Rotor spectra, berry phases, and monopole fields: From antiferromagnets to QCD

The order parameter of a finite system with a spontaneously broken continuous global symmetry acts as a quantum mechanical rotor. Both antiferromagnets with a spontaneously broken $SU(2{)}_s$ spin symmetry and massless QCD with a broken $SU(2{)}_L\times SU(2{)}_R$ chiral symmetry have rotor spectra when considered in a finite volume. When an electron or hole is doped into an antiferromagnet or when a nucleon is propagating through the QCD vacuum, a Berry phase arises from a monopole field and the angular momentum of the rotor is quantized in half-integer units.