A formalism is presented for treating strongly-correlated graphene quantum Hall states
in terms of an SO(8) fermion dynamical symmetry that includes pairing as well as
particle--hole generators. The graphene SO(8) algebra is isomorphic to an SO(8) algebra
that has found broad application in nuclear physics, albeit with physically very different
generators, and exhibits a strong formal similarity to SU(4) symmetries that have been
proposed to describe high-temperature superconductors. The well-known SU(4)
symmetry of quantum Hall ferromagnetism for graphene is recovered as one subgroup
of SO(8), but the dynamical symmetry structure associated with the full set of SO(8)
subgroup chains allows analytical many-body solutions for a rich set of collective states
exhibiting spontaneously-broken symmetry that may be important for the low-energy
physics of graphene in strong magnetic fields. The SO(8) symmetry permits a natural
definition of generalized coherent states that correspond to symmetry-constrained
Hartree--Fock--Bogoliubov solutions exhibiting the interplay between competing
spontaneously broken symmetries in determining the ground state.
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