Fermions, strings, and gauge fields in lattice spin models

In this paper, we investigate the general properties of lattice spin models with emerging fermionic excitations in any dimensions. We argue that fermions always come in pairs and their creation operator always has a string-like structure with the newly created particles appearing at the endpoints 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 2 and 3 dimensions. Our analysis is based on a novel algebraic formula which allows one to compute the statistics of a lattice particle in terms of its hopping operators. This algebra allows us to calculate the statistics of a particle from the properties of the corresponding string operator.