Recently, there has been significant interest in understanding the macroscopic quantum transport in a many-body system of chiral fermions. A natural framework for describing such a system that is generally out of equilibrium is the transport equation for its phase space distribution function. In this paper, we obtain a complete solution of the covariant chiral transport for massless fermions, by starting from the general Wigner function formalism and carrying out a complete and consistent semiclassical expansion up to $\widehat{\mathbf{O}}(\hslash )$ order. In particular, we clarify certain subtle and confusing issues surrounding the Lorentz noninvariance and frame dependence associated with the three-dimensional chiral kinetic theory. We prove that such frame dependence is uniquely and completely fixed by an unambiguous definition of the $\widehat{\mathbf{O}}(\hslash )$ correction to the distribution function in each reference frame.

The BEC-BCS (Bose-Einstein condensation–Bardeen-Cooper-Shriffer) crossover in QCD at finite baryon and isospin chemical potentials is investigated in the Nambu-Jona-Lasinio model. The diquark condensation in two color QCD and the pion condensation in real QCD would undergo a BEC-BCS crossover when the corresponding chemical potential increases. We determined the crossover chemical potential as well as the BEC and BCS regions. The crossover is not triggered by increasing the strength of attractive interaction among quarks but driven by changing the charge density. The chiral symmetry restoration at finite temperature and density plays an important role in the BEC-BCS crossover. For real QCD, strong couplings in diquark and vector meson channels can induce a diquark BEC-BCS crossover in color superconductor, and in the BEC region the chromomagnetic instability is fully cured and the ground state is a uniform phase.

We study fermion mass correction to chiral kinetic equations in electromagnetic fields. Different from the chiral limit where fermion number density is the only independent distribution, the number and spin densities are coupled to each other for massive fermion systems. To the first order in $\hslash$, we derived the quantum correction to the classical on-shell condition and the Boltzmann-type transport equations. To the linear order in the fermion mass, the mass correction does not change the structure of the chiral kinetic equations and behaves like additional collision terms. While the mass correction exists already at classical level in general electromagnetic fields, it is only a first-order quantum correction in the study of the chiral magnetic effect.

Four different phases of the two-flavor quark system, i.e., the normal quark matter and color superconducting phases with and without the charge neutrality condition, are investigated in the framework of the SU(2) Nambu–Jona-Lasinio model. It is found that the color superconducting phase without charge neutrality has the lowest thermodynamic potential, and the charge neutral systems have higher thermodynamic potentials. However, the BCS pairing always lowers the system’s thermodynamic potential, i.e., for both the charged and charge neutral systems, the superconducting phases always have lower thermodynamic potentials. Compared with the BCS gap ${\Delta }_0$ for the color superconducting phase without charge neutrality, the BCS gap $\Delta$ in the charge neutral color superconducting phase has been largely reduced. When the thermodynamic potential for the charge neutral color superconducting phase equals that in the neutral normal quark matter, the diquark condensate disappears.

A general analysis for the competition between breached pairing (BP) and LOFF pairing mechanisms in asymmetric fermion superfluids is presented in the frame of a four fermion interaction model. Two physical conditions which can induce mismatched Fermi surfaces are considered: (1) fixed chemical potential asymmetry $\delta \mu$ and (2) fixed fermion number asymmetry $\alpha$. In case (1), the BP state is ruled out because of Sarma instability, and the LOFF state is thermodynamically stable in a narrow window of $\delta \mu$. In case (2), while the Sarma instability can be avoided and both the BP and LOFF states can survive provided $\alpha$ is less than the corresponding critical value, the BP state suffers magnetic instability and the LOFF state is always thermodynamically stable. While the LOFF window in case (2) is much larger than the one in the conventional case (1), the longitudinal superfluid density of the LOFF state is negative for small $\alpha$ and it suffers magnetic instability too.

The Green-function approach is extended to the moderate baryon density region in the framework of an extended Nambu–Jona-Lasinio model, and the thermodynamic potential with both chiral and diquark condensates evaluated by using the massive quark propagator. The phase structure along the chemical potential direction is investigated and the strong competition between the chiral and diquark condensate is analyzed by investigating the influence of the diquark condensate on the sharp Fermi surface. The influence of the diquark condensate on the quark properties is investigated, even though the quarks in the color-breaking phase are very different from those in the chiral-breaking phase; the difference between quarks in different colors is very small.