We present a theoretical study of the ground state of the BCS-BEC crossover in dilute two-dimensional Fermi gases. While the mean-field theory provides a simple and analytical equation of state, the pressure is equal to that of a noninteracting Fermi gas in the entire BCS-BEC crossover, which is not consistent with the features of a weakly interacting Bose condensate in the BEC limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of the 2D mean-field theory indicates that the quantum fluctuations are much more pronounced than those in 3D. In this work, we show that the inclusion of the Gaussian quantum fluctuations naturally recovers the above features in both the BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on the boson chemical potential is recovered by the quantum fluctuations. Near the quantum phase transition from the vacuum to the BEC phase, we compare our equation of state with the known grand canonical equation of state of 2D Bose gases and determine the ratio of the composite boson scattering length a B to the fermion scattering length a 2D . We find a B ≃ 0.56a 2D , in good agreement with the exact four-body calculation. We compare our equation of state in the BCS-BEC crossover with recent results from the quantum Monte Carlo simulations and the experimental measurements and find good agreements.

The effects of chiral imbalance and external magnetic field on pion superfluidity and color superconductivity are investigated in extended Nambu-Jona-Lasinio models. We take Schwinger approach to treat the interaction between charged pion condensate and magnetic field at finite isospin density and include simultaneously the chiral imbalance and magnetic field at finite baryon density. For the superfluidity, the chiral imbalance and magnetic field lead to catalysis and inverse catalysis effects, respectively. For the superconductivity, the chiral imbalance enhances the critical baryon density, and the magnetic field results in a de Haas-van Alphan oscillation on the phase transition line.

We study the QCD vacuum structure under the influence of an electromagnetic field with a nonzero second Lorentz invariant I 2 = E · B. We show that the presence of I 2 can induce neutral pion (π 0 ) condensation in the QCD vacuum through the electromagnetic triangle anomaly. Within the frameworks of chiral perturbation theory at leading small-momenta expansion as well as the Nambu-Jona-Lasinio model at leading 1/N c expansion, we quantify the dependence of the π 0 condensate on I 2 . The stability of the π 0 -condensed vacuum against the Schwinger charged pair production due to electric field is also discussed.