Tube algebras, excitations statistics and compactification in gauge models of topological phases

Abstract

We consider lattice Hamiltonian realizations of (d+1)-dimensional Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian yields point-like excitations classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalisation of this strategy that is valid in any dimensions. We then apply this generalisation to derive the algebraic structure of loop-like excitations in (3+1)d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1)d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an R-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a manifold that is n-times compactified can be expressed in terms of another model in n-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues. arXiv:1905.08673v2 [cond-mat.str-el]

Excitations in strict 2-group higher gauge models of topological phases

“…At this point, the two 2-holonomies have matching source and target 1-paths so that they can be composed. The resulting 2-holonomy is labelled by h(g 02 h ) and is such that bp(0145) = (0), s(0145) = (04) ∪ (45) and t(0145) = (01) ∪ (15). 3 Such factors can be induced by the requirement that the normalisation of the states is preserved under such isomorphisms, or equally by considering the isomorphism as a cobordism operator in the corresponding Yetter homotopy 2-type topological theory.…”

“…This lattice Hamiltonian presents so-called open boundary conditions since graphstates with different G-colourings on ∂Σ o are not mixed. In this case, the corresponding ground state subspace admits a decomposition in terms of boundary G-colourings: [45], this gluing operation can be extended to a symmetry of the ground state subspace V G [Σ o ]. It follows from the discussion in sec.…”

“…Under this assumption, the model (2.12) reduces to a gauge model, namely the Hamiltonian realization of Dijkgraaf-Witten theory with trivial cohomology class in H 4 (BG, U(1)) [50,51]. The authors showed in [43,45] that in this case the tube algebra for loop-like excitations is isomorphic to the (untwisted) quantum triple algebra, which we reproduce below for convenience:…”

“…The simple modules of this algebra are labelled by equivalence classes that correspond to sets of Gcolourings of T 2 × {0} related via simultaneous conjugation, and irreducible representations of the corresponding stabiliser groups. The physical interpretation of these simple modules goes as follows [45]: Given the three-disk D 3 , removing a solid torus D 2 × S 1 from it creates a loop-like defect. After this operation, we can find a non-contractible 1-cycle, starting and ending at a given basepoint, that winds once around the hole left by the torus.…”

“…The tube algebra approach, which is a generalization of Ocneanu's tube algebra [35,36], is particularly intuitive and has proven very successful [37][38][39][40][41][42][43][44][45]. More specifically, the authors showed in [45] that for lattice Hamiltonian realizations of Dijkgraaf-Witten theory, it was possible to apply the tube algebra approach in any dimensions in order to classify the excitations of the corresponding model. For instance, when applied to the case of bulk point-like excitations in (2+1)d and bulk loop-like excitations in (3+1)d, we can confirm that ∼ 2 ∼ elementary excitations are classified by the simple modules of the twisted quantum double algebra and the twisted quantum triple algebra, respectively.…”