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 × [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.Contents arXiv:1606.06639v2 [cond-mat.str-el]

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...

We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schrödinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called δ interaction.