Electric-magnetic duality of lattice systems with topological order

TL;DR

This work identifies electric-magnetic duality as a unifying principle for lattice systems with topological order, extending beyond Abelian toric codes to non-Abelian quantum doubles by placing them within the Hopf-algebra framework $[\mathrm{D}(H)]_\Lambda$ and their duals $[\mathrm{D}(H^*)]_{\Lambda^*}$. It introduces extended string-net models $[\mathrm{ESN}_{(\mathcal{C},\omega)}]_\Lambda$ via a fiber functor $\omega$, connects them to tensor-categorical duals $\mathcal{C}^*$ through Tannaka-Krein duality, and shows EM duality is implemented by invertible domain walls between dual lattice systems. By projecting ESN models to ordinary string-nets, the paper derives a duality between magnetic and electric projections, relates these to Morita equivalences, and identifies the conditions under which all SN models may be accommodated (via weak $C^*$-Hopf algebras). It culminates in a framework where topology is measurable through dual tensor-network states, linking lattice models, tensor categories, and domain-wall physics, with implications for experimental realizations and further dualities among topological phases.

Abstract

We investigate the duality structure of quantum lattice systems with topological order, a collective order also appearing in fractional quantum Hall systems. We define electromagnetic (EM) duality for all of Kitaev's quantum double models based on discrete gauge theories with Abelian and non-Abelian groups, and identify its natural habitat as a new class of topological models based on Hopf algebras. We interpret these as extended string-net models, whereupon Levin and Wen's string-nets, which describe all intrinsic topological orders on the lattice with parity and time-reversal invariance, arise as magnetic and electric projections of the extended models. We conjecture that all string-net models can be extended in an analogous way, using more general algebraic and tensor-categorical structures, such that EM duality continues to hold. We also identify this EM duality with an invertible domain wall. Physical applications include topology measurements in the form of pairs of dual tensor networks.

Figures (7)

• Figure 1: The duality landscape of quantum double models. We represent: Quantum doubles $\mathrm{D} (A)$ based on Abelian groups, including the toric code, all of which are self-dual; general group quantum double models $\mathrm{D} (G)$, related by duality to quantum double models $\mathrm{D} (G^\ast)$ based on algebras of functions on groups, whose intersection with $\mathrm{D} (G)$ is $\mathrm{D} (A)$; these are all instances of quantum double models $\mathrm{D} (H)$ based on $C^\ast$-Hopf algebras, a larger class closed under duality, with cases of self-duality beyond groups BMCA. We conjecture that EM duality can be defined for all extended string-net models $\mathrm{ESN}$, and that these can be identified as quantum double models $\mathrm{D} (W)$ based on weak $C^\ast$-Hopf algebras. Arrows denote model identifications by EM duality.
• Figure 2: Projectors in the $\mathrm{D} (G)$ model. Vertex ($A_v$) and plaquette ($B_p$) projectors in the Hamiltonian of Kitaev's $\mathrm{D} (G)$ model are defined by their action on a basis of the corresponding multi-edge Hilbert space. Oriented edges are labelled by group elements, denoting kets in the orthonormal computational basis $\{ | g \rangle \}_{ g \in G }$; change of orientation is implemented by applying the inversion map $g \mapsto g^{-1}$. The vertex operator projects onto gauge-invariant configurations, that is, symmetrises over simultaneous multiplication with group elements. The plaquette operator involves a Kronecker delta on the unit element $e \in G$ and projects onto trivial magnetic flux across $p$.
• Figure 3: Structure and dualities in Hopf algebras. The first row represents the structure maps of a Hopf algebra $H$. The second and third rows illustrate the algebraic duality within the class of Hopf algebras, taking as examples a group algebra $H = \mathbb{C} G$ and its dual $H^\ast$, whose structure is defined implicitly in the figure.
• Figure 4: The core of $\mathrm{D} (H)$ models. Topological models based on Hopf algebras are defined via these vertex and plaquette representations of elements $u$ of Hopf algebra $H$ and elements $f$ of the dual $H^\ast$.
• Figure 5: Going over to the dual lattice. The mapping $U\colon H \rightarrow H^\ast$ is here associated with a lattice dualisation $\Lambda \rightarrow \Lambda^\ast$.