Infinite volume ground states in the non-Abelian quantum double model

Hamdan, Mahdie 2024. Infinite volume ground states in the non-Abelian quantum double model. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

We study a particular class of ground states of the non-abelian Kitaev quantum double model on an infinite plane. These states are associated to the irreducible representations of the quantum double of a finite group G, and are called anyonic excitations. Anyonic excitations are a particular feature of topological phases of matter. Recall that if G is a finite group, the irreducible representations of the quantum double D(G) can be labelled by pairs (π, C), where C is a conjugacy class of G and π is an irreducible representation of the centralizer subgroup of a fixed element r ∈ C, the choice of which is irrelevant. Using the notion of ribbon operators as in [Kit03], we consider for each irreducible representation α := (π, C) ∈ D\(G), each label I = 1, . . . , dimα and semi-infinite ribbon ξ, the amplimorphisms χ II,α ξ defined as in [Naa15, Eq 5.3] and show that the states ω0 ◦ χ II,α ξ define pure states, where ω0 is the vacuum state of the model. Given two irreducible representations α, β ∈ D\(G) and two semi-infinite ribbons ξ1, ξ2, we show that the GNS representations of ω II,α ξ1 and ω JJ,β ξ2 are unitarily equivalent if and only if α ∼= β. Furthermore, if either π ̸= triv or |C| = 1 holds, then ω II,α ξ is a ground state for a semi-infinite ribbon ξ in the infinite plane. We interpret ω II,α ξ as a state creating a single localized excitation that cannot be removed by local observables. We also prove that the states ω II,α ξ are indeed non-ground states in the other case and construct alternative non-pure ground states corresponding to these anyon sectors. We conclude this work with an exposition on a work in progress. The amplimorphisms described in [Naa15] are transportable and localized, which is why they give rise to representations satisfying a superselection criterion for cones. These localized and transportable amplimorphisms form a category, and we conjecture that they are equivalent to the category rep(D(G)) of representations of D(G) as a monoidal tensor category. In this thesis, we present some steps towards this conjecture.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Funders:	EPSRC
Date of First Compliant Deposit:	20 February 2025
Last Modified:	20 Feb 2025 15:32
URI:	https://orca.cardiff.ac.uk/id/eprint/176342

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