltonian to describe the system and the dynamics. For more precision, and depending on the regime of interest, we can use either the Kohn-Sham Hamiltonian to describe system and dynamics, or use the standard zero-order non-interacting Hamiltonian for the dynamics but refine the approximation of the initial and final energy spectra (here done up to first order perturbation, FOP): the implementations of both options have similar degrees of difficulty. Finally, we can combine the Kohn-Sham Hamiltonian for the dynamics and the FOP (or higher order precision) for the initial and final spectra.
