In the Hartree-Fock method, one computes the energy of an interacting quantum-many body system, described by $H$, via taking a non-interacting trial ground state, $|\psi_{\mathrm{HF}}\rangle$, and minimizing the total Hartree-Fock energy, $E_{\mathrm{HF}} = \langle\psi_{\mathrm{HF}}|H|\psi_{\mathrm{HF}}\rangle$ with respect to the atomic orbitals. Doing so then yields a set of self-consistent Hartree-Fock equations which allows you to determine both the Hartree-Fock energy and precise form of the atomic orbitals.
However, I am confused how one uses this technique to do anything other than compute the total Hartree-Fock energy. For example, I was reading this paper, https://arxiv.org/abs/2012.05255, and the authors used Hartree-Fock to detect the presence of different orders in this material, WTe2. But how exactly does computing the Hartree-Fock energy allow one to explore this type of physics? How does one use this method to predict phase transitions and different ordering phases based off the interaction strength?
Is the idea that, once you've solved the Hartree-Fock equations and constructed the optimal atomic orbitals and Hartree-Fock potential, you've essentially reduced the interacting electron problem back to an independent electron problem, and, from there, you can apply the usual machinery of solid-state physics to compute whatever quantities you’re interested in?
