Fig. 7: Main procedures of the numerical tensor-network methods.

a In the variational uniform matrix-product-state (VUMPS) algorithm, the environment (light blue background) of an infinite tensor network is approximated by the infinite matrix-product-state (iMPS), which consists of local tensors (light blue squares) with a bond dimension D. The iMPS is optimized by maximizing the tensor network with the variational method. b The eigen-equation \({\mathbb{T}}\left|j\right.> ={\lambda }_{j}\left|j\right.> \) of the column-to-column transfer operator \({\mathbb{T}}\) is approximated by the local tensors, where the green box is the eigenvector \(\left|j\right.> \) with the (j + 1)th largest eigenvalue λ_j. c In the corner-transfer-matrix renormalization group (CTMRG) algorithm, the environment (light blue background) of an infinite tensor network is approximated by the edge fixed point tensors (light blue squares) and corner fixed point tensors (light blue snip single corner squares) with a bond dimension D. d The eigen-equation \({{\mathbb{T}}}^{2}\left|j\right.> ={\lambda }_{j}\left|j\right.> \) of the column-to-column transfer operator \({{\mathbb{T}}}^{2}\) is approximated by the edge tensors and local double tensors. e The tensor network representation of the reduced density matrix ρ in the CTMRG algorithm, where Z is the normalization factor. The red dashed line emphasizes that the up/down cut bonds are regarded as the row/column of the reduced density matrix.