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

Abstract

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.

Figures (5)

• Figure 1: Block-tensor transformation and its spin picture with $b=2$. The "Block" step in this Figure is the Step (i), while the "Tensor contraction" step corresponds to Step (ii) and (iii). The triangular tensor in (a) is the unitary transformation that identifies important states. In numerical calculations, truncations of the state are according to the eigenvalue spectrum of the density matrix of the larger square block after the "Block" step. When truncations happen, the triangular tensors are known as isometric tensors tnr2015.
• Figure 2: (Upper left) A tensor $A^{\text{EDL}}(e)$ with an EDL structure is the tensor product of 12 copies of edge-matrix $e$ located at 12 edges of a cube block; every 4 matrix-indices on the same face are grouped together to be a thicker leg of the 6-leg $A^{\text{EDL}}(e)$ . (Lower right) A block-tensor RG with rescaling factor $b=2$. A total of $12 \times 8 = 96$ copies of edge-matrix $e$ form 3 types: (i) Around each of the 6 axes inside the block, 4 copies of $e$ form a loop and becomes a number, so there are $4 \times 6$ copies; (ii) On each of 6 faces, copies of $e$ form 4 entanglement pairs inside the face, so there are $2\times 4 \times 6$ copies; (iii) The remaining $2\times 12$ copies sit on the 12 edges of the larger block, each edge having 2 copies side by side. Only type-(iii) copies survive the exact block-tensor RG; the other two types become an overall multiplication factor. The coarser tensor has an EDL structure built from $e \otimes e$: $A' = A^{\text{EDL}}(e \otimes e)$.
• Figure 3: The rapid growth of the RG error with respect to the RG step. The temperature is set to be the estimated critical temperature obtained by studying the RG flows, as is explained in Refs. lyu-kawashima2021Lyu:Kawashima:2024. The RG error grows to more than $10\%$ just after the first RG step. Near the critical fixed point, which is around RG steps $3 \leq n \leq 5$, the RG error increases from $10\%$ to $30\%$ when the bond dimension increases from $4$ to $10$. At the largest bond dimension we reach, $\chi=22$ (not shown in the Figure), the RG truncation error near the critical fixed point has grown to about $38\%$.
• Figure 4: Drifting of the estimates of scaling dimensions that happens at most bond dimensions $\chi \leq 22$. The example shown here is at $\chi = 8$. The spin and energy density operators are denoted as $\sigma, \epsilon$, respectively, while $\partial_i \sigma, \partial_{i} \partial_j \sigma$ and $\partial_i \epsilon$ are their descendant operators. The energy-momentum operator is denoted as $T_{ij}$. The estimate of $\epsilon$ drifts with respect to the RG step.
• Figure 5: Estimation of 3D Ising scaling dimensions $\Delta$ using high-order tensor renormalization group hotrg2012hotrg-paral2022 and its linearization lyu-kawashima2021. Results are organized as per spin-flip $\mathbb{Z}_2$ symmetry. Dashed lines are results from conformal bootstrap bootstrap-3dising. Most approximation errors of $\sigma$ and $\epsilon$ operators lie between $10\%$ and $20\%$ and fail to decrease with more couplings retained.