The thermal and quantum phase transitions are studied basing on theCornwall-Jackiw-Tomboulis (CJT) effective action approach for the relativisticlinear sigma model of the two-component mixing system. After obtaining theexpression of the thermodynamic potential in Hartree-Fock (HF) approximation,which preserves the Goldstone theorem, the numerical results show that there maybe two phase transition scenarios in the system. The first scenario is the thermalphase transition which can only occur in one component. The second scenario isquantum phase transitions which can occur in both two components. In addition,both these two types of phase transitions belong the second order phase transition.