Emergent spin

• The spin–statistics theorem is not required for particles on a lattice.
• Spin emerges dynamically when spinless fermions have a relativistic continuum limit.
• Graphene and staggered fermions are examples of this phenomenon.
• The phenomenon is intimately tied to chiral symmetry and fermion doubling.
• Anomaly cancellation is a crucial feature of any valid lattice fermion action.

Quantum mechanics and relativity in the continuum imply the well known spin–statistics connection. However for particles hopping on a lattice, there is no such constraint. If a lattice model yields a relativistic field theory in a continuum limit, this constraint must “emerge” for physical excitations. We discuss a few models where a spin-less fermion hopping on a lattice gives excitations which satisfy the continuum Dirac equation. This includes such well known systems such as graphene and staggered fermions.