Anyonic Chains, Topological Defects, and Conformal Field Theory

TL;DR

This work develops a universal framework linking one-dimensional anyonic chains to two-dimensional CFTs via non-local topological symmetries that descend to topological defects in the emergent theory. It establishes a precise map between input fusion data $\mathcal{C}_{\rm in}$ and a defect-augmented output theory $\mathcal{T}_{\rm out}$, showing that defects commuting with the chiral algebra protect the theory from certain relevant deformations. By analyzing defect Hilbert spaces and defect-operator spectra, the authors derive conditions under which perturbations are forbidden and prove several theorems relating defect eigenvalues to bulk field properties. The framework is illustrated with Fibonacci chains, $su(2)_k^{int}$ generalizations, and input/output constraints, highlighting a robust topological mechanism for preserving criticality and delineating how the space of 2D CFTs can be accessed via anyonic chains.

Abstract

Motivated by the three-dimensional topological field theory / two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, that can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic "topological symmetries" that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry / topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones.

Figures (17)

• Figure 1: (a) The usual graphical representation of the Heisenberg chain. Each arrow indicates a state in the local Hilbert space, $\mathcal{H}_i$. (b) The graphical representation of the ferromagnetic ground state. Every spin pair forms a spin-$1$ representation. (c) The graphical representation of the anti-ferromagnetic groundstate. Every spin pair forms a spin-$\frac{1}{2}$ representation.
• Figure 2: Fusion trees label states in the Hilbert space. For every collection of $\{x_i\}$ allowed by the fusion rules, there is a state in the Hilbert space. This chain wraps a topologically non-trivial cycle.
• Figure 3: It is more convenient to define the action of the Hamiltonian in a different basis, related by an $F-$move to the original one.
• Figure 4: We choose the projector that forces the fusion outcome of nearest neighbors to be equal to $\ell^\prime$.
• Figure 5: An $F$-move back to the original basis gives an explicit formula for the Hamiltonian.