Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

Abstract

Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group $G$, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded into module 2-categories over the input spherical fusion 2-category. Individual tensors are characterised by symmetry conditions with respect to non-local operators acting on entanglement degrees of freedom. In the case of Dirichlet and Neumann boundary conditions, we show that the symmetry operators form the fusion 2-categories ${2Vec}_G$ of $G$-graded 2-vector spaces and $2Rep(G)$ of 2-representations of $G$, respectively. In virtue of the Morita equivalence between ${2Vec}_G$ and $2Rep(G)$ -- which we explicitly establish -- the topological order can be realised as the Drinfel'd centre of either 2-category of operators; this is a realisation of the electromagnetic duality of the theory. Specialising to the case $G={\mathbb{Z}}_2$, we recover tensor network representations that were recently introduced, as well as the relation between the electromagnetic duality of a pure (3+1)d ${\mathbb{Z}}_2$ gauge theory and the Kramers-Wannier duality of a boundary (2+1)d Ising model.

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• Higher categorical symmetries and gauging in two-dimensional spin systems
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• String Condensations in 3+1D and Lagrangian Algebras
• Rigid and Separable Algebras in Fusion 2-Categories
• On weak Hopf symmetry and weak Hopf quantum double model
• Low-depth unitary quantum circuits for dualities in one-dimensional quantum lattice models
• Entanglement and the density matrix renormalisation group in the generalised Landau paradigm
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories
• Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models
• Systematic construction of stabilizer codes via gauging abelian boundary symmetries
• 2-Drinfel'd double symmetry of the 4d Kitaev model
• Les Houches Lecture Notes on Tensor Networks