Topological phases from higher gauge symmetry in 3+1D

Abstract

We propose an exactly solvable Hamiltonian for topological phases in $3+1$ dimensions utilising ideas from higher lattice gauge theory, where the gauge symmetry is given by a finite 2-group. We explicitly show that the model is a Hamiltonian realisation of Yetter's homotopy 2-type topological quantum field theory whereby the groundstate projector of the model defined on the manifold $M^3$ is given by the partition function of the underlying topological quantum field theory for $M^3\times [0,1]$. We show that this result holds in any dimension and illustrate it by computing the ground state degeneracy for a selection of spatial manifolds and 2-groups. As an application we show that a subset of our model is dual to a class of Abelian Walker-Wang models describing $3+1$ dimensional topological insulators.

28 pages, 4 figures

References in corpus (10)

• Local stabilizer codes in three dimensions without string logical operators
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• Topological invariants for gauge theories and symmetry-protected topological phases
• Hamiltonian models for topological phases of matter in three spatial dimensions
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations
• Topological Higher Gauge Theory - from BF to BFCG theory

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