SO(5) as a Critical Dynamical Symmetry in the SU(4) Model of   High-Temperature Superconductivity
  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. 
Lian-Ao Wu, Mike Guidry, Yang Sun, Cheng-Li Wu