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We construct an exactly solvable Hamiltonian acting on a 3-dimensionallattice of spin-$\frac 1 2$ systems that exhibits topological quantum order.The ground state is a string-net and a membrane-net condensate. Excitationsappear in the form of quasiparticles and fluxes, as the boundaries of stringsand membranes, respectively. The degeneracy of the ground state depends uponthe homology of the 3-manifold. We generalize the system to $D\geq 4$, weredifferent topological phases may occur. The whole construction is based oncertain special complexes that we call colexes.Comment: Revtex4 file, color figures, minor correction

physical review. b, condensed matter and materials physics

Exact topological quantum order in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math>and beyond: Branyons and brane-net condensates

Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates