Limits on dynamically generated spin-orbit coupling: Absence of $l=1$ Pomeranchuk instabilities in metals

An ordered state in the spin sector that breaks parity without breaking time-reversal symmetry, i.e., that can be considered dynamically generated spin-orbit coupling, was proposed to explain puzzling observations in a range of different systems. Here, we derive severe restrictions for such a state that follow from a Ward identity related to spin conservation. It is shown that $l=1$ spin-Pomeranchuk instabilities are not possible in nonrelativistic systems since the response of spin-current fluctuations is entirely incoherent and nonsingular. This rules out relativistic spin-orbit coupling as an emergent low-energy phenomenon. We illustrate the exotic physical properties of the remaining higher-angular-momentum analogs of spin-orbit coupling and derive a geometric constraint for spin-orbit vectors in lattice systems.

Figure 1

Distortion of the spin-up (green) and spin-down (blue) Fermi surface for the $l=1$ Pomeranchuk instability in the (a) charge and (b) spin channels.

Figure 2

Two examples of complex trajectories $z\left(t\right)$ that are allowed by Eq. (17) are shown in (b) and (d), with corresponding spin textures (for $\eta =+1)$ given in (a) and (c), respectively. In the limit of a spherical Fermi surface, the texture of (a) coincides with the $l=3$ Dresselhaus coupling in Eq. (16).