Reentrant behaviour in Landau Fermi liquids with spin-split Pomeranchuk instabilities

We study the effects of spin-antisymmetric interactions on the stability of a Landau-Fermi liquid on the square lattice, using the generalized Pomeranchuk method for two-dimensional lattice systems. In particular, we analyze interactions that could induce instabilities of the so called spin-split type, that is when spin-up and spin-down Fermi surfaces are displaced with respect to each other. The phase space is studied as a function of the strength of the interaction $V$, the electron chemical potential $\mu$ and an external magnetic field $h$. We find that such interactions produce in general an enhancement of the instability region of the Landau-Fermi liquid. More interestingly, in certain regions of the $V$-$\mu$ phase space, we find a reentrant behaviour as a function of the magnetic field $h$, similar to that found in recent experiments, e.g. in URu${}_2$Si${}_2$ and Sr${}_3$Ru${}_2$O${}_7$.