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arXiv:2204.05341 · Physical review. B./Physical review. B

Excitations in the higher-lattice gauge theory model for topological phases. II. The (2+1) -dimensional case

Joe Huxford, Steven H. Simon

Published Dec 12, 2023• 4 citations
arXivHTMLPDF

Abstract

In this work, the second paper of this series, we study the $(2+1)$-dimensional version of a Hamiltonian model for topological phases based on higher-lattice gauge theory. We construct the ribbon operators that produce the pointlike excitations. These ribbon operators are used to find the braiding properties and topological charge carried by the pointlike excitations. The model also hosts looplike excitations, which are produced by membrane operators. By considering a change of basis, we show that, in certain cases, some looplike excitations represent domain walls between patches corresponding to different symmetry-related ground states, and we find this symmetry. We also map the higher-lattice gauge theory Hamiltonian to the symmetry-enriched string-net model for symmetry-enriched topological phases described by Heinrich, Burnell, Fidkowski, and Levin [Phys. Rev. B 94, 235136 (2016)], again in a subset of cases.

DOI
10.1103/physrevb.108.245133

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In this work, the second paper of this series, we study the (2+1)(2+1)(2+1)-dimensional version of a Hamiltonian model for topological phases based on higher-lattice gauge theory. We construct the ribbon operators that produce the pointlike excitations. These ribbon operators are used to find the braiding properties and topological charge carried by the pointlike excitations. The model also hosts looplike excitations, which are produced by membrane operators. By considering a change of basis, we show that, in certain cases, some looplike excitations represent domain walls between patches corresponding to different symmetry-related ground states, and we find this symmetry. We also map the higher-lattice gauge theory Hamiltonian to the symmetry-enriched string-net model for symmetry-enriched topological phases described by Heinrich, Burnell, Fidkowski, and Levin [Phys. Rev. B 94, 235136 (2016)], again in a subset of cases.
In this series of papers, we study a Hamiltonian model for (3+1)-dimensional topological phases introduced in [Bullivant et al., Phys. Rev. B 95, 155118 (2017)], based on a generalization 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, in addition to ordinary 1-gauge fields which describe the parallel transport of points. In this series we explicitly construct the creation operators for the pointlike and looplike excitations supported by the model. We use these creation operators to examine the properties of the excitations, including their braiding statistics. These creation operators also reveal that some of the excitations are confined, costing energy to separate that grows linearly with the length of the creation operator used. 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 rederive 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 looplike excitations. This first paper provides an accessible summary of our findings, with more detailed results and proofs to be presented in the other papers in the series.
In this, the third paper in our series describing the excitations of the higher-lattice gauge theory model for topological phases, we will examine the (3+1)-dimensional case in detail. We will explicitly construct the ribbon and membrane operators which create the topological excitations, and use these creation operators to find the pattern of condensation and confinement. We also use these operators to find the braiding relations of the excitations, and to construct charge measurement operators which project to states of definite topological charge. Published by the American Physical Society 2024
Topological phases of matter are characterised by long-range entanglement between their constituent degrees of freedom, which allows them to host excitations with non-trivial exchange statistics. While topological phases in 2+1d are relatively well studied, less is known about their 3+1d counterparts. In this thesis, we examine features of these higher-dimensional phases in a tractable example model, based on a generalisation of lattice gauge theory called higher lattice gauge theory. As well as hosting point-like excitations, the model supports loop-like excitations with non-trivial loop-loop and point-loop braiding statistics. 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, while others are condensed and can be produced locally. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also explain a method for measuring topological charge 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 higher lattice gauge theory 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.
(3+1)(3+1)(3+1)-dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for nontrivial loop-loop and point-loop braiding statistics not permitted to point-like excitations alone. Furthermore, these loop-like excitations can be linked together, changing their properties. In particular, this can lead to distinct three-loop braiding, involving two loops undergoing an exchange process while linked to a third loop. In this work, we investigate the loop-like excitations in a (3+1)(3+1)(3+1)-dimensional Hamiltonian realization of Dijkgraaf-Witten theory through direct construction of their membrane operators, for a general finite Abelian group and 4-cocycle twist. Using these membrane operators, we find the braiding relations and fusion rules for the loop-like excitations, including those linked to another loop-like excitation. Furthermore, we use these membrane operators to construct projection operators that measure the topological charge and show that the number of distinct topological charges measured by the 2-torus matches the ground-state degeneracy of the model on the 3-torus, explicitly confirming a general expectation for topological phases. This direct construction of the membrane operators sheds significant light on the key properties of the loop-like excitations in (3+1)(3+1)(3+1)-dimensional topological phases.
We discuss several bosonic topological phases in (3+1) dimensions enriched by a global Z2{\mathbb{Z}}_{2}Z2​ symmetry, and gauging the Z2{\mathbb{Z}}_{2}Z2​ symmetry. More specifically, following the spirit of the bulk-boundary correspondence, expected to hold in topological phases of matter in general, we consider boundary (surface) field theories and their orbifold. From the surface partition functions, we extract the modular S\mathcal{S}S and T\mathcal{T}T matrices and compare them with (2+1)(2+1)(2+1)-dimensional toplogical phase after dimensional reduction. As a specific example, we discuss topologically ordered phases in (3+1)(3+1)(3+1) dimensions described by the BF topological quantum field theories, with abelian exchange statistics between pointlike and looplike quasiparticles. Once the Z2{\mathbb{Z}}_{2}Z2​ charge conjugation symmetry is gauged, the Z2{\mathbb{Z}}_{2}Z2​ flux becomes non-Abelian excitation. The gauged topological phases we are considering here belong to the quantum double model with non-Abelian group in (3+1)(3+1)(3+1) dimensions.
Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.\n\n\n\nThe operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.