Rotor Spectra, Berry Phases, and Monopole Fields: from Graphene to Antiferromagnets and QCD

Nonrelativistic electrons hopping on the honeycomb lattice of graphene emerge as massless Dirac fermions. When the on-site repulsion between electrons on a honeycomb lattice exceeds a critical value, as it is the case for the dehydrated precursor of the high-temperature superconductor Na2CoO2 × yH2O, the system spontaneously breaks its SU(2)s spin symmetry and becomes an antiferromagnet. The emergence of antiferromagnetism is analogous to the spontaneous breakdown of the SU(2)L ×SU(2)R chiral symmetry in QCD. Just as the low-energy physics of pions and nucleons is described by baryon chiral perturbation theory, magnons and holes in an antiferromagnet are also described by a systematic low-energy effective theory. Both the chiral condensate in QCD and the staggered magnetization in an antiferromagnet act as a quantum mechanical rotor when the theory is put in a finite volume. When a nucleon is propagating through the QCD vacuum or when a hole is doped into an antiferromagnet, a Berry phase arises from a geometric monopole gauge field and the angular momentum of the rotor is quantized in half-integer units. The finite-size effects of the rotor spectrum depend on the low-energy parameters of the corresponding effective theories.