Microscopic models of two-dimensional 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-$\frac{1}{2}$ (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. A magnetic moment coexists with the topological order. Our model is a member of a family of topologically ordered models, one of which is integrable and realizes the toric quantum error correction code but does not conserve any component of the spin. We relate our model to a dimer model which could be a spin $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}$ symmetric realization of topological order and its concomitant quantum number fractionalization.