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.
