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We study a 3D generalization of the toric code model introduced recently byChamon. This is an exactly solvable spin model with six-qubit nearest neighborinteractions on an FCC lattice whose ground space exhibits topological quantumorder. The elementary excitations of this model which we call monopoles can begeometrically described as the corners of rectangular-shaped membranes. Weprove that the creation of an isolated monopole separated from other monopolesby a distance R requires an operator acting on at least R^2 qubits. Compositeparticles that consist of two monopoles (dipoles) and four monopoles(quadrupoles) can be described as end-points of strings. The peculiar featureof the model is that dipole-type strings are rigid, that is, such strings mustbe aligned with face-diagonals of the lattice. For periodic boundary conditionsthe ground space can encode 4g qubits where g is the greatest common divisor ofthe lattice dimensions. We describe a complete set of logical operators actingon the encoded qubits in terms of closed strings and closed membranes.

Topological order in an exactly solvable 3D spin model