On 2-form gauge models of topological phases

We explore various aspects of 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space $B^2G$ of the symmetry group $G$, and they are classified by cohomology classes of $B^2G$. Discrete topological gauge theories can typically be embedded into continuous quantum field theories. In the 2-form case, the continuous theory is shown to be a strict 2-group gauge theory. This embedding is studied by carefully constructing the space of $q$-form connections using the technology of Deligne-Beilinson cohomology. The same techniques can then be used to study more general models built from Postnikov towers. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of $B^2G$ as provided by the so-called $W$-construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of $G$-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.Comment: 78 page

From gauge to higher gauge models of topological phases

We consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition of the unitary map performing a local change of triangulation is equivalent to the coherence relation of the pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of such 2-category, we obtain a cohomological model whose underlying 1-category is a 2-group. Topological models from 2-groups together with their lattice realization are then studied from a higher gauge theory point of view. Symmetry protected topological phases protected by higher symmetry structures are explicitly constructed, and the gauging procedure which yields the corresponding topological gauge theories is discussed in detail. We finally study the correspondence between symmetry protected topological phases and 't Hooft anomalies in the context of these higher group symmetries.Comment: 38 pages, v2: minor revisio

Entanglement entropy of (3+1)D topological orders with excitations

Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how different types of topological excitations contribute to the entanglement entropy, or alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\omega\in\mathcal{H}^4[G;U(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant $\ln d_i$ to the entanglement entropy, where $d_i$ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\,\omega)$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\,\omega)$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\omega$ from the others.Comment: 12 pages, 4 figures; v2: minor changes, published versio

Intertwined symmetry and topology in 3 + 1 dimensional gapped quantum phases of matter

This thesis is an investigation of various topological properties of 3 + 1 dimensional Bosonic gapped phases of matter. We divide our study into two main themes, the first involving short-range entangled phases with global symmetries also known as symmetry protected topological phases and the second involving phases with long range entanglement also known as topologically ordered phases