Quantum crystallization in a Sombrero Metal

Recent experiments on lightly-doped multilayer graphene, where the conduction band assumes a "Mexican Hat" shape in the presence of a strong perpendicular displacement field, motivated us to propose a model 2D Hamiltonian called the sombrero metal. This model consists of isotropic Coulomb interactions and a kinetic energy dispersion ε_k = -αk²+βk⁴+α²/4β (α,β > 0) that takes the shape of a sombrero. We study the process of quantum crystallization and the intermediate phases of the sombrero metal at zero-temperature using the Hartree-Fock approximation. We find that quantum crystallization occurs sequentially as we decrease the electron density or increase the interaction strength. Firstly, it develops local orientation by forming a nematic Fermi liquid; then, due to anisotropy in dispersion, it forms an electron liquid-crystal where translational order is broken only in one direction and eventually becomes a Wigner crystal where translational symmetry is fully broken. We find that in the electron liquid-crystal phase, the electron density exhibits a periodicity nearly half that of the unit cell to significantly lower its Hartree energy. Our findings provide a foundational understanding of the crystallization process and phase transitions in the Sombrero Metal.