Spin Conservation and Fermi Liquid near a Ferromagnetic Quantum Critical Point

We propose a new low-energy theory for itinerant fermions near a ferromagnetic quantum critical point. We show that the full low-energy model includes, in addition to conventional interaction via spin fluctuations, another type of interaction, whose presence is crucial for the theory to satisfy SU(2) spin conservation. We demonstrate the consistency between a loopwise expansion and a Fermi liquid description for the full model. We further show that, prior to the ferromagnetic instability, the system develops a Pomeranchuk-type instability into a state with zero magnetization but with $p$-wave deformations of the Fermi surfaces of spin-up and spin-down electrons (a spin nematic).

Figure 1

(a) Ladder diagrams for ${\Gamma }^{\Omega }$. Summing up these diagrams for a contact interaction $U$ (dashed line) and rearranging spin indices, one obtains the effective spin-spin interaction near the critical point (wavy line), where $Um/2\pi \approx 1$. The rest of the diagrams (not shown) eliminate the instability in the charge channel. (b) Renormalization of the spin-fermion vertex. The last two diagrams are the Aslamazov-Larkin—type (AL) contributions which restore spin conservation. (c) The full effective interaction ${\Gamma }^{\Omega ,\mathrm{full}}$ with contributions from the direct spin-spin interaction and AL-type processes.

Figure 2

Schematic evolution of the partial components of the full Landau function [Eq. (7)] vs the inverse mass renormalization parameter ${\lambda }^{-1}$. All charge and even spin Landau components are positive but diverge near a FM QCP, while odd spin components are negative. The spin-nematic component $g_{s,1}$ reaches the Pomeranchuk critical value of $-1$ first.