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 as 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 non-relativistic systems since the response of spin-current fluctuations is entirely incoherent and non-singular. 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 analogues of spin-orbit coupling and derive a geometric constraint for spin-orbit vectors in lattice systems.