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Anyon-Permuting Symmetry

Steven H. Simon

Published Sep 29, 2023• 0 citations
Publisher

Abstract

Abstract A theory with anyon permuting symmetry is unchanged under permutation of the names of the particle types. The toric code is an example of a theory with such a symmetry. The toric code, and all theories with such symmetries can be viewed as arising as a condensation of anyons from some more complicated parent theory. Given that such a symmetry exists, we consider defects of this symmetry which generalize the mathematical structure of the TQFT to a so-called G-crossed extension. To clarify how this works we use a particularly tranparent example of the toric code where the symmetry of particle types is reduced to a lattice symmetry, and the defects correspond to dislocations in the lattice. It can be shown that these defects harbor Majorana zero modes. We discuss the general structure of a parent theory condensing via a G-crossed theory to a theory with permutation symmetry. Finally we mention theories with both on-site symmetries and anyons, which are known as symmetry enriched topological phases.

DOI
10.1093/oso/9780198886723.003.0036

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Abstract A theory with anyon permuting symmetry is unchanged under permutation of the names of the particle types. The toric code is an example of a theory with such a symmetry. The toric code, and all theories with such symmetries can be viewed as arising as a condensation of anyons from some more complicated parent theory. Given that such a symmetry exists, we consider defects of this symmetry which generalize the mathematical structure of the TQFT to a so-called G-crossed extension. To clarify how this works we use a particularly tranparent example of the toric code where the symmetry of particle types is reduced to a lattice symmetry, and the defects correspond to dislocations in the lattice. It can be shown that these defects harbor Majorana zero modes. We discuss the general structure of a parent theory condensing via a G-crossed theory to a theory with permutation symmetry. Finally we mention theories with both on-site symmetries and anyons, which are known as symmetry enriched topological phases.
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The global eee-mmm exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.
The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature. To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in the toric code. A more speculative and interesting direction is the study of Banados–Teitelboim–Zanelli (BTZ) black holes and quantum gravity in 3d. The clearly defined physical and mathematical issues require a far-reaching generalization of anyons and seem to be within reach. In this short survey, I will first cover the extensions of anyon theory to symmetry defects and gapped boundaries. Then, I will discuss a desired generalization of anyons to anyon-like objects — the BTZ black holes — in 3d quantum gravity.
Abstract In this work, we explore a microscopic realization of three types of anyonic symmetries in a Z 2 × Z 2 topological order, corresponding to a double toric code. These symmetries act as nontrivial permutations on the anyon labels of the parent state. We consider a setup consisting of two decoupled Wen plaquette models stacked on top of each other and introduce dislocations that modify the Hamiltonian, giving rise to localized twist defects, eventually inducing interactions between the layers. In this context, branch cuts act as sources of anyon permutations when they cross it. We characterize the defects by calculating their quantum dimensions, and we also consider double loop operators around them that allow us to determine the non-Abelian fusion rules between the defects, including when they carry different anyon permutations.
The global eee-mmm exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.
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.