This paper introduces R-TNR, a version of tensor-network renormalization that builds algebraic topological rules directly into the small building blocks (the local tensors) and keeps those rules as the network is coarse-grained. The aim is to use a discrete spacetime network to recover the long-distance physics of conformal field theories (CFTs). A key advance is that the method can separate contributions from different “anyon” sectors — different types of topological charge — and produce sector-resolved quantities that behave correctly under modular transformations of the torus.

What the authors did was to encode the data of a fusion category R — the mathematical rules that say how topological charges combine — into the local tensor spaces. They then run a version of Loop-TNR (Loop Tensor Network Renormalization), an algorithm that systematically coarse-grains a tensor network, while preserving those categorical constraints. For suitable choices of the category R, many different initial tensors flow under R-Loop-TNR to the same stable, gapless fixed point. The paper gives explicit examples using the Ising model, the tricritical Ising model, and the three-state Potts model.

At a high level the idea is simple. A fusion category is a short way to encode how topological excitations, or anyons, fuse and split. By building those fusion rules into each tensor and preserving them during coarse-graining, the algorithm keeps track of which part of the network belongs to which anyon sector. That makes it possible to compute sector partition functions and sector-resolved observables. The authors report that the method resolves conformal towers by anyon sector and extracts scaling dimensions, conformal spins, and magnitudes of sector-resolved structure constants with very high accuracy compared to previous tensor-network approaches.