The Conditions for $l=1$ Pomeranchuk Instability in a Fermi Liquid

We analyze the forms of static spin and charge susceptibilities in $l >0$ channels in a Galilean-invariant Fermi liquid, to verify recent claim that $l=1$ Pomeranchuk instability in a Fermi liquid is absent. We first show that, because a charge or a spin order parameter with $l>0$ is, in general not a conserved quantity, the corresponding susceptibility, $\chi_l$, has contributions from states near and away from the Fermi surface. We discuss a generic form of $\chi_l$ and show the results for $l=1$ and $l=2$ to second order in the Hubbard $U$. For $l=1$, we show that $\chi_1$ changes qualitatively between the special case when the order parameter coincides with the current (the form-factor is $A {\bf k}$), and a generic case when the form-factor is ${\bf k} f(|k|)$ and $f(|k|)$ is not a constant. In the first case, the corrections to free-fermion result for $\chi_{l=1}$ disappear, even in the spin case, and $p-$wave Pomeranchuk instability does not occur. In perturbative calculations, this comes about due to a particular cancellation between contributions to $\chi_1$ from states near and away from the Fermi surface. In a generic case, we found that such cancellation does not occur and p-wave spin Pomeranchuk instability develops for strong enough interaction.