Flop Transitions in Cuprate and Color Superconductors: From SO(5) to SO(10) Unification?

The phase diagrams of cuprate superconductors and of QCD at non-zero baryon chemical potential are qualitatively similar. The Neel phase of the cuprates corresponds to the chirally broken phase of QCD, and the high-temperature superconducting phase corresponds to the color superconducting phase. In the SO(5) theory for the cuprates the $SO(3)_s$ spin rotational symmetry and the $U(1)_{em}$ gauge symmetry of electromagnetism are dynamically unified. This suggests that the $SU(2)_L \otimes SU(2)_R \otimes U(1)_B$ chiral symmetry of QCD and the $SU(3)_c$ color gauge symmetry may get unified to SO(10). Dynamical enhancement of symmetry from $SO(2)_s \otimes \Z(2)$ to $SO(3)_s$ is known to occur in anisotropic antiferromagnets. In these systems the staggered magnetization flops from an easy 3-axis into the 12-plane at a critical value of the external magnetic field. Similarly, the phase transitions in the SO(5) and SO(10) models are flop transitions of a ``superspin''. Despite this fact, a renormalization group flow analysis in $4-\epsilon$ dimensions indicates that a point with full SO(5) or SO(10) symmetry exists neither in the cuprates nor in QCD.