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

Abstract

We consider an exactly solvable model for topological phases in (3+1)d whose input data is a strict 2-group. This model, which has a higher gauge theory interpretation, provides a lattice Hamiltonian realisation of the Yetter homotopy 2-type topological quantum field theory. The Hamiltonian yields bulk flux and charge composite excitations that are either point-like or loop-like. Applying a generalised tube algebra approach, we reveal the algebraic structure underlying these excitations and derive the irreducible modules of this algebra, which in turn classify the elementary excitations of the model. As a further application of the tube algebra approach, we demonstrate that the ground state subspace of the three-torus is described by the central subalgebra of the tube algebra for torus boundary, demonstrating the ground state degeneracy is given by the number of elementary loop-like excitations.arXiv:1909.07937v1 [cond-mat.str-el]

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview

“…Then in Ref. [59], the excitations were studied using a tube algebra approach, through which the loop-like excitations in the model were classified and the simple types were counted. Furthermore, it was shown in Ref.…”

“…Furthermore, it was shown in Ref. [59] that there is a relationship between the number of types of elementary excitation and the ground-state degeneracy of the model on a 3-torus. In addition, it was shown in Ref.…”

“…As far as we know, this is the first time that this method has been applied in 3+1d using explicit membrane operators, although this method has some similarity to the tube algebra methods used in Ref. [59]. In the tube algebra method, the excitations are described in terms of the allowed degrees of freedom on the boundary of a given region, which is presumably equivalent to finding the allowed measurement operators for the surface forming the boundary of the region.…”

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

“…In order to reveal the algebraic structure underlying the bulk excitations in arbitrary spatial dimension, several strategies exist. Our focus is on the so-called tube algebra approach [5,33,44,[46][47][48][49][50], which is a generalization of Ocneanu's tube algebra [51,52]. In general, the 'tube' refers to the manifold ∂Σ × [0, 1], where ∂Σ is the boundary left by JHEP07(2021)025 removing a regular neighbourhood of the excitation in question, and the 'algebra' to an algebraic extension of the gluing operation (∂Σ × [0, 1]) ∪ ∂Σ (∂Σ × [0, 1]) (∂Σ × [0, 1]) to the Hilbert space of states on the tube.…”

“…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]. This strategy has been extensively applied to general two-dimensional models, and more recently to gauge and higher gauge models in three dimensions [33,50].…”