Braiding statistics of strings with general linking in 3$+$1D topological order

Yi Zhang, Tarun Grover, Ari M. Turner +2 more

Pavel Putrov, Juven Wang, Shing–Tung Yau

Juven Wang, Xiao-Gang Wen

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist ${\ensuremath{\omega}}_{4}$ of $G$'s cohomology group ${\mathcal{H}}^{4}(G,\mathbb{R}/\mathbb{Z})$ in three-dimensional space and one-dimensional time $(3+1\mathrm{D})$. We establish the topological spin and the spin-statistics relation for the closed strings and their multistring braiding statistics. The $3+1\mathrm{D}$ twisted gauge theory can be characterized by a representation of a modular transformation group, $\mathrm{SL}(3,\mathbb{Z})$. We express the $\mathrm{SL}(3,\mathbb{Z})$ generators ${\mathsf{S}}^{xyz}$ and ${\mathsf{T}}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle ${\ensuremath{\omega}}_{4}$. As we compactify one of the spatial directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators ${\mathsf{S}}^{xy}$ and ${\mathsf{T}}^{xy}$ of an $\mathrm{SL}(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order ${\mathcal{C}}^{3\mathrm{D}}$ to a direct sum of degenerate states of 2D topological orders ${\mathcal{C}}_{b}^{2\mathrm{D}}$ in different flux $b$ sectors: ${\mathcal{C}}^{3\mathrm{D}}={\ensuremath{\bigoplus}}_{b}{\mathcal{C}}_{b}^{2\mathrm{D}}$. The 2D topological orders ${\mathcal{C}}_{b}^{2\mathrm{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycle ${\ensuremath{\omega}}_{3(b)}$, dimensionally reduced from the 4-cocycle ${\ensuremath{\omega}}_{4}$. We show that the $\mathrm{SL}(2,\mathbb{Z})$ generators, ${\mathsf{S}}^{xy}$ and ${\mathsf{T}}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into a three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

Xiaofeng Shi, J. Q. You, Franco Nori

Chao‐Ming Jian, Xiao-Liang Qi

While the topological order in two dimensions has been studied extensively since the discovery of the integer and fractional quantum Hall systems, topological states in three spatial dimensions are much less understood.In this paper, we propose a general formalism for constructing a large class of threedimensional topological states by stacking layers of 2D topological states and introducing coupling between them.Using this construction, different types of topological states can be obtained, including those with only surface topological order and no bulk topological quasiparticles, and those with topological order both in the bulk and at the surface.For both classes of states, we study its generic properties and present several explicit examples.As an interesting consequence of this construction, we obtain example systems with nontrivial braiding statistics between string excitations.In addition to studying the string-string braiding in the example system, we propose a topological field-theory description for the layer-constructed systems, which captures not only the string-particle braiding statistics but also the string-string braiding statistics when the coupling is twisted.Last, we provide a proof of a general identity for Abelian string statistics and discuss an example system with non-Abelian strings.

Heidar Moradi, Xiao-Gang Wen

Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground-state wave functions. For systems with gapped boundaries, these quantities are representations of the mapping class group $\mathtt{MCG}(\mathcal{M})$ of the space manifold $\mathcal{M}$ on which the systems live. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to the fusion algebra and statistics of particles and string excitations. In particular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced 2+1D topological data contain information on the fusion and the braiding of non-Abelian string excitations in 3D. These universal quantities generalize the well-known modular $S$ and $T$ matrices to any dimension.