Inspired by observations of generalized Wigner crystal states in moiré materials, we systematically investigate charge orderings of electrons on a square lattice with long-range interactions $V\left(r\right)=U{r}^{-\alpha }$. Using simulated annealing Monte Carlo simulations in the classical limit without kinetic energy, we obtain the generalized Wigner crystal states for almost all rational fillings $\nu =\frac{p}{q}$ ($q\le 20$) and different $\alpha$. For large $\alpha$, we find the ground states can be obtained by the so-called Farey construction method by compositing different fundamental stripes with $\frac{1}{n}$-filling (except $\frac{1}{7}, \frac{1}{13}, \frac{1}{17}, \frac{1}{19}, ...$), largely consistent with the greedy lattice algorithm for fast-decay interactions. For small $\alpha$, we further identify the $\frac{3}{8}$-filling as a new fundamental state such that the states for $\frac{1}{3}<\nu <\frac{3}{8}$ can be constructed by the Farey construction method. For $\frac{3}{8}<\nu <\frac{1}{2}$, a dual lattice subtraction applies: removing $(1-2\nu )$-filling states from the half-filled checkerboard state. Further incorporating kinetic energy via self-consistent mean-field calculations shows that most of these Wigner crystal states are stable against quantum fluctuations until melting. Our work establishes a complete classical-to-quantum framework to understand the generalized Wigner crystals on the square lattice, which can be generalized to other underlying lattices.

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• Long-range interactions
• Wigner crystal
• Mean-field & cluster methods
• Monte Carlo methods