Critical density for Wigner crystallization of the clean 2D electron gas
In plain words
Computer simulations predict the electron sheet crystallizes once repulsion exceeds kinetic energy by a factor of roughly 30 to 40, but the exact threshold and whether the change is abrupt are not pinned down.
Precise statement
For the 2D homogeneous electron gas with $1/r$ repulsion and a uniform neutralizing background at $T = 0$, determine the critical r_s_c for the liquid to triangular Wigner crystal transition, whether it is first order, and the spin state of each phase. Diffusion Monte Carlo gave r_s_c ~ 37 (Tanatar and Ceperley 1989), $31 \pm 1$ (Drummond and Needs 2009) and $35 \pm 1$ with no stable ferromagnetic fluid (Azadi, Drummond and Vinko 2024, Physical Review B 110, 245145); an answer is r_s_c with controlled fixed-node and finite-size errors.
What would settle it
A calculation with controlled nodal error (for example neural-network variational wavefunctions benchmarked against released-node or auxiliary-field results) at sizes that allow a thermodynamic extrapolation.
Status in the literature
Unverified note
Fixed-node diffusion Monte Carlo estimates differ by more than their quoted errors ($31 \pm 1$ in 2009, $35 \pm 1$ in 2024), so nodal and finite-size errors are not under control.