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

Abstract

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.

37 pages, additional 12 page appendix, 51 figures, Revisions: some restructuring of sections, additional references

References in corpus (13)

• Non-Abelian Anyons and Topological Quantum Computation
• Surface codes: Towards practical large-scale quantum computation
• Anyons and the quantum Hall effect - a pedagogical review
• Condensate induced transitions between topologically ordered phases
• A classification of symmetry enriched topological phases with exactly solvable models
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Generalized string-net models: A thorough exposition
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• Twisted gauge theories in 3D Walker-Wang models
• Generalized string-nets for unitary fusion categories without tetrahedral symmetry
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases II: The (2+1)-Dimensional Case

Cited by in corpus (5)

• 2-Form U(1) Spin Liquids: Classical Model and Quantum Aspects
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases II: The (2+1)-Dimensional Case
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases III: the (3+1)-Dimensional Case
• Finite 2-group gauge theory and its 3+1D lattice realization
• Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge