Generalized string-net model for unitary fusion categories without tetrahedral symmetry

Hahn, Alexander; Wolf, Ramona

doi:10.1103/physrevb.102.115154

Cited by 22 publications

(15 citation statements)

References 62 publications

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“…[33] (see Refs. [31,32,[41][42][43] for such generalizations). In both of these classes of models, it is well understood how to find the ground-state degeneracy [13,44] and the properties of the excitations, such as braiding statistics [13,32,33] (as we will elaborate on shortly).…”

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confidence: 99%

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview

Huxford¹,

Simon²

2022

Preprint

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In this series of papers, we study a Hamiltonian model for 3+1d topological phases, based on a generalisation of lattice gauge theory known as "higher lattice gauge theory". Higher lattice gauge theory has so called "2-gauge fields" describing the parallel transport of lines, just as ordinary gauge fields describe the parallel transport of points. In the Hamiltonian model this is represented by having labels on the plaquettes of the lattice, as well as the edges. In this paper we summarize our findings in an accessible manner, with more detailed results and proofs to be presented in the other papers in the series. The Hamiltonian model supports both point-like and loop-like excitations, with non-trivial braiding between these excitations. We explicitly construct operators to produce and move these excitations, and use these to find the loop-loop and point-loop braiding relations. These creation operators also reveal that some of the excitations are confined, costing energy to separate. This is discussed in the context of condensation/confinement transitions between different cases of this model. We also discuss the topological charges of the model and use explicit measurement operators to re-derive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the model on the 3-torus. From these measurement operators, we can see that the ground state degeneracy on the 3-torus is related to the number of types of linked loop-like excitations.

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confidence: 99%

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview

Huxford¹,

Simon²

2022

Preprint

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“…The matrix elements of this operator are worked out in detail in [1,6]. Every operator B a p is a 18-body operator in the sense that its action explicitly depends on the six string types living on the edges of the plaquette p, the six multiplicities and the six edges adjacent to the plaquette, but acts diagonally on the latter.…”

Section: A Levin and Wen's String-netsmentioning

confidence: 99%

“…Ever since their conception, the string-net models as originally proposed by Levin and Wen [1] and their subsequent generalizations [2][3][4][5][6][7] have provided a rich playground for studying microscopic realisations of non-chiral topologically ordered phases of matter in 2+1 dimensions. Taking a unitary fusion category (UFC) D [8] as input, these exactly solvable models allow for the explicit realization of several key features of topologically ordered systems, such as ground state degeneracies that depend on the topology of the space and anyonic quasi-particle excitations that satisfy non-trivial braiding statistics.…”

Section: Introductionmentioning

confidence: 99%

Mapping between Morita equivalent string-net states with a constant depth quantum circuit

Lootens,

Cuiper,

Schuch

et al. 2021

Preprint

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We construct a constant depth quantum circuit that maps between Morita equivalent string-net models. Due to its constant depth and unitarity, the circuit cannot alter the topological order, which demonstrates that Morita equivalent string-nets are in the same phase. The circuit is constructed from an invertible bimodule category connecting the two input fusion categories of the relevant string-net models, acting as a generalized Fourier transform for fusion categories. The circuit does not only act on the ground state subspace, but acts unitarily on the full Hilbert space when supplemented with ancillas.

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“…Since we drop the unitary condition, things are sightly different: the Hamiltonian is symmetric (it is easy to see that the matrix element is invariant under the exchange of abc and a b c ) but not Hermitian. However, the spectrum of the Hamiltonian is still entirely real for it is a projector (the possible eigenvalue for a projector is either 1 or 0), which means that B 2 p = B p at each plaquette p. In order to prove it as an projector, we need another relation, called completeness relation in [9]:…”

Section: A String-net Modelmentioning

confidence: 99%

Galois conjugates of String-net Model

Chen,

Lao,

et al. 2021

Preprint

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We revisit a class of non-Hermitian topological models that are Galois conjugates of their Hermitian counter parts. Particularly, these are Galois conjugates of unitary string-net models. We demonstrate these models necessarily have real spectra, and that topological numbers are recovered as matrix elements of operators evaluated in appropriate bi-orthogonal basis, that we conveniently reformulate as a concomitant Hilbert space here. We also compute in the bi-orthogonal basis the topological entanglement entropy, demonstrating that its real part is related to the quantum dimension of the topological order. While we focus mostly on the Yang-Lee model, the results in the paper apply generally to Galois conjugates. a

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Generalized string-net model for unitary fusion categories without tetrahedral symmetry

Cited by 22 publications

References 62 publications

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview

Mapping between Morita equivalent string-net states with a constant depth quantum circuit

Galois conjugates of String-net Model

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