In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear
sigma model
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...
s. It was originally proposed as a model of the
nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number).
Until the 1960s, nucleons were ...
by (and named after)
Tony Skyrme in 1961. As a
topological soliton in the
pion
In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
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
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
ic or
polariton fluids the skyrmion topology corresponds to a full Poincaré beam (a
spin vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...
comprising all the states of
polarization mapped by a stereographic projection of the
Poincaré sphere Poincaré sphere may refer to:
* Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light
** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system
* Poincaré homology s ...
to the real plane). A dynamical pseudospin skyrmion results from the stereographic projection of a rotating polariton Bloch sphere in the case of dynamical full Bloch beams.
Skyrmions have been reported, but not conclusively proven, to be in
Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.6 ...
s, thin magnetic films and in chiral nematic
liquid crystals
Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. The ...
.
As a model of the
nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number).
Until the 1960s, nucleons were ...
, 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 (Cheshire 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
Daniel Stuart Freed (born 17 April 1959) is an American mathematician, specializing in global analysis and its applications to supersymmetry, string theory, and quantum field theory. Since 1989, he has been a professor at the University of Texas at ...
. This can be interpreted as a foundation for the duality between a
quantum chromodynamics (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
In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
, where the target manifold is a
homogeneous space of the
structure group
:
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
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
, for ''N'' = 2, the chiral symmetries are understood to be the
isospin
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
symmetry of the
nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number).
Until the 1960s, nucleons were ...
. 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
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
has the topology S
3×R, then classical configurations can be classified by an integral
winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
because the third
homotopy group
:
is equivalent to the ring of integers, with the congruence sign referring to
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
.
A topological term can be added to the chiral Lagrangian, whose integral depends only upon the
homotopy class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
; 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
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
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
:
where
,
,
are the
isospin
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
,