In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
and
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, a
topological soliton or a
topological defect is a solution of a system of
partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s or of a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of
homotopically distinct solutionsIn topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
. Typically, this occurs because the boundary on which the boundary conditions are specified has a non-trivial
homotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
which is preserved in
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s; the solutions to the differential equations are then topologically distinct, and are classified by their homotopy class.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
and
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, a
topological soliton or a
topological defect is a solution of a system of
partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
s or of a
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of
homotopically distinct solutionsIn topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
. Typically, this occurs because the boundary on which the boundary conditions are specified has a non-trivial
homotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
which is preserved in
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s; the solutions to the differential equations are then topologically distinct, and are classified by their homotopy class. Topological defects are not only stable against small perturbations, but cannot decay or be undone or be de-tangled, precisely because there is no continuous transformation that will map them (homotopically) to a uniform or "trivial" solution.
Examples include the
solitonIn mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. "Dispersive effects" refer to dispersion relations between the frequency...
or
solitary waveIn mathematics and physics, a solitary wave can refer to* The solitary wave or wave of translation, as observed by John Scott Russell in a barge canal in 1834...
which occurs in many exactly solvable models, the screw dislocations in crystalline materials, the
SkyrmionIn theoretical physics, a skyrmion, conceived by Tony Skyrme, is a mathematical model used to model baryons .A skyrmion is a homotopically non-trivial classical solution of a nonlinear sigma model with a non-trivial target manifold topology: a particular case of a topological soliton...
and the
Wess-Zumino-Witten modelIn theoretical physics and mathematics, the Wess-Zumino-Witten model, also called the Wess-Zumino-Novikov-Witten model, is a simple model of conformal field theory whose solutions are realized by affine Kac-Moody algebras. It is named after Julius Wess, Bruno Zumino, Sergei P...
in quantum field theory.
Topological defects are believed to drive
phase transitionA phase transition is a natural physical process. It has the characteristic of taking a given medium with given properties and transforming some or all of that medium, into a new medium with new properties. Phase transitions occur frequently and are found everywhere in the natural world...
s in
condensed matterThere are at least 2 publications named Condensed Matter.-Ukrainian journal:Condensed Matter, scientific journal,publication of the Institute for Condensed Matter Physics ofNational Academy of Sciences of Ukraine....
physics. Notable examples of topological defects are observed in
Lambda transitionThe λ universality class is probably the most important group in condensed matter physics. It regroups several systems possessing strong analogies, namely, superfluids, superconductors and smectics...
universality class systems including: screw/edge-dislocations in liquid crystals, magnetic flux tubes in superconductors, vortices in superfluids.
Cosmology
Certain grand unified theories predict topological defects to have formed in the early
universeThe Universe comprises everything that physically exists, the entirety of space and time, all forms of matter and energy, and the physical laws and constants that govern them...
. According to the
Big BangThe Big Bang is the cosmological model of the initial conditions and subsequent development of the Universe that is supported by the most comprehensive and accurate explanations from current scientific evidence and observation...
theory, the universe cooled from an initial hot, dense state triggering a series of phase transitions much like what happens in condensed-matter systems.
In
physical cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of our universe and is concerned with fundamental questions about its formation and evolution. Cosmology involves itself with studying the motions of the celestial bodies and the first cause....
, a topological defect is an (often) stable configuration of matter predicted by some theories to form at
phase transitionA phase transition is a natural physical process. It has the characteristic of taking a given medium with given properties and transforming some or all of that medium, into a new medium with new properties. Phase transitions occur frequently and are found everywhere in the natural world...
s in the very early universe.
Symmetry breakdown
Depending on the nature of Symmetry breakdown, various
solitonIn mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. "Dispersive effects" refer to dispersion relations between the frequency...
s are believed to have formed in the early universe according to the Higgs-Kibble mechanism. The well-known topological defects are
magnetic monopoleIn physics, a magnetic monopole is a hypothetical particle that is a magnet with only one pole . In more technical terms, it would have a net "magnetic charge"...
s,
cosmic stringA cosmic string is a hypothetical 1-dimensional topological defect in various fields. Cosmic strings are hypothesized to form when the field undergoes a phase change in different regions of spacetime, resulting in condensations of energy density at the boundaries between regions...
s,
domain wallA domain wall is a term used in physics which can have one of two distinct but similar meanings in either magnetism or string theory. It is also used as technobabble in science fiction....
s,
SkyrmionIn theoretical physics, a skyrmion, conceived by Tony Skyrme, is a mathematical model used to model baryons .A skyrmion is a homotopically non-trivial classical solution of a nonlinear sigma model with a non-trivial target manifold topology: a particular case of a topological soliton...
s and
texturesIn cosmology, a texture is a type of topological defect in the structure of spacetime that forms when larger, more complicated symmetry groups are completely broken. They are not as localized as the other defects, and are unstable...
.
As the universe expanded and cooled, symmetries in the laws of physics began breaking down in regions that spread at the
speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
; topological defects occur where different regions came into contact with each other. The matter in these defects is in the original symmetric phase, which persists after a phase transition to the new asymmetric new phase is completed.
Types of topological defects
Various different types of topological defects are possible, with the type of defect formed being determined by the symmetry properties of the matter and the nature of the phase transition. They include:
- Domain wall
A domain wall is a term used in physics which can have one of two distinct but similar meanings in either magnetism or string theory. It is also used as technobabble in science fiction....
s, two-dimensional membranes that form when a discrete symmetry is broken at a phase transition. These walls resemble the walls of a closed-cell foamThe most general definition of foam is a substance that is formed by trapping many gas bubbles in a liquid or solid. It can also refer to anything that is analogous to such a phenomenon, such as quantum foam. Often the term is used in reference to polyurethane foam , XPS foam, Polystyrene, or many...
, dividing the universe into discrete cells.
- Cosmic string
A cosmic string is a hypothetical 1-dimensional topological defect in various fields. Cosmic strings are hypothesized to form when the field undergoes a phase change in different regions of spacetime, resulting in condensations of energy density at the boundaries between regions...
s are one-dimensional lines that form when an axial or cylindrical symmetry is broken.
- Monopole
In physics, a magnetic monopole is a hypothetical particle that is a magnet with only one pole . In more technical terms, it would have a net "magnetic charge"...
s, point-like defects that form when a spherical symmetry is broken, are predicted to have magnetic charge, either north or south (and so are commonly called "magnetic monopoleIn physics, a magnetic monopole is a hypothetical particle that is a magnet with only one pole . In more technical terms, it would have a net "magnetic charge"...
s").
- Texture
In cosmology, a texture is a type of topological defect in the structure of spacetime that forms when larger, more complicated symmetry groups are completely broken. They are not as localized as the other defects, and are unstable...
s form when larger, more complicated symmetry groups are completely broken. They are not as localized as the other defects, and are unstable. Other more complex hybrids of these defect types are also possible.
- Extra-dimensions and higher dimensions
Dimensions is a French project that makes educational movies about mathematics, focusing on spatial geometry. It uses POV-Ray to render some of the animations, and the films are release under a Creative Commons licence....
.
Observation
Topological defects, of the cosmological type, are extremely high-energy phenomena and are likely impossible to produce in artificial Earth-bound physics experiments, but topological defects that formed during the universe's formation could theoretically be observed.
No topological defects of any type have yet been observed by astronomers, however, and certain types are not compatible with current observations; in particular, if domain walls and monopoles were present in the observable universe, they would result in significant deviations from what astronomers can see. Theories that predict the formation of these structures
within the observable universe (see:
inflation) can therefore be largely ruled out. On the other hand,
cosmic stringA cosmic string is a hypothetical 1-dimensional topological defect in various fields. Cosmic strings are hypothesized to form when the field undergoes a phase change in different regions of spacetime, resulting in condensations of energy density at the boundaries between regions...
s have been suggested as providing the initial 'seed'-gravity around which the
large-scale structure of the cosmosIn physical cosmology, the large-scale structure of the universe refers to the characterization of observable distributions of matter and light on the largest scales...
of matter has condensed. Textures are similarly benign. In late 2007, a
cold spotThe WMAP Cold Spot or CMB Cold Spot is a region of the sky seen in microwaves which analysis found to be unusually large and cold relative to the expected properties of the cosmic microwave background radiation...
in the cosmic microwave background was interpreted as possibly being a sign of a
textureIn cosmology, a texture is a type of topological defect in the structure of spacetime that forms when larger, more complicated symmetry groups are completely broken. They are not as localized as the other defects, and are unstable...
lying in that direction.
Images
See also
- quantum vortex
In physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. The existence of these quantum vortices were independently predicted by Richard Feynman and Alexei Alexeyevich Abrikosov in the 1950s...
- dislocation
In materials science, a dislocation is a crystallographic defect, or irregularity, within a crystal structure. The presence of dislocations strongly influences many of the properties of materials. The theory was originally developed by Vito Volterra in 1905....
- vector soliton
In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one polarization component, while vector solitons have two distinct...
- Quantum topology
Quantum topology is a subfield of topology/geometry/knot theory whereby existing quantum invariants or newly developed quantum invariants are used to determine if simple arrangements in manifolds are equal...
- Topological entropy in physics
- Topological order
In physics, topological orderis a new kind of order in aquantum state that is beyond the Landau symmetry-breakingdescription...
- Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
- Topological quantum number
In physics, a topological quantum number is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations...
- Topological string theory
In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry...
External links