Essential difference between 2D and 3D from the perspective of real-space renormalization group

We point out that area laws of quantum-information concepts indicate limitations of block transformations as well-behaved real-space renormalization group (RG) maps, which in turn guides the design of better RG schemes. Mutual-information area laws imply the difficulty of Kadanoff's block-spin method in two dimensions (2D) or higher due to the growth of short-scale correlations among the spins on the boundary of a block. A leap to the tensor-network RG, in hindsight, follows the guidance of mutual information and is efficient in 2D, thanks to its mixture of quantum and classical perspectives and the saturation of entanglement entropy in 2D. In three dimensions (3D), however, entanglement grows according to the area law, posing a threat to 3D block-tensor map as an apt RG transformation. As a numerical evidence, we show that estimations of 3D Ising critical exponents fail to improve by retaining more couplings. As a guidance to proceed, a tensor-network toy model is proposed to capture the 3D entanglement-entropy area law.