  An SU(4) model of high-temperature superconductivity and antiferromagnetism
has recently been proposed. The SO(5) group employed by Zhang is embedded in
this SU(4) as a subgroup, suggesting a connection between our SU(4) model and
the Zhang SO(5) model. In order to understand the relationship between the the
two models, we have used generalized coherent states to analyze the nature of
the SO(5) subgroup. By constructing coherent-state energy surfaces, we
demonstrate explicitly that the SU(4) supset SO(5) symmetry can be interpreted
as a critical dynamical symmetry interpolating between superconducting and
antiferromagnetic phases, and that this critical dynamical symmetry has many
similarities to critical dynamical symmetries identified previously in other
fields of physics. More generally, we demonstrate with this example that the
mathematical techniques associated with generalized coherent states may have
powerful applications in condensed matter physics because they provide a clear
connection between microscopic many-body theories and their broken-symmetry
approximate solutions. In addition, these methods may be interpreted as
defining the most general Bogoliubov transformation subject to a Lie group
symmetry constraint, thus providing a mathematical connection between algebraic
formulations and the language of quasiparticle theory. Finally, we suggest that
the identification of the SO(5) symmetry as a critical dynamical symmetry
implies deep algebraic connections between high-temperature superconductors and
seemingly unrelated phenomena in other field of physics.
