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

TL;DR

This paper demonstrates that in non-relativistic metals, $l=1$ spin-Pomeranchuk instabilities, which could lead to dynamically generated spin-orbit coupling, are forbidden due to fundamental conservation laws, constraining emergent spin-orbit phenomena.

Contribution

It establishes theoretical restrictions on dynamically generated spin-orbit coupling, specifically ruling out $l=1$ instabilities in non-relativistic systems based on Ward identities.

Findings

$l=1$ spin-Pomeranchuk instabilities are impossible in non-relativistic metals.

Relativistic spin-orbit coupling cannot emerge as a low-energy phenomenon in these systems.

Higher angular momentum analogues exhibit exotic physical properties.

Abstract

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.