Yi-Ming Wu,Avraham Klein,Andrey V. Chubukov

We perform a microscopic analysis of how the constraints imposed by conservation laws affect $q=0$ Pomeranchuk instabilities in a Fermi liquid. The conventional view is that these instabilities are determined by the static interaction between low-energy quasiparticles near the Fermi surface, in the limit of vanishing momentum transfer $q$. The condition for a Pomeranchuk instability is set by ${F}_{l}^{c\left(s\right)}=-1$, where ${F}_{l}^{c\left(s\right)}$ (a Landau parameter) is a properly normalized partial component of the antisymmetrized static interaction $F(k,k+q;p,p-q)$ in a charge (c) or spin (s) subchannel with angular momentum $l$. However, it is known that conservation laws for total spin and charge prevent Pomeranchuk instabilities for $l=1$ spin- and charge-current order parameters. Our study aims to understand whether this holds only for these special forms of $l=1$ order parameters or is a more generic result. To this end we perform a diagrammatic analysis of spin and charge susceptibilities for charge and spin density order parameters, as well as perturbative calculations to second order in the Hubbard $U$. We argue that for $l=1$ spin-current and charge-current order parameters, certain vertex functions, which are determined by high-energy fermions, vanish at ${F}_{l=1}^{c\left(s\right)}=-1$, preventing a Pomeranchuk instability from taking place. For an order parameter with a generic $l=1$ form factor, the vertex function is not expressed in terms of ${F}_{l=1}^{c\left(s\right)}$, and a Pomeranchuk instability may occur when ${F}_{1}^{c\left(s\right)}=-1$. We argue that for other values of $l$, a Pomeranchuk instability may occur at ${F}_{l}^{c\left(s\right)}=-1$ for an order parameter with any form factor.