Microscopic Models of 2D Magnets with Fractionalized Excitations

We demonstrate that spin-charge separation can occur in two dimensions and note its confluence with superconductivity, topology, gauge theory, and fault-tolerant quantum computation. We construct a microscopic Ising-like model and, at a special coupling constant value, find its exact ground state as well as neutral spin 1/2 (spinon), spinless charge e (holon), and $Z_2$ vortex (vison) states and energies. The fractionalized excitations reflect the topological order of the ground state which is evinced by its fourfold degeneracy on the torus -- a degeneracy which is unrelated to translational or rotational symmetry -- and is described by a $Z_2$ gauge theory. Our model is related to the quantum dimer model and is a member of a family of topologically-ordered models, one of which is integrable and realizes the toric quantum error correction code.