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We obtain a unified theory of discrete minimal surfaces based on discreteholomorphic quadratic differentials via a Weierstrass representation. Ourdiscrete holomorphic quadratic differential are invariant under M\"{o}biustransformations. They can be obtained from discrete harmonic functions in thesense of the cotangent Laplacian and Schramm's orthogonal circle patterns.  We show that the corresponding discrete minimal surfaces unify the earliernotions of discrete minimal surfaces: circular minimal surfaces via theintegrable systems approach and conical minimal surfaces via the curvatureapproach. In fact they form conjugate pairs of minimal surfaces.  Furthermore, discrete holomorphic quadratic differentials obtained fromdiscrete integrable systems yield discrete minimal surfaces which are criticalpoints of the total area.

Discrete minimal surfaces: critical points of the area functional from integrable systems