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

TL;DR

This paper investigates the specific conditions under which $l=1$ Pomeranchuk instabilities occur in a Fermi liquid, revealing that conservation laws prevent such instabilities for certain current order parameters but not for generic forms.

Contribution

The study clarifies how conservation laws influence $l=1$ Pomeranchuk instabilities, showing they prevent instability for specific current order parameters but allow it for generic form-factors.

Findings

Conservation laws prevent $l=1$ Pomeranchuk instabilities for certain current order parameters.

Generic $l=1$ form-factors can lead to instabilities when $F^{c(s)}_1=-1$.

Pomeranchuk instabilities occur at $F^{c(s)}_l=-1$ for other angular momenta $l$.

Abstract

We perform a microscropic 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 $F^{c(s)}_l =-1$, where $F^{c(s)}_l$ (a Landau parameter) is a properly normalized partial component of the anti-symmetrized static interaction $F(k,k+q; p,p-q)$ in a charge (c) or spin (s) sub-channel 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 $F^{c(s)}_{l=1}=-1$, 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 $F^{c(s)}_{l=1}$, and a Pomeranchuk instability does occur when $F^{c(s)}_1=-1$. We argue that for other values of $l$, a Pomeranchuk instability occurs at $F^{c(s)}_{l} =-1$ for an order parameter with any form-factor