Competition Between Induced Symmetry Breaking, Cooper Pairing and Chiral Condensate at Finite Density

We study the competition between induced symmetry breaking (ISB), Cooper pairing (superconductivity) and chiral condensation at finite non-asymptotic density. Using a quantum mechanical model studied recently by Ilieva and Thirring, we analyze the expectation value of the fermion number density $\la a^*a\ra$ in the presence of chemical potential and discuss relevance of the ISB phenomenon in the model. By Bogoliubov transformation, we obtain the ground state energy and find that a phase with both superconducting and mean field gap is a true ground state. We also study the effective potentials of a two-dimensional field theory model for competitions between $\la\bar\psi\psi\ra$, $\la\psi\psi\ra$ and $\la\psi^\dagger\psi\ra$. We find that in the regime $\mu <\mu_c$, where $\mu_c$ is found to be around $0.4 m_F$ with $m_F$ being the renormalizaton point of chiral condensate, we have a chiral symmetry-breaking phase with almost zero fermion number density and in the case of $\mu >\mu_c$ the system sits in the mixed phase characterized by $\la \psi^\dagger\psi\ra\neq 0$ and $\la \psi\psi\ra\neq 0$ as long as the renormalization point, $M_0$, corresponding to a superconducting gap is not zero, $M_0\neq 0$.