Spin-orbit coupling is conventionally viewed as a single-electron property
inherited from relativity but not directly related to many-body interactions.
We with (S. C. Zhang) proposed a new concept for spontaneous generation
of spin-orbit coupling purely from electron interactions
(
[Ref. 1] ).
It is through an unconventional type of magnetic phase transitions
(e.g. e-wave) based on Fermi surface instabilities.
( [Ref. 2] ).
Uncoventional magnetism of itinerant electrons
In ferromagnetic metals, spin rotational symmetry is broken because of
the spontaneous spin polarization.
However, orbital rotational symmetry remains, i.e., spin of each
momentum polarizes along the same direction around Fermi surfaces.
This is similar to the situation in conventional s-wave superconductors
where the phase of the pairing gap functions keeps constant over
the Fermi surface.
Therefore ferromagnetism can be considered as ``s-wave" magnetism.
In the context of superconductivity or fermion pairing superfluidity,
in addition to the conventional s-wave pairing,
there are unconventional pairing structures,
including the d-wave pairing in high Tc cuprates
and the p-wave pairing in superfluid-He3 and
Sr2RuO4.
In analogy to unconventional superconductivity, we have proposed
``non-$s$-wave'' generalizations of ferromagnetic states in which
spin no longer polarizes along a unique direction but varies with momentum
( [Ref. 2] ).
These unconventional magnetic states have close connections to many
research focuses in condensed matter physics, including unconventional
superconductivity, spin-orbit coupling and spintronics, and
electron liquid crystal states in strongly correlated systems.
The unconventional magnetic states include both isotropic and
anisotropic cases.
The isotropic phases still have circular or spherical Fermi surfaces
with topologically non-trivial spin configurations in momentum space,
providing a mechanism for dynamic generation of spin-orbit coupling
through many-body interactions; the anisotropic phases are electron
liquid crystal states with spin degree of freedom, exhibiting
anisotropic Fermi surface distortions.
Both types of phases arise from a general class of Fermi liquid
instabilities of the Pomeranchuk type in the spin channel, which
include ferromagnetism as a specital example.
Pomeranchuk instability of Fermi liquids
Most of our current understanding of interacting electron systems are
based on the Landau theory of Fermi lqiuids.
In this theory, the interactions among quasiparticles are captured by
a few Landau parameters $F^{s,a}_l$, where $l$ denotes the orbital
angular momentum partial-wave channel, and $s$, $a$ denote
spin-singlet and -triplet channels, respectively.
Physical quantities, such as the spin susceptibility, and properties of
collective excitations, such as the dispersion relation of zero
sound collective modes, acquire significant but finite renormalizations
due to the Landau interactions.
It has, however, long been known that the stability of
Fermi liquids requires that the Landau parameters cannot be too
negative, $F^{s,a}_l>-(2l+1)$, a result derived by Pomeranchuk.
Otherwise, Fermi surfaces will be distorted. Such a class of
Fermi surface instabilities are named Pomeranchuk instabilities.
The most familiar examples of these Pomeranchuk instabilities are
found in the s-wave channel: the Stoner ferromagnetism at
$F^a_1<-1$ channel and phase separation at $F^s_1<-1$.
Dynamic spin-orbit ordering: Alpha and beta-phases
The unconventional magetism arises from the Pomeranchuk instabilities
in the spin triplet channel with high orbital partical waves $(F^a_l (l>0))$.
The resultant ordered phases are classified into two classes, dubbed the
alpha and beta phases by analogy to the superfluid He3-A and B-phases,
respectively.
The Fermi surfaces in the $\alpha$-phases exhibit spontaneous anisotropic
distortions, which are the electronic nematic phases augmented by the spin
degree of freedom as shown in Fig. \ref{fig:phase} ($C$).
This phase in the $p$-wave channel was proposed by J. Hirsch
(PRB 41, 6820; PRB 41, 6828) under the name of "spin-split" state in
the Chromium system.
The Fermi surfaces in the beta-phases remain circular or spherical
with topologically non-trivial spin configurations in momentum space.
The beta-phases are isotropic, but exhibit relative spin-orbit symmetry
breaking, a concept first proposed by Leggett in the superfluid
He-3 systems. The mean-field band structures in both phases exhibit
various types of spin-orbit couplings.
These states have close connections to many research directions in
condensed matter physics, including unconventional superconductivity,
spin-orbit coupling and spintronics, and electron liquid crystal states
in strongly correlated systems.
In particular, the recent experiment by D. Hsieh's group found
a parity-breaking electronic nematic state in Cd$_2$Re$_2$O$_7$,
which may be related to this class of phase.
Spin-orbit coupled Fermi liquid theory
We have investigated Fermi liquid states of the ultracold magnetic dipolar Fermi gases
in the simplest two-component case
[Ref. 3].
The magnetic dipolar interaction is invariant under the simultaneous spin-orbit
rotation but not under either the spin or the orbit one.
Therefore, the corresponding Fermi liquid theory is intrinsically spin-orbit coupled.
This is a fundamental feature of magnetic dipolar Fermi gases different from
electric dipolar ones.
The Landau interaction matrix is calculated and is diagonalized in terms of
the spin-orbit coupled partial-wave channels of the total angular momentum J.
An exotic propagating collective mode is identified as spin-orbit coupled Fermi
surface oscillations in which spin distribution on the Fermi surface exhibits
a topologically nontrivial hedgehog configuration.
References and talks
Phys. Rev. B 85, 205126 (2012) ,
pdf file
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Last modified: July 15, 2007.