Anyon condensation and tensor categories

By Liang Kong. 2013

290 citations recorded by the source corpus.

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.

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Citations in the corpus

Fusion 2-categories and a state-sum invariant for 4-manifolds (2018, background): “ontext of a modular tensor category representing excitations of a 2-dimensional topological phase of matter, the splitting of a commutative separable algebra can be thought of as ‘anyon condensation’ [Kon14]. 4Finite semisimple 1-categories are a categorication of nite-dimensional vector spaces, and”
Higher Gauging and Non-invertible Condensation Defects (2022, background): “It is known that the topological boundary conditions of a TQFT are in one-to-one correspondence with Lagrangian algebras of the underlying UMTC C [42,85,102,129].”
State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter (2016, methodology): “inus sign implies that the symmetry is anomalous. In the condensed matter language, our objective is to condense the anyon B. The general mathematical formalism for anyon condensation is described in [35]. It should be applied to the commutative separable algebra A= I+ B. We will use a somewhat simp”
On finite symmetries and their gauging in two dimensions (2017, background): “This operation is in fact known under the name of anyon condensation [22].”