Nonlinear $O(3)$ sigma model in discrete complex analysis

We present a discrete version of the two-dimensional nonlinear $O(3)$ sigma model examined by Belavin and Polyakov. We formulate it by means of Mercat's discrete complex analysis and its elaboration by Bobenko and G\"unther. We define a weighted discrete Dirichlet energy and area on a planar quad-graph and derive an inequality between them. We write $f$ for the complex function obtained from the unit vector field of the model. The inequality is saturated if and only if the $f$ is discrete (anti-)holomorphic. By using a weight $W$ obtained from a kind of tiling of the sphere $S^2$, the weighted discrete area ${\cal A}^{W}_{\diamondsuit}(f)$ admits a geometrical interpretation, namely, ${\cal A}^{W}_{\diamondsuit}(f)=4 \pi N$ for a topological quantum number $N \in \pi_2(S^2)$. This ensures the topological stability of the solution described by the $f$, and we have the quantized energy $E^{W}_{\diamondsuit}(f)=|{\cal A}^{W}_{\diamondsuit}(f)|=4 \pi |N|$. For quad-graphs with orthogonal diagonals, we show that the discrete (anti-)holomorphic function $f$ satisfies the Euler--Lagrange equation derived from the weighted discrete Dirichlet energy. On some rhombic lattices, the discrete power functions $z^{(N)}$ give the topological quantum number $N$. Moreover, the weighted discrete Dirichlet energy, area, and Euler--Lagrange equation tend to their continuous forms as the lattice spacings tend to zero.