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arXiv:1502.02026 · Journal of High Energy Physics

Generalized ADE classification of topological boundaries and anyon condensation

Ling-Yan Hung, Yidun Wan

Published Jul 1, 2015• 63 citations
arXivHTMLPDF

Abstract

In this paper we would like to demonstrate how the known, physically-motivat-ed rules of anyon condensation proposed by Bais et al. can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized ADE classification of CSFA’s and modular invariants to classify anyon condensation, characterize the topological boundaries between topological field theories and thus describe all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classification is equivalent to the classification we proposed in ref. [1].

DOI
10.1007/jhep07(2015)120

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In this paper we would like to demonstrate how the known, physically-motivat-ed rules of anyon condensation proposed by Bais et al. can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized ADE classification of CSFA’s and modular invariants to classify anyon condensation, characterize the topological boundaries between topological field theories and thus describe all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classification is equivalent to the classification we proposed in ref. [1].
Instead of constructing 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 relations 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. More general situation is also discussed 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.
Abstract We classify condensable E 2-algebras in a modular tensor category C up to 2-Morita equivalence. Physically, this classification provides an explicit criterion to determine when distinct condensable E 2-algebras yield the same condensed topological phase under a two-dimensional anyon condensation process. The relations between different condensable algebras can be translated into their module categories, interpreted physically as gapped domain walls in topological orders. As concrete examples, we interpret the categories of quantum doubles of finite groups and examples beyond group symmetries. Our framework fully elucidates the interplay among condensable E 1-algebras in C , condensable E 2-algebras in C up to 2-Morita equivalence, and Lagrangian algebras in C ⊠ C ¯ .
The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.
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.
Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in Z(VecGω)\mathcal{Z}(\text{Vec}_G^ω)Z(VecGω​), i.e. in group-theoretical topological orders for finite groups GGG with anomaly ωωω. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.
We propose a formula for the transformation law of anyons in topologically ordered phases or topological quantum field theories (TQFTs) through a gapped or symmetry-preserving domain wall. Our formalism is based on the ring homomorphism between the C\mathbb{C}C-linear commutative fusion rings, also known as symmetry topological field theories (SymTFTs). The fundamental assumption in our formalism is the validity of the Verlinde formula, applicable to commutative fusion rings. By combining it with more specific data of the settings, our formula provides classifications of anyons compatible with developing categorical formulations. It also provides the massless renormalization group (RG) flows between conformal field theories (CFTs), or a series of measurement-induced quantum phase transitions, in the language of SymTFT, through the established correspondence between CFTs and TQFTs. Moreover, by studying the correspondence between the ideal structure in the massless RG and the module in the related massive RG, one can make the Nambu-Goldstone-type arguments for generalized symmetry. By combining our formula with orbifolding, extension, and similarity transformation, one can get a series of classifications for the corresponding extended models, or symmetry-enriched topological orders and quantum criticalities.