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

TLDR: In this article, 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.

About: This article is published in Annals of Physics.The article was published on 2008-04-01 and is currently open access. It has received 40 citations till now. The article focuses on the topics: Spin–orbit interaction & Quantum entanglement.

Figures

Fig. 5. The destruction of the spin vortex by dipolar locking. The U(1) degree of freedom is indicated by an arrow. In the center where the electric field is located, the angle follows a vortex configuration of unitwinding number, corresponding to one charge quantum. Since the electric field, decaying as 1r, is not able to compete with the dipolar locking at long distances, the U(1) angle becomes fixed at the Leggett angle, indicated by a horizontal arrow.

Fig. 7. The phase diagram of the highly frustrated j-(ET)2Cu2(CN)3, as proposed by Kanoda [35]. The spin liquid state shows linear specific heat, which might signal the presence of a spinon Fermi surface. This would amount to making a spinon Fermi liquid out of an insulator. Then the interesting possibility is that this spinon metal might be unstable against an S = 1 spin superfluid.

Fig. 1. A superfluid 3He container acts as a capacitor capable of trapping a quantized electrical line charge density via the electric field generated by persistent spin-Hall currents. This is the analog of magnetic flux trapping in superconductors by persistent charge supercurrents.

Fig. 6. View from the top of our container. The container radius is R, and the wire has radius a. Now, the Coulomb energy of the wire has to make a tiny region of superfluid normal again, in order to make phase slips happen, removing the topological constraint. The region in which this should happen, needs to be of the width of the coherence length n, but it has to extend over the whole radius of the container.

Fig. 3. The laminar flow of a parallel transported spin current, Fig. 2, can also be viewed as a static spin spiral magnet.

Fig. 2. Laminar flow of a classical spin fluid in an electric field. The fluid elements (blue) carry non-Abelian charge, the red arrows indicating the spin direction. The flow lines are directed to the right, and the electric field is pointing outwards of the paper. Due to Eq. (49), the spin precesses as indicated.

Frequently asked questions

Q1. What are the contributions in "Non-abelian hydrodynamics and the flow of spin in spin- orbit coupled substances" ?

The authors 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. The authors present an extension of Jackiw ’ s scheme: non-Abelian hydrodynamical currents can be factored in a ‘ noncoherent ’ 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. The authors analyze the quantum-mechanical single particle currents of relevance to mesoscopic 0003-4916/ $ see front matter 2007 Elsevier Inc. 908 B. W. A. Leurs et al. / Annals of Physics 323 ( 2008 ) 907–945 transport with as highlight the Ahronov–Casher effect, where the authors 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. The authors identify this to be a peculiarity of coherent nonAbelian 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 the authors 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. 

Q2. What is the modern twist of this argument?

The modern twist of this argument is [9]: the spin spiral can be caused by magnetic frustration as well, and it now acts as a cause (instead of effect) for an induced ferroelectric polarization. 

Q3. What is the meaning of the absence of order parameter rigidity?

The absence of order parameter rigidity means that the authors are considering classical spin fluids as they are realized at higher temperatures, i.e. away from the mesoscopic regime of the previous section and the superfluids addressed in Section 9. 

Q4. What is the way to make a spin superfluid?

So the authors need something made out of electrons, having however a huge gap for charge excitations: the authors need a spin superfluid made out of a Mott insulator. 

Q5. What is the term containing the non-Abelian velocity in this electromagnetic current?

The term containing the non-Abelian velocity (the coherent spin current) in this electromagnetic current will only contribute when there is magnetic order ÆSaæ „ 0. 

Q6. What is the definition of the 'gauge' field in the Abelian realms?

In the Abelian realms of electrical charge or mass a universal description of this transport is available in the form of hydrodynamics, be it the hydrodynamics of water, the magneto-hydrodynamics of charged plasmas, or the quantum-hydrodynamics of superfluids and superconductors. 

Q7. What is the simplest example of a non-Abelian ‘hydrodynamics?

This elementary example highlights the essence of the problem dealing with non-Abelian ‘hydrodynamics’: the covariant conservation principle underlying everything is good enough to ensure a local conservation of non-Abelian charge so that one can reliably predict how the spin current evolves over infinitesimal times and distances. 

Q8. What is the lack of hydrodynamics in the spintronics community?

The lack of hydrodynamics is well understood in the spintronics community: after generating a spin current is just disappears after a time called the spin-relaxation time. 

Q9. What is the inverse of the charge density in superconductors?

This is of course a very large line-charge density, and this is of course rooted in the fact that this quantum is ‘dual’ to the tiny spin–orbit coupling of helium, in the same way that the flux quantum in superconductors is inversely proportional to the electrical charge. 

Q10. What is the connection between the flow of spin in the presence of spin–orbit coupling?

Goldhaber [13] and later Fröhlich and Studer [14], Balatsky and Altshuler [15] and others realized that in the presence of spin–orbit coupling spin is subjected to a parallel transport principle that is quite similar to the parallel transport of matter fields in Yang–Mills non-Abelian gauge theory, underlying for instance QCD. 

Q11. Why is there no hydrodynamical description of color transport?

The description of the color currents in the quark–gluon plasma is suffering from a fatal flaw: because of the lack of a hydrodynamical conservation law there is no hydrodynamical description of color transport.