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We explore the Potts model on the generalized decorated square lattice, withboth nearest (J1) and next-neighbor (J2) interactions. Using the tensorrenormalization-group method augmented by higher-order singular valuedecompositions, we calculate the spontaneous magnetization of the Potts modelwith q = 2, 3, and 4. The results for q = 2 allow us to benchmark our numericsusing the exact solution. For q = 3, we find a highly degenerate ground statewith partial order on a single sublattice, but with vanishing entropy per site,and we obtain the phase diagram as a function of the ratio J2=J1. There is nofinite-temperature transition for the q = 4 case when J1 = J2, but the magneticsusceptibility diverges as the temperature goes to zero, showing that the modelis critical at T = 0.

Partial order in Potts models on the generalized decorated square lattice