In particle theory, the skyrmion () is a topologically
stable field configuration of a certain class
of non-linear sigma models. It was originally
proposed as a model of the nucleon by Tony
Skyrme in 1962. As a topological soliton in
the pion field, it has the remarkable property
of being able to model, with reasonable accuracy,
multiple low-energy properties of the nucleon,
simply by fixing the nucleon radius. It has
since found application in solid state physics,
as well as having ties to certain areas of
string theory.
Skyrmions as topological objects are important
in solid state physics, especially in the
emerging technology of spintronics. A two-dimensional
magnetic skyrmion, as a topological object,
is formed, e.g., from a 3D effective-spin
"hedgehog" (in the field of micromagnetics:
out of a so-called "Bloch point" singularity
of homotopy degree +1) by a stereographic
projection, whereby the positive north-pole
spin is mapped onto a far-off edge circle
of a 2D-disk, while the negative south-pole
spin is mapped onto the center of the disk.
In a spinor field such as for example photonic
or polariton fluids the skyrmion topology
corresponds to a full Poincaré beam (which
is, a quantum vortex of spin comprising all
the states of polarization).Skyrmions have
been reported, but not conclusively proven,
to be in Bose-Einstein condensates, superconductors,
thin magnetic films and in chiral nematic
liquid crystals.As a model of the nucleon,
the topological stability of the Skyrmion
can be interpreted as a statement that the
baryon number is conserved; i.e. that the
proton does not decay. The Skyrme Lagrangian
is essentially a one-parameter model of the
nucleon. Fixing the parameter fixes the proton
radius, and also fixes all other low-energy
properties, which appear to be correct to
about 30%. It is this predictive power of
the model that makes it so appealing as a
model of the nucleon.
Hollowed-out skyrmions form the basis for
the chiral bag model (Chesire cat model) of
the nucleon. Exact results for the duality
between the fermion spectrum and the topological
winding number of the non-linear sigma model
have been obtained by Dan Freed. This can
be interpreted as a foundation for the duality
between a QCD description of the nucleon (but
consisting only of quarks, and without gluons)
and the Skyrme model for the nucleon.
The skyrmion can be quantized to form a quantum
superposition of baryons and resonance states.
It could be predicted from some nuclear matter
properties.
== Topological soliton ==
In field theory, skyrmions are homotopically
non-trivial classical solutions of a nonlinear
sigma model with a non-trivial target manifold
topology – hence, they are topological solitons.
An example occurs in chiral models of mesons,
where the target manifold is a homogeneous
space of the structure group
(
S
U
(
N
)
L
×
S
U
(
N
)
R
S
U
(
N
)
diag
)
{\displaystyle \left({\frac {SU(N)_{L}\times
SU(N)_{R}}{SU(N)_{\text{diag}}}}\right)}
where SU(N)L and SU(N)R are the left and right
chiral symmetries, and SU(N)diag is the diagonal
subgroup. In nuclear physics, for N=2, the
chiral symmetries are understood to be the
isospin symmetry of the nucleon. For N=3,
the isoflavor symmetry between the up, down
and strange quarks is more broken, and the
skyrmion models are less successful or accurate.
If spacetime has the topology S3×R, then
classical configurations can be classified
by an integral winding number because the
third homotopy group
π
3
(
S
U
(
N
)
L
×
S
U
(
N
)
R
S
U
(
N
)
diag
≅
S
U
(
N
)
)
{\displaystyle \pi _{3}\left({\frac {SU(N)_{L}\times
SU(N)_{R}}{SU(N)_{\text{diag}}}}\cong SU(N)\right)}
is equivalent to the ring of integers, with
the congruence sign referring to homeomorphism.
A topological term can be added to the chiral
Lagrangian, whose integral depends only upon
the homotopy class; this results in superselection
sectors in the quantised model. In 1+1 dimensional
spacetime, a skyrmion can be approximated
by a soliton of the Sine-Gordon equation;
after quantisation by the Bethe ansatz or
otherwise, it turns into a fermion interacting
according to the massive Thirring model.
== Lagrangian ==
The Lagrangian for the Skyrmion, as written
for the original chiral SU(2) effective Lagrangian
of the nucleon-nucleon interaction (in 3+1-dimensional
spacetime), can be written as
L
=
−
f
π
2
4
tr
⁡
(
L
μ
L
μ
)
+
1
32
g
2
tr
⁡
[
L
μ
,
L
ν
]
[
L
μ
,
L
ν
]
{\displaystyle {\mathcal {L}}={\frac {-f_{\pi
}^{2}}{4}}\operatorname {tr} \left(L_{\mu
}L^{\mu }\right)+{\frac {1}{32g^{2}}}\operatorname
{tr} [L_{\mu },L_{\nu }][L^{\mu },L^{\nu }]}
where
L
μ
=
U
†
∂
μ
U
{\displaystyle L_{\mu }=U^{\dagger }\partial
_{\mu }U}
and
U
=
exp
⁡
i
τ
→
⋅
θ
→
{\displaystyle U=\exp i{\vec {\tau }}\cdot
{\vec {\theta }}}
and
τ
→
{\displaystyle {\vec {\tau }}}
are the isospin Pauli matricies, and
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
is the Lie bracket commutator, and tr is the
matrix trace. The meson field (pion field,
up to a dimensional factor) at spacetime coordinate
x
{\displaystyle x}
is given by
θ
→
=
θ
→
(
x
)
{\displaystyle {\vec {\theta }}={\vec {\theta
}}(x)}
.
When written this way, the
U
{\displaystyle U}
is clearly an element of the Lie group SU(2),
and
θ
→
{\displaystyle {\vec {\theta }}}
an element of the Lie algebra su(2). The pion
field can be understood abstractly to be a
section of the tangent bundle of the principal
fiber bundle of SU(2) over spacetime. This
abstract interpretation is characteristic
of all non-linear sigma models.
The first term,
tr
⁡
(
L
μ
L
μ
)
{\displaystyle \operatorname {tr} \left(L_{\mu
}L^{\mu }\right)}
is just an unusual way of writing the quadratic
term of the non-linear sigma model; it reduces
to
−
tr
⁡
(
∂
μ
U
†
∂
μ
U
)
{\displaystyle -\operatorname {tr} \left(\partial
_{\mu }U^{\dagger }\partial ^{\mu }U\right)}
. When used as a model of the nucleon, one
writes
U
=
(
σ
+
i
τ
→
⋅
π
→
)
/
f
π
{\displaystyle U=(\sigma +i{\vec {\tau }}\cdot
{\vec {\pi }})/f_{\pi }}
with the dimensional factor of
f
π
{\displaystyle f_{\pi }}
being the pion decay constant. (In 1+1 dimensions,
this constant is not dimensional and can thus
be absorbed into the field definition.)
The second term establishes the characteristic
size of the lowest-energy soliton solution;
it determines the effective radius of the
soliton. As a model of the nucleon, it is
normally adjusted so as to give the correct
radius for the proton; once this is done,
other low-energy properties of the nucleon
are automatically fixed, to within about 30%
accuracy. It is this result, of tying together
what would otherwise be independent parameters,
and doing so fairly accurately, that makes
the Skyrme model of the nucleon so appealing
and interesting. Thus, for example constant
g
{\displaystyle g}
in the quartic term is interpreted as the
vector-pion coupling
ρ
−
π
−
π
{\displaystyle \rho -\pi -\pi }
between the rho meson (the nuclear vector
meson) and the pion; the skyrmion relates
the value of this constant to the baryon radius.
== Noether current ==
The local winding number density is given
by
B
μ
=
ϵ
μ
ν
α
β
tr
⁡
L
ν
L
α
L
β
{\displaystyle B^{\mu }=\epsilon ^{\mu \nu
\alpha \beta }\operatorname {tr} L_{\nu }L_{\alpha
}L_{\beta }}
where
ϵ
μ
ν
α
β
{\displaystyle \epsilon ^{\mu \nu \alpha \beta
}}
is the totally antisymmetric Levi-Civita symbol
(equivalently, the Hodge star, in this context).
As a physical quantity, this can be interpreted
as the baryon current; it is conserved:
∂
μ
B
μ
=
0
{\displaystyle \partial _{\mu }B^{\mu }=0}
, and the conservation follows as a Noether
current for the chiral symmetry.
The corresponding charge is the baryon number:
B
=
∫
d
3
x
B
0
(
x
)
{\displaystyle B=\int d^{3}xB^{0}(x)}
As a conserved charge, it is 
time-independent:
d
B
/
d
t
=
0
{\displaystyle dB/dt=0}
, the physical interpretation of which is
that protons do not decay.
In the chiral bag model, one cuts a hole out
of the center and fills it with quarks. Despite
this obvious "hackery", the total baryon number
is conserved: the missing charge from the
hole is exactly compensated by the spectral
asymmetry of the vacuum fermions inside the
bag!
== Magnetic materials/data storage ==
One particular form of skyrmions is magnetic
skyrmions, found in magnetic materials that
exhibit spiral magnetism due to the Dzyaloshinskii-Moriya
interaction, double-exchange mechanism or
competing Heisenberg exchange interactions.
They form "domains" as small as 1 nm (e.g.
in Fe on Ir(111)). The small size and low
energy consumption of magnetic skyrmions make
them a good candidate for future data storage
solutions and other spintronics devices.
Researchers could read and write skyrmions
using scanning tunneling microscopy. The topological
charge, representing the existence and non-existence
of skyrmions, can represent the bit states
"1" and "0". Room temperature skyrmions were
reported.Skyrmions operate at current densities
that are several orders of magnitude weaker
than conventional magnetic devices. In 2015
a practical way to create and access magnetic
skyrmions under ambient room-temperature conditions
was announced. The device used arrays of magnetized
cobalt disks as artificial Bloch skyrmion
lattices atop a thin film of cobalt and palladium.
Asymmetric magnetic nanodots were patterned
with controlled circularity on an underlayer
with perpendicular magnetic anisotropy (PMA).
Polarity is controlled by a tailored magnetic
field sequence and demonstrated in magnetometry
measurements. The vortex structure is imprinted
into the underlayer's interfacial region via
suppressing the PMA by a critical ion-irradiation
step. The lattices are identified with polarized
neutron reflectometry and have been confirmed
by magnetoresistance measurements
