Shape of Wigner Crystals and Hole Self-Doping in a Mexican-Hat Dispersion

Abstract

We study Wigner crystals (WCs) induced by a strong Coulomb interaction from the ring-like Fermi surface of a Mexican-hat dispersion $\epsilon_k= c_2k^2+c_4k^4$. We design orbital shape in order to minimize the energy of the WC, and find that a low ground-state energy requires an orbital shape with a depletion of electrons near $k=0$. To capture the Coulomb-induced correlations, we include a Jastrow factor as well as a factor describing the correlation between electrons and doped vacancies. Using variational Monte Carlo calculations, we calibrate the effective band parameters $c_2$ and $c_4$ to reproduce the two transitions observed experimentally as the electron density is lowered: from a spin-valley-polarized Fermi liquid with a disk-like Fermi surface, to one with a ring-like Fermi surface, and finally to a WC. We find that, even with an optimized orbital shape that depletes electrons near $k=0$, a WC with hole self-doping near $k=0$ can still be energetically favorable near the WC transition, provided that the electron-vacancy correlation is included. We also estimate the dispersion of the doped hole.