Abstract
We investigate a family of invertible phases of matter with higher-dimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev’s chain in 1+1d. The excitation has \mathbb{Z}_2ℤ2 higher-form symmetry that mixes with the spacetime Lorentz symmetry to form a higher group spacetime symmetry. We focus on the invertible exotic loop topological phase in 3+1d. This invertible phase is protected by the \mathbb{Z}_2ℤ2 one-form symmetry and the time-reversal symmetry, and has surface thermal Hall conductance not realized in conventional time-reversal symmetric ordinary bosonic systems without local fermion particles and the exotic loops. We describe a UV realization of the invertible exotic loop topological order using the SO(3)_-SO(3)− gauge theory with unit discrete theta parameter, which enjoys the same spacetime two-group symmetry. We discuss several applications including the analogue of ``fermionization’’ for ordinary bosonic theories with \mathbb{Z}_2ℤ2 non-anomalous internal higher-form symmetry and time-reversal symmetry.
Highlights
Let us first discuss the property of the symmetry, we will describe an invertible phase with spacetime two-group symmetry realized by a 2 two-form gauge theory
The invertible topological phase that is constructed from the exotic loop, namely that is protected by the two-group symmetry (2)[w1w2], has a partition function that depends on the background ρ2, i.e. the v3 Wu structure
The theory is equivalent to SO(3)+ theory with θ = 2π, where the subscript + indicates that it has zero discrete theta parameter.28. This UV model on a spin manifold gives essentially the same physics as the combination of the 2 one-form symmetry SPT phase that belongs to the class m = 3 in the 4 classification [37, 38] (whose partition function is given by (E.1)) and the ν = 2 class DIII topological superconductor discussed in Subsection 3.7
Summary
Invertible or short-range entangled phases of matter have a unique ground state with an energy gap. The invertible exotic loop topological order phase has a boundary thermal Hall conductance that equals ±1, which distinguishes the exotic loop phase from ordinary time-reversal symmetric bosonic systems. The second model is the SO(3) gauge theory with θ = 2π, which has the two-group spacetime symmetry and the exotic loop excitation. By coupling the gauge theory to matter fields, we find a theory with exotic loops, describing a plausible deconfined quantum critical point with the spacetime twogroup symmetry which includes the 2 one-form symmetry and time-reversal symmetry. The spacetime n-group symmetry implies that the (n − 1)-dimensional membranes, which are charged under the 2 (n − 1)-form symmetry, obey properties similar to the exotic loop in the iELTO phase. The time-reversal symmetry is essential to distinguish these invertible phases from other invertible phases discussed in the literature
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