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Exactly soluble spin-$\frac{1}2$ models on three-dimensional lattices areproposed by generalizing Kitaev model on honeycomb lattice to three dimensionswith proper periodic boundary conditions. The simplest example is spins on adiamond lattice which is exactly soluble. The ground state sector of the modelmay be mapped into a p-wave paired state on cubic lattice. We observe for thefirst time a topological phase transition from a gapless phase to a gappedphase in an exactly soluble spin model. Furthermore, the gapless phase can notbe gapped by a perturbation breaking the time reversal symmetry. Unknotted andunlinked Wilson loops arise as eigen excitations, which may evolute into linkedand knotted loop excitations. We show that these closed string excitations obeyabelian statistics in the gapped phase and non-abelian statistics in thegapless phase.

Exactly soluble spin-1/2 models on three-dimensional lattices and non-abelian statistics of closed string excitations