Tensor complex renormalization with generalized symmetry and topological bootstrap

Abstract

Recent progress in generalized symmetry and topological holography has shown that, in conformal field theory (CFT), topological data from one dimensional higher can play a key role in determining local dynamics. Based on this insight, a fixed-point (FP) tensor complex (TC) for CFT has recently been constructed. In this work, we develop a TC renormalization (TCR) algorithm adapted to this CFT-based structure, forming a renormalization-group (RG) framework with generalized symmetry. We show that the full FP tensor can emerge from the RG flow starting with only the three-point function of the primary fields. Remarkably, even when starting solely from topological data, the RG process can still reconstruct the full FP tensor--a method we call as topological bootstrap. This approach deepens the connection between the topological and dynamical aspects of CFT and suggests pathways toward a fully algebraic description of gapless quantum states, with potential extensions to higher dimensions.

Figures (36)

• Figure 1: Representing three-point and four-point correlation functions in the form of the TC as triple-line tensors. Red dots/lines denote CBCs, while black lines denote BCOs labeled by primary and descendant fields.
• Figure 2: One RG step of the Loop-TNR algorithm for the TC. Red dots denote CBCs. (a) Each rank-4 tensor is decomposed into two rank-3 tensors by performing SVD in two different directions. (b) Loop optimization is carried out on the octagons indicated by the blue arrows. (c) Each shaded region containg four rank-3 tensors is contracted, with the internal CBC summed over, to produce a new coarse-grained rank-4 tensor.
• Figure 3: Two types of SVDs are performed on each rank-4 tensor block to obtain two rank-3 tensors. For the SVD shown on the right, we fix the boundary conditions $b,d$ and group together all tensor blocks with different $a,c$ into a single large matrix, on which the SVD is performed for each pair $(b,d)$. Similarly, for the SVD shown on the left, we fix $a,c$ and group over $b,d$.
• Figure 4: Cost function defined as the distance between two periodic MPS wave functions with TC structure. In both MPS wave functions, the leftmost CBCs are identified with the rightmost CBCs.
• Figure 5: Ising CFT spectrum in the disk gauge, with conformal block initialized using only primary components, $\chi=16$.