Non-Abelian hydrodynamics and the flow of spin in spin–orbit coupled substances

Abstract

Motivated by the heavy ion collision experiments there is much activity in studying the hydrodynamical properties of non-Abelian (quark–gluon) plasmas. A major question is how to deal with color currents. Although not widely appreciated, quite similar issues arise in condensed matter physics in the context of the transport of spins in the presence of spin–orbit coupling. The key insight is that the Pauli Hamiltonian governing the leading relativistic corrections in condensed matter systems can be rewritten in a language of SU(2) covariant derivatives where the role of the non-Abelian gauge fields is taken by the physical electromagnetic fields: the Pauli system can be viewed as Yang–Mills quantum-mechanics in a ‘fixed frame’, and it can be viewed as an ‘analogous system’ for non-Abelian transport in the same spirit as Volovik’s identification of the He superfluids as analogies for quantum fields in curved space time. We take a similar perspective as Jackiw and coworkers in their recent study of non-Abelian hydrodynamics, twisting the interpretation into the ‘fixed frame’ context, to find out what this means for spin transport in condensed matter systems. We present an extension of Jackiw’s scheme: non-Abelian hydrodynamical currents can be factored in a ‘non-coherent’ classical part, and a coherent part requiring macroscopic non-Abelian quantum entanglement. Hereby it becomes particularly manifest that non-Abelian fluid flow is a much richer affair than familiar hydrodynamics, and this permits us to classify the various spin transport phenomena in condensed matter physics in an unifying framework. The “particle based hydrodynamics” of Jackiw et al. is recognized as the high temperature spin transport associated with semiconductor spintronics. In this context the absence of faithful hydrodynamics is well known, but in our formulation it is directly associated with the fact that the covariant conservation of non-Abelian currents turns into a disastrous non-conservation of the incoherent spin currents of the high temperature limit. We analyze the quantum-mechanical single particle currents of relevance to mesoscopic transport with as highlight the Ahronov–Casher effect, where we demonstrate that the intricacies of the non-Abelian transport render this effect to be much more fragile than its abelian analog, the Ahronov–Bohm effect. We subsequently focus on spin flows protected by order parameters. At present there is much interest in multiferroics where non-collinear magnetic order triggers macroscopic electric polarization via the spin–orbit coupling. We identify this to be a peculiarity of coherent non-Abelian hydrodynamics: although there is no net particle transport, the spin entanglement is transported in these magnets and the coherent spin ‘super’ current in turn translates into electric fields with the bonus that due to the requirement of single valuedness of the magnetic order parameter a true hydrodynamics is restored. Finally, ‘fixed-frame’ coherent non-Abelian transport comes to its full glory in spin–orbit coupled ‘spin superfluids’, and we demonstrate a new effect: the trapping of electrical line charge being a fixed frame, non-Abelian analog of the familiar magnetic flux trapping by normal superconductors. The only known physical examples of such spin superfluids are the 3He A- and B-phase where unfortunately the spin–orbit coupling is so weak that it appears impossible to observe these effects.