The effect of finite isospin chemical potential to the effective masses of the mesons at finite temperature is investigated in the framework of the O(4) linear sigma model with explicit chiral symmetry breaking. We present a mechanism to include the isospin chemical potential in the model. By using the Cornwall-Jackiw-Tomboulis method of composite operators, we obtain a set gap equations for the effective masses of the mesons and get the numerical results in the Hartree approximation. We find that the introduction of the chemical potential only affects the mass of the charged pions and sigma, while there is almost NO effects on the mass of neutral pions.

In mean field approximation, we study a chiral soliton of the linear sigma model with two flavors at finite temperature and density. The stable soliton solutions are calculated with some appropriate boundary conditions. Energy and radius of the soliton are determined in a hot medium of constituent quarks. It is found that for $TT_c$, there is a sharp delocalization phase transition from hadron matter to quark matter coincident with the restoration of chiral symmetry. In the transition region, the thermodynamic properties show large discontinuities which is an indication for a first-order phase transition.Comment: 14 pages, 7 figures, version accepted for publication in Physical Review

The chiral condensate and dressed Polyakov loop at finite temperature and density have been investigated in the framework of the $N_f=2+1$ Nambu-Jona-Lasinio (NJL) model with two degenerate $u$, $d$ quarks and one strange quark. In the case of explicit chiral symmetry breaking with physical quark masses, it is found that the phase transitions for light $u$, $d$, and $s$ quark are sequentially happened, and the separation between the transition lines for different flavors becomes wider and wider with the increase of baryon density. For each flavor, the pseudocritical temperatures for chiral condensate and dressed Polyakov loop differ in a narrow transition range in the lower baryon density region, and the two transitions coincide in the higher baryon density region.

The improved quark mass density-dependent model (IQMDD) based on soliton bag model is studied at finite temperature. Applying the finite temperature field theory, the effective potential of the IQMDD model and the bag constant $B(T)$ have been calculated at different temperatures. It is shown that there is a critical temperature $T_C\simeq 110\mathrm{MeV}$. We also calculate the soliton solutions of the IQMDD model at finite temperature. It turns out that when $T<T_C$, there is a bag constant $B(T)$ and the soliton solutions are stable. However, when $T>T_C$ the bag constant $B(T)=0$ and there is no soliton solution, therefore, the confinement of quarks are removed quickly.