  We have recently shown that dense quark matter possesses a color
ferromagnetic phase in which a stable color magnetic field arises
spontaneously. This ferromagnetic state has been known to be Savvidy vacuum in
the vacuum sector. Although the Savvidy vacuum is unstable, the state is
stabilized in the quark matter. The stabilization is achieved by the formation
of quantum Hall states of gluons, that is, by the condensation of the gluon's
color charges transmitted from the quark matter. The phase is realized between
the hadronic phase and the color superconducting phase. After a review of
quantum Hall states of electrons in semiconductors, we discuss the properties
of quantum Hall states of gluons in quark matter in detail. Especially, we
evaluate the energy of the states as a function of the coupling constant. We
also analyze solutions of vortex excitations in the states and evaluate their
energies. We find that the states become unstable as the gauge coupling
constant becomes large, or the chemical potential of the quarks becomes small,
as expected. On the other hand, with the increase of the chemical potential,
the color superconducting state arises instead of the ferromagnetic state. We
also show that the quark matter produced by heavy ion collisions generates
observable strong magnetic field $\sim 10^{15}$ Gauss when it enters the
ferromagnetic phase.
