Nonconservation of the fermion number in a cold dense fermion medium in $V-A$ gauge theories

It is shown that in four-dimensional Abelian $V-A$ theories the ground state of cold neutral fermionic matter at sufficiently high densities is an essentially inhomogeneous state in which there is a domain of anomalous matter surrounded by normal vacuum. Within the domain, the condensate of the gauge field is nonzero; its appearance reduces effectively to zero the number of real fermions both within and without the domain. In four-dimensional non-Abelian $V-A$ theories the anomalous state of cold neutral fermionic matter is, in its turn, absolutely unstable, and the system goes over to the normal state with a small number of fermions above a topologically nontrivial vacuum. Thus, in a number of non-Abelian models the fermion density cannot exceed a certain critical value.