New phenomenology from an old theory--new equilibrium states in the BCS model

I analyze the low temperature limit of the BCS theory of s-wave single-band superconductors, when the attraction band may be asymmetric with respect to the chemical potential. I discuss equilibrium systems, taking consistently into account the variation of the energy and of the total number of particles with the populations of the quasiparticle energy levels. I show that the equation for the energy gap has two solutions, one of which is stable and the other one is metastable. When the chemical potential is the center of the attraction band (the standard BCS assumption), the energy gap in the stable solution is $\Delta_0$, whereas in the metastable one is $\Delta_0/3$. If the chemical potential is not in the center of the attraction band, then a quasiparticle imbalance appears. If the absolute value of the difference between the chemical potential and center of the attraction band is bigger than $2\Delta_0$, then the superconducting energy gap cannot be formed. If the number of particles is conserved and the attraction band is asymmetric, then the stable solution is unphysical and only metastable solutions are realized.