Physics-Constrained Self-Energy Warm Starts for Charge-Self-Consistent DFT+DMFT: Application to Iron at Core Conditions

Charge self-consistent DFT+DMFT quantitatively captures dynamical electronic correlations in real materials, but its cost precludes the large-scale thermodynamic sampling required for phase boundaries and equations of state. Here, we develop a physics-constrained machine-learning warm start for realistic DFT+DMFT: E(3)-equivariant graph neural networks predict a compact, real-valued representation of the local self-energy and Fermi level -- \{\,$\Sigma(\infty),\,\Sigma_\ell,\,E_f\,$\} -- tied to the known high-frequency and analytic structure of $\Sigma(i\omega_n)$, and used to initialize the full DFT+DMFT self-consistency cycle. Across metallic Fe, correlated FeO, and Mott-insulating NiO, the scheme yields a 2--4 times reduction in the number of DMFT iterations required to reach self-consistency. As a demanding application, we leverage this capability to generate correlated energies and forces for Fe at core pressures, train an equivariant machine-learned interatomic potential, and determine the hcp-Fe melting curve by solid--liquid coexistence simulations in the NVE ensemble in 9216-atom cells. We obtain a melting temperature of 6225 K at 330 GPa, in agreement with recent experimental constraints and consistent with the view that dynamical electronic correlations contribute to the discrepancy between DFT-based predictions and experiment.