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arXiv:1404.7854 · Physical Review B

Non-Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3+1) -dimensional twisted gauge theory

Juven Wang, Xiao-Gang Wen

Published Jan 29, 2015• 123 citations
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Abstract

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.

DOI
10.1103/physrevb.91.035134

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String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group GGG and a 4-cocycle twist \ensuremathω4{\ensuremath{\omega}}_{4}\ensuremathω4​ of GGG's cohomology group H4(G,R/Z){\mathcal{H}}^{4}(G,\mathbb{R}/\mathbb{Z})H4(G,R/Z) in three-dimensional space and one-dimensional time (3+1D)(3+1\mathrm{D})(3+1D). We establish the topological spin and the spin-statistics relation for the closed strings and their multistring braiding statistics. The 3+1D3+1\mathrm{D}3+1D twisted gauge theory can be characterized by a representation of a modular transformation group, SL(3,Z)\mathrm{SL}(3,\mathbb{Z})SL(3,Z). We express the SL(3,Z)\mathrm{SL}(3,\mathbb{Z})SL(3,Z) generators Sxyz{\mathsf{S}}^{xyz}Sxyz and Txy{\mathsf{T}}^{xy}Txy in terms of the gauge group GGG and the 4-cocycle \ensuremathω4{\ensuremath{\omega}}_{4}\ensuremathω4​. As we compactify one of the spatial directions zzz into a compact circle with a gauge flux bbb inserted, we can use the generators Sxy{\mathsf{S}}^{xy}Sxy and Txy{\mathsf{T}}^{xy}Txy of an SL(2,Z)\mathrm{SL}(2,\mathbb{Z})SL(2,Z) subgroup to study the dimensional reduction of the 3D topological order C3D{\mathcal{C}}^{3\mathrm{D}}C3D to a direct sum of degenerate states of 2D topological orders Cb2D{\mathcal{C}}_{b}^{2\mathrm{D}}Cb2D​ in different flux bbb sectors: C3D=\ensuremath⨁bCb2D{\mathcal{C}}^{3\mathrm{D}}={\ensuremath{\bigoplus}}_{b}{\mathcal{C}}_{b}^{2\mathrm{D}}C3D=\ensuremath⨁b​Cb2D​. The 2D topological orders Cb2D{\mathcal{C}}_{b}^{2\mathrm{D}}Cb2D​ are described by 2D gauge theories of the group GGG twisted by the 3-cocycle \ensuremathω3(b){\ensuremath{\omega}}_{3(b)}\ensuremathω3(b)​, dimensionally reduced from the 4-cocycle \ensuremathω4{\ensuremath{\omega}}_{4}\ensuremathω4​. We show that the SL(2,Z)\mathrm{SL}(2,\mathbb{Z})SL(2,Z) generators, Sxy{\mathsf{S}}^{xy}Sxy and Txy{\mathsf{T}}^{xy}Txy, 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.
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.
(3+1)(3+1)(3+1)-dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for nontrivial loop-loop and point-loop braiding statistics not permitted to point-like excitations alone. Furthermore, these loop-like excitations can be linked together, changing their properties. In particular, this can lead to distinct three-loop braiding, involving two loops undergoing an exchange process while linked to a third loop. In this work, we investigate the loop-like excitations in a (3+1)(3+1)(3+1)-dimensional Hamiltonian realization of Dijkgraaf-Witten theory through direct construction of their membrane operators, for a general finite Abelian group and 4-cocycle twist. Using these membrane operators, we find the braiding relations and fusion rules for the loop-like excitations, including those linked to another loop-like excitation. Furthermore, we use these membrane operators to construct projection operators that measure the topological charge and show that the number of distinct topological charges measured by the 2-torus matches the ground-state degeneracy of the model on the 3-torus, explicitly confirming a general expectation for topological phases. This direct construction of the membrane operators sheds significant light on the key properties of the loop-like excitations in (3+1)(3+1)(3+1)-dimensional topological phases.
Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be ``twisted,'' in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.
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 MCG(M)\mathtt{MCG}(\mathcal{M})MCG(M) of the space manifold M\mathcal{M}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 SSS and TTT matrices to any dimension.
We investigate whether the braid group representation on NNN anyons uniquely determines the modular data (SSS and TTT matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local NNN-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global SSS-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon RRR-matrix with the asymptotic Hecke relation σi2=1\sigma_i^2 = 1σi2​=1 (up to phase), which rigidly fixes the local FFF and RRR symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ), precluding alternative non-Abelian fusion rules. The TTT-matrix is determined by topological spins, but the full SSS-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory. ## 1. Introduction Topological quantum computation exploits the non-Abelian statistics of anyons—quasiparticle excitations in two-dimensional systems whose braiding operations are represented by non-commuting unitary matrices on a degenerate ground-state manifold [1]. A central question is whether the local braiding data of NNN anyons suffices to reconstruct the full modular data of the underlying topological order (TO), or whether additional global information is required. This question has direct physical consequences for 2D topological superconductors (TSCs) hosting Majorana zero modes (MZMs), where the anyonic excitations are vortices binding MZMs and the fermionic quasiparticle is a transparent (local) fermion [2,3]. The modular data of a TO consists of the SSS-matrix, which encodes the mutual braiding statistics of anyons, and the TTT-matrix, which encodes the topological spins. Together, these matrices form a projective representation of t
We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+13+13+1 dimensions, based on a generic finite group GGG and a 4-cocycle $\ensuremath{\omega}$ over GGG. 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 quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3,Z)SL(3,\mathbb{Z})SL(3,Z) generators as the modular SSS and TTT matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the (3+1)(3+1)(3+1)-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group GGG. This work is presented to be accessible for a wide range of physicists and mathematicians.