Itinerant ferromagnetism is one of the major subjects in condensed matter
physics.
In conventional ferromagnets, spin rotational symmetry is certainly
broken because of the spontaneous spin polarization.
However, orbital rotational symmetry is still maintained, 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 Cooper pairing order parameter
keeps constant over the Fermi surface.
Therefore from the symmetry point of view,
ferromagnetism can be considered as ``s-wave" magnetism.
In the context of superconductivity or superfluidity,
in addition to the conventional s-wave pairing,
there are unconventional Cooper pairing structures,
including the d-wave pairing in high Tc cuprates
and the p-wave pairing in superfluid-He3 and
Sr2RuO4.
In analogy with 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.
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 $F^s_1<-1$.
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.
Alpha and beta-phases
The Fermi surfaces of spin up and down electrons in the alpha-phases exhibit
spontaneous anisotropic distortions. 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, and also
by C. Varma et al (PRL 96, 36405) for the hidden order behavior in the
$URu_2Si_2$ 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. This may have potential applications in controlling electron spins in the emerging science of spintronics.
Spontaneous chiral instability
In the $p$-wave beta-phase, a Lifshitz-like instability develops, leading to an
inhomogeneous ground state in this originally nonchiral system. This effect
does not occur in the ferromagnetic transition or the density channel
instabilities. It bears some similarity to helimagnets and cholesteric
liquid crystals, except that parity is explicitly broken there.
References and talks
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Last modified: July 15, 2007.