Abstract
We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.
Highlights
A prominent class of gapped quantum phases of matter are given by so-called topological phases of matter
Focusing on lattice Hamiltonian realisations of Dijkgraaf-Witten theory, a.k.a gauge models of topological phases, we studied gapped boundaries and their excitations in (2+1)d and (3+1)d
As explained in detail in [33], local operators of lattice Hamiltonian realisations of Dijkgraaf-Witten theory can be conveniently expressed in terms of the partition function of the theory applied to so-called pinched interval cobordisms
Summary
A prominent class of gapped quantum phases of matter are given by so-called topological phases of matter. The corresponding state-sum invariant is referred to as the Dijkgraaf-Witten invariant [38] In this context, (bulk) anyonic excitations, defined as a region whose energy is higher than that of the ground state, are described in terms of the so-called twisted quantum double of the group, whose irreducible representations provide the simple objects of the Drinfel’d centre of the category of G-graded vector spaces [39, 40]. The twisted quantum double and the twisted quantum triple are found to be isomorphic to the tube algebras associated with the manifolds S1 × [0, 1] and T2 × [0, 1], respectively This approach relies on the fact that properties of a given excitation, which, let us recall, is defined as a region for which the energy is higher than that of the ground state, are encoded into the boundary conditions that the model assigns to the boundary ∂Σ [33]. The correspondence with the centre construction of the bicategory of group-graded 2-vector spaces is established
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