Fermions, strings, and gauge fields in lattice spin models

TL;DR

This paper explores how fermionic excitations in lattice spin models inherently involve string-like operators and gauge fields, providing exactly solvable examples in two and three dimensions and a general algebraic framework applicable across dimensions.

Contribution

It introduces a universal algebraic method linking particle statistics to hopping operators, revealing the string and gauge field structure of fermions in lattice models.

Findings

Fermions in lattice models always appear in pairs with string-like creation operators.

Fermions couple to nontrivial gauge fields due to their string-like operators.

Exactly solvable models demonstrate these properties in 2D and 3D.

Abstract

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 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 an algebraic formula that relates the statistics of a lattice particle to the properties of its hopping operators. This approach has the advantage that it works in any number of dimensions - unlike the flux-binding picture developed in FQH theory.