Abstract
Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.
Highlights
Instead of studying anyon condensation in concrete models, we take an abstract approach
The main goal of this paper is to provide a detailed explanation of how each ingredient of the complete mathematical structures emerge naturally from concrete and natural physical requirements
We assume that an anyon condensation happens in a 2d region inside of a 2d phase C as depicted in Fig. 1, and the anyons in the condensed phase form another modular tensor category (MTC) D, which is equipped with a tensor product ⊗D, a tensor unit 1D, an associator αDL,M,N : (L ⊗D M) ⊗D N → − L ⊗D (M ⊗D N), a braiding cDM,N : M ⊗D N → − N ⊗D M and a twist θDM : M → − M for all L, M, N ∈ D
Summary
Anyon condensation is an important subject to study in the field of topological orders. The main goal of this paper is to provide a detailed explanation of how each ingredient of the complete mathematical structures emerge naturally from concrete and natural physical requirements We hope that this analysis can convince physicists that the tensor-categorical language, abstract, is a powerful and necessary language for anyon condensation. The layout of the paper is: in Section 2, we carry out this bootstrap analysis and derive our main results; in Section 3, we discuss how to use physical macroscopic data to determine the condensation; in Section 4, we provide examples; in Section 5, we discuss the Witt equivalence between 2d topological orders; Appendix contains the definitions of all tensor-categorical notions appeared in this work.
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