Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

Abstract

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we will continue the study of Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. In particular, we show that a previously proposed construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialised CWdecompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group.The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretised 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.1 Due to a lack of local observables, experimentally distinguishing different topological phases can be a difficult task from a microscopic point of view. Instead the characterising properties of topological phases are most efficiently described by their emergent behaviours. Signatures for the presence of topological order include ground state degeneracies which depend on the spatial topology of the material in question [69,43], universal negative corrections to the entanglement entropy [44,37,48] and the presence of stable topological excitations which provide non-trivial representations of their respective motion groups [47,41]. This means the braid group for point particles (anyons) in 2+1D [54] and the loop braid group for loop excitations in 3+1D [66].In 2+1D there exist several constructions for TQFTs (see for instance [64]). Path-integral models arise from Chern-Simons-Witten theory [73] and from BF theory [1], while the discrete realisation of BF-theory coincides with the Turaev-Viro [64]/Barrett-Westbury [6] state-sum (see [2]). In contrast, in 3+1D a framework general enough to capture all features of 4D topology is still lacking. Nevertheless, we have the Crane-Yetter TQFT [23,64] and its generalisations [71,25]; and the Yetter homotopy 2-type TQFT [74,59,33], derived from a strict finite 2-group [4]. All of these 3+1D TQFTs give rise to topological invariant...

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

“…The tube algebra is then given by the twisted groupoid algebra: The simple modules can then be found, analogously to the case of the twisted quantum double and twisted quantum triple, by first reducing the algebra to subalgebras given by objects related by conjugation and then resolving each such algebra by the irreducible representations of the stabiliser group of a representative object [33]. Furthermore, it is possible to enrich the present constructions to accommodate lattice models that have a higher gauge theory interpretation [44,[50][51][52][53][54][55][56][57][58]. This generalization was formally stated in [20] in the strict 2-group setting using the language of groupoids for general spacetime dimensions and choices of boundary manifold.…”

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

“…which must be trivial given that the fake-flatness constraint (2.5) is enforced. Applying the whiskering rules, it is always possible to compose 2-holonomies associated with adjacent plaquettes, and it was shown in [32] that given a 2-path the corresponding 2-holonomy is uniquely defined. Let us for instance consider the cube c ≡ (01234567) ⊂ Σ depicted below , ∼ 5 ∼ we can show by means of the whiskering rules that the (closed) 2-holonomy associated with its boundary 2-path reads…”

“…Let us now define the higher gauge model. More details can be found in [32,33]. The microscopic Hilbert space H G [Σ ] is spanned by graph-states |g where g is a G-labelling of Σ as defined earlier.…”

“…In general, we define a Hamiltonian realization on a d-dimensional hypersurface Σ as a sum of mutually commuting projectors such that the ground state subspace is equal to the image of the corresponding (d+1)-dimensional partition function on the manifold Σ×[0, 1], where the partition function is thought as an Hermitian projector. In this manuscript, we are interested in lattice Hamiltonian models that correspond to Yetter's homotopy 2-type topological quantum field theories [29][30][31][32][33], whose input data are so-called (finite) strict 2-groups.…”

2-group actions and moduli spaces of higher gauge theory

“…A careful study of HGT along similar lines to ours was carried out by Bullivant, Calcada, Kádár, Faria Martins and Martin [8] in the context of their investigation of topological phases of matter in 3+1 dimensions. They also discretize the higher connections using manifolds with an adapted cell structure, termed a 2-lattice structure.…”