Global well-posedness in a Hartree-Fock model for graphene
William Borrelli, Umberto Morellini
TL;DR
The paper establishes global well-posedness for the time-dependent Bogoliubov-Dirac-Fock mean-field model of graphene, treating electrons as 2D massless Dirac fermions with instantaneous Coulomb interactions under external time-dependent potentials. It develops a UV-cutoff regularized framework, derives the Hartree-Fock/BDF energy, and proves a self-consistent, unitary evolution equation for the density matrix with a conserved, bounded energy functional. Under a velocity/coupling threshold $v_F \ge v_c$ (equivalently $0\le \alpha \le \alpha_c$), the authors prove a unique global solution in the appropriate operator space, with the solution remaining a projection and preserving the no-photon HF structure. This work provides a rigorous foundation for non-perturbative, time-dependent graphene dynamics and opens avenues toward scattering theory in 2D massless Dirac systems.
Abstract
Graphene is a monolayer graphitic film in which electrons behave like two-dimensional Dirac fermions without mass. Its study has attracted a wide interest in the domain of condensed matter physics. In particular, it represents an ideal system to test the comprehension of 2D massless relativistic particles in a laboratory, the Fermi velocity being 300 times smaller than the speed of light. In this work, we present a global well-posedness result for graphene in the Hartree- Fock approximation. The model allows to describe the time evolution of graphene in the presence of external time-dependent electric potentials, such as those induced by local charge defects in the monolayer of carbon atoms. Our approach is based on a well established non-perturbative framework originating from the study of three-dimensional quantum electrodynamics.