Fermions, strings, and gauge fields in lattice spin models

We investigate the general properties of lattice spin models with emerging fermionic excitations. We argue that fermions always come in pairs and their creation operator always has a stringlike structure with the newly created particles appearing at the end points of the string. The physical implication of this structure is that the fermions always couple to a nontrivial gauge field. We present exactly soluble examples of this phenomenon in two and three dimensions. Our analysis is based on an algebraic formula that relates the statistics of a lattice particle to the properties of its hopping operators. This approach has the advantage in that it works in any number of dimensions—unlike the flux-binding picture developed in fractional quantum Hall theory.