  We study the formation of spontaneous spin polarization in inhomogeneous
electron systems with pair interaction localized in a small region that is not
separated by a barrier from surrounding gas of non-interacting electrons. Such
a system is interesting as a minimal model of a quantum point contact, in which
the electron-electron interaction is strong in a small constriction coupled to
electron reservoirs without barriers. Based on the analysis of the grand
potential within the self-consistent field approximation, we find that the
formation of the polarized state strongly differs from the Bloch or Stoner
transition in homogeneous interacting systems. The main difference is that a
metastable state appears in the critical point in addition to the globally
stable state, so that when the interaction parameter exceeds a critical value,
two states coexist. One state has spin polarization and the other is
unpolarized. Another feature is that the spin polarization increases
continuously with the interaction parameter and has a square-root singularity
in the critical point. We study the critical conditions and the grand
potentials of the polarized and unpolarized states for one-dimensional and
two-dimensional models in the case of extremely small size of the interaction
region.
