  We develop coarse-graining tensor renormalization group algorithms to compute
physical properties of two-dimensional lattice models on finite periodic
lattices. Two different coarse-graining strategies, one based on the tensor
renormalization group and the other based on the higher-order tensor
renormalization group, are introduced. In order to optimize the tensor-network
model globally, a sweeping scheme is proposed to account for the
renormalization effect from the environment tensors under the framework of
second renormalization group. We demonstrate the algorithms by the classical
Ising model on the square lattice and the Kitaev model on the honeycomb
lattice, and show that the finite-size algorithms achieve substantially more
accurate results than the corresponding infinite-size ones.
