Contact potential instability in the path-integral description of itinerant ferromagnetism

Abstract: It has long been predicted that a two-component non-localized Fermi gas will exhibit spontaneous polarization for sufficiently strong repulsive interactions, a phenomenon which is called itinerant ferromagnetism. Recent experiments with ultracold atomic gases have reached the interaction strength for which theoretical models have predicted the occurrence of the normal-to-itinerant-ferromagnetic phase transition, but so far this transition has not been observed. The instability of the repulsive branch of the Feshbach resonance prevents the formation of the itinerant ferromagnetic state, but it is not clear whether this is the only instability impeding its experimental realization. In this article, we use the path-integral formalism with density fields in the Hubbard-Stratonovich transformation to study the stability of a homogeneous two-component Fermi gas with contact interactions. Within the saddle-point approximation we show that none of the extrema of the action are minima, meaning all extrema are unstable to small density fluctuations. This implies a more general mechanical instability of the polarized (itinerant ferromagnetic) and normal states of the system in the path-integral formalism. We find that it is important to consider the stability of the system when studying itinerant ferromagnetism. Since (mechanical) stability may be influenced by the details of the interaction potential, we suggest the use of a more realistic potential than the contact potential in future theoretical descriptions.