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.