Condensation of fermion zero modes in the vortex

The energy levels of the fermions bound to the vortex are considered for vortices in the superfluid/superconducting systems that contain the symmetry protected plane of zeroes in the gap function in bulk. The Caroli–de Gennes–Matricon branches with different approach zero energy level at $p_z \to 0$. The density of states of the bound fermions diverges at zero energy giving rise to the $\sqrt{\Omega}$ dependence of the density of states in the polar phase of superfluid $^3$He rotating with the angular velocity $\Omega$ and to the $\sqrt{B}$ dependence of the density of states for superconductors in the ($d_{xz}+id_{yz}$)-wave pairing state.