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We develop coarse-graining tensor renormalization group algorithms to computephysical properties of two-dimensional lattice models on finite periodiclattices. Two different coarse-graining strategies, one based on the tensorrenormalization group and the other based on the higher-order tensorrenormalization group, are introduced. In order to optimize the tensor-networkmodel globally, a sweeping scheme is proposed to account for therenormalization effect from the environment tensors under the framework ofsecond renormalization group. We demonstrate the algorithms by the classicalIsing model on the square lattice and the Kitaev model on the honeycomblattice, and show that the finite-size algorithms achieve substantially moreaccurate results than the corresponding infinite-size ones.

Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices