Anyonic Chains - a-Induction - CFT - Defects - Subfactors

Givenaunitaryfusioncategory,onecandefinetheHilbertspaceofaso-called anyonic spin-chain and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to1 + 1-dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, thes pin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. Oneof our main results is the construction of a novel algebra of MPOs acting on a bipartiteanyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of1+1 CFTs as constructed by Frohlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called double triangle algebra arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Bockenhauer et al.based on the braided structure of the categories and on alpha-induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.

Automated summary

This paper investigates the relationship between fusion categories and quantum spin chains, and discovers an isomorphism between the algebra of matrix-product operators of the spin chain and the defect algebra of 1+1 CFTs. It further conjectures that the central projections of the algebra are associated with the irreducible vertical (transparent) defects in the scaling limit of the model.