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Color Magnetism in Non-Abelian Vortex Matter

We propose color magnetism as a generalization of the ordinary Heisenberg (anti-)ferro magnets on a triangular lattice. Vortex matter consisting of an Abrikosov lattice of non-Abelian vortices with color magnetic fluxes shows a color ferro or anti-ferro magnetism, depending on the interaction among the vortex sites. A prime example is a non-Abelian vortex lattice in rotating dense quark matter, showing a color ferromagnetism. We show that the low-energy effective theory for the vortex lattice system in the color ferromagnetic phase is described by a 3+1 dimensional $CP^{N-1}$ nonlinear sigma model with spatially anisotropic couplings. We identify gapless excitations independent from Tkachenko modes as color magnons, that is, Nambu-Goldstone modes propagating in the vortex lattice with an anisotropic linear dispersion relation $ω_p^2 = c_{xy}^2(p_x^2+p_y^2) + c_z^2 p_z^2$. We calculate the transition temperature between the ordered and disordered phases, and apply it to dense quark matter. We also identify the order parameter spaces for color anti-ferromagnets.