Quantum Fluctuations and Excitations in Two-Dimensional Wigner Crystals

It is widely believed that there is no direct transition between a Fermi liquid and a Wigner crystal as a function of density in a two-dimensional electron gas. However, the intermediate phase between them remains elusive, and recent reports of Wigner crystals in two-dimensional materials such as rhombohedral graphene and transition metal dichalcogenides provide a new opportunity to study this outstanding problem in condensed matter physics. In this work, we develop a microscopic theory for these intermediate phases, employing constrained Hartree-Fock theory with a quasi-boson approximation to evaluate the correlation energies. We focus on intermediate values of $r_s$ ranging from 4 to 10, where we observe that the Hartree-Fock energy of the Wigner crystal is almost an order of magnitude lower than its correlation energy. However, partial melting of the Wigner crystal leads to a decrease in correlation energy, which offsets the increase in Hartree-Fock energy, resulting in a lower overall energy in this intermediate state. This intermediate phase exhibits dual characteristics: its ground state breaks translational symmetry and forms a periodic lattice akin to a crystal, yet its excitation spectrum resembles that of a liquid, characterized by a plasmon mode and a particle-hole continuum. Notably, it lacks a transverse sound mode, a definitive trait of solids. Our findings suggest that this phase represents a metallic charge density wave, embodying properties of both crystalline and liquid states.