Electric-magnetic duality and topological order on the lattice

Institute for Advanced Study (Science Hall), Tsinghua University, Beijing 100084, China(Dated: July 1, 2010)Dualities are deep conceptual tools in many areas of physics and mathematics; in particular,electric-magnetic (EM) duality has a long tradition ranging from Dirac’s magnetic poles to S-dualityin string and field theory. Here we investigate the duality structure in quantum lattice systemswith topological order, a collective order also appearing in fractional quantum Hall systems. Wedefine EM duality for all of Kitaev’s quantum double models based on discrete gauge theories withAbelian and non-Abelian groups, and identify its natural habitat as a new class of topological modelsbased on Hopf algebras. We interpret these as extended string-net models, whereupon Levin andWen’s string-nets, which describe all topological orders on the lattice with parity and time-reversalinvariance, arise as magnetic and electric projections of the extended models. We conjecture thatall string-net models can be extended in an analogous way, using a more general algebraic structure,such that EM duality continues to hold. Physical applications include topology probes in the formof pairs of dual tensor networks.