Twisted Gauge Theory Model of Topological Phases in Three Dimensions

TL;DR

We construct an exactly solvable 3+1D lattice model of topological order defined by a finite group $G$ and a 4-cocycle $\omega$, yielding gapped phases with ground-state degeneracy on $\mathbb{T}^3$ and loop-like excitations labeled by two flux indices and a charge. The topological data are encoded in two flux labels via doubly-twisted 2-cocycles $\beta_{k,g}$ and in projective representations of centralizers $Z_{k,g}$, with modular $\mathcal{S}$ and $\mathcal{T}$ matrices providing a robust 3D analogue of 2D anyon data. The phase classification depends on $[\omega]\in H^4(G,U(1))$ and the induced $\beta_{k,g}$, with equivalence under coboundaries ensuring that coboundary-related cocycles describe the same phase. By connecting to Dijkgraaf-Witten theory, the work establishes a Hamiltonian extension of 3+1D DW gauge theory and clarifies the relationship to twisted BF theory, offering concrete tools and examples for understanding 3D topological orders and their excitations. These results open pathways to studying higher-dimensional topological phases, their boundary theories, and potential realizations in quantum computation contexts.

Abstract

We propose an exactly solvable lattice Hamiltonian model of topological phases in $3+1$ dimensions, based on a generic finite group $G$ and a $4$-cocycle $ω$ over $G$. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the $3$-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasi-excitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$ generators as the modular $S$ and $T$ matrices of the ground states, which yield a set of topological quantum numbers classified by $ω$ and quantities derived from $ω$. Our model fulfills a Hamiltonian extension of the $3+1$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group $G$. This work is presented to be accessible for a wide range of physicists and mathematicians.

Figures (9)

• Figure 1: A crop of a graph that represents the basis vectors in the Hilbert space. Each edge $ab$, with $a<b$, is oriented from the larger enumeration to the smaller and is assigned a group element, $ab\in G$.
• Figure 2: (a) The defining graph of the $4$-cocycle $[v_1v_2, v_2v_3,v_3v_4,v_4v_5]$. (b) For $[v_1v_2, v_2v_3,v_3v_4,v_4v_5]^{-1}$.
• Figure 3: The topology of the action of $A_{v_2}^g$. The arrows are omitted for simplicity because the vertex ordering is explicit.
• Figure 4: The simplest triangulation of a $3$-torus. Vertices are in the order of enumerations $1<2<3<4<5<6<7<8$, so the arrows are omitted for simplicity. Any two squares opposite to each other are identified along the edges with the same arrow. Restricted to the subspace $\mathcal{H}^{B_f=1}$, there are only three independent group degrees of freedom. (a) A natural presentation of the ($zxy$) basis: $k=15,g=12,h=13\in G$. (b) A physical presentation of the ($zxy'$) basis: $k$, $g$, and $h'=23=\bar{g}h$.
• Figure 5: $\mathcal{S}$ and $\mathcal{T}$ transformations of a $3$-torus.

Theorems & Definitions (5)

• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Theorem 8