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.

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