Topological defect

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A topological soliton or "toron" occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological soliton occurs in old-fashioned coiled telephone handset cords, which are usually coiled clockwise. Years of picking up the handset can end up coiling parts of the cord in the opposite counterclockwise direction, and when this happens there will be a distinctive larger loop that separates the two directions of coiling. This odd looking transition loop, which is neither clockwise nor counterclockwise, is an excellent example of a topological soliton. No matter how complex the context, anything that qualifies as a topological soliton must at some level exhibit this same simple issue of reconciliation seen in the twisted phone cord example.

Contents

Topological solitons arise with ease when creating the crystalline semiconductors used in modern electronics, and in that context their effects are almost always deleterious. For this reason such crystal transitions are called topological defects. However, this mostly solid-state terminology distracts from the rich and intriguing mathematical properties of such boundary regions. Thus for most non-solid-state contexts the more positive and mathematically rich phrase "topological soliton" is preferable.

A more detailed discussion of topological solitons and related topics is provided below.

In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution.

Overview

The existence of a topological defect can be demonstrated whenever the boundary conditions entail the existence of homotopically distinct solutions. Typically, this occurs because the boundary on which the conditions are specified has a non-trivial homotopy group which is preserved in differential equations; 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

Topological defects occur in partial differential equations and are believed[ according to whom? ] to drive[ how? ] phase transitions in condensed matter physics.

The authenticity[ further explanation needed ] of a topological defect depends on the nature of the vacuum in which the system will tend towards if infinite time elapses; false and true topological defects can be distinguished if the defect is in a false vacuum and a true vacuum, respectively.[ clarification needed ]

Solitary wave PDEs

Examples include the soliton or solitary wave which occurs in exactly solvable models, such as

Lambda transitions

Topological defects in lambda transition universality class[ clarification needed ] systems including:

Cosmological defects

Topological defects, of the cosmological type, are extremely high-energy[ clarification needed ] phenomena which are deemed impractical to produce[ according to whom? ] in Earth-bound physics experiments. Topological defects created during the universe's formation could theoretically be observed without significant energy expenditure.

In the Big Bang theory, the universe cools from an initial hot, dense state triggering a series of phase transitions much like what happens in condensed-matter systems such as superconductors. Certain[ which? ] grand unified theories predict the formation of stable topological defects in the early universe during these phase transitions.

Symmetry breakdown

Depending on the nature of symmetry breakdown, various solitons are believed to have formed in the early universe according to the Kibble-Zurek mechanism. The well-known topological defects are:

  • Cosmic strings are one-dimensional lines that form when an axial or cylindrical symmetry is broken.
  • Domain walls, two-dimensional membranes that form when a discrete symmetry is broken at a phase transition. These walls resemble the walls of a closed-cell foam, dividing the universe into discrete cells.
  • Monopoles, cube-like defects that form when a spherical symmetry is broken, are predicted to have magnetic charge,[ why? ] either north or south (and so are commonly called "magnetic monopoles").
  • Textures form when larger, more complicated symmetry groups[ which? ] are completely broken. They are not as localized as the other defects, and are unstable.[ clarification needed ]
  • Skyrmions
  • Extra dimensions and higher dimensions.

Other more complex hybrids of these defect types are also possible.

As the universe expanded and cooled, symmetries in the laws of physics began breaking down in regions that spread at the speed of light; topological defects occur at the boundaries of adjacent regions.[ how? ] The matter composing these boundaries is in an ordered phase, which persists after the phase transition to the disordered phase is completed for the surrounding regions.

Biochemistry

Defects[ which? ] have also been found in biochemistry, notably in the process of protein folding.

Formal classification

An ordered medium is defined as a region of space described by a function f(r) that assigns to every point in the region an order parameter , and the possible values of the order parameter space constitute an order parameter space. The homotopy theory of defects uses the fundamental group of the order parameter space of a medium to discuss the existence, stability and classifications of topological defects in that medium. [1]

Suppose R is the order parameter space for a medium, and let G be a Lie group of transformations on R. Let H be the symmetry subgroup of G for the medium. Then, the order parameter space can be written as the Lie group quotient [2] R = G/H.

If G is a universal cover for G/H then, it can be shown [2] that πn(G/H) = πn−1(H), where πi denotes the i-th homotopy group.

Various types of defects in the medium can be characterized by elements of various homotopy groups of the order parameter space. For example, (in three dimensions), line defects correspond to elements of π1(R), point defects correspond to elements of π2(R), textures correspond to elements of π3(R). However, defects which belong to the same conjugacy class of π1(R) can be deformed continuously to each other, [1] and hence, distinct defects correspond to distinct conjugacy classes.

Poénaru and Toulouse showed that [3] crossing defects get entangled if and only if they are members of separate conjugacy classes of π1(R).

Observation

Topological defects have not been observed by astronomers; however, 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.

Because of these observations, the formation of defects within the observable universe is highly constrained, requiring special circumstances (see Inflation (cosmology)). On the other hand, cosmic strings have been suggested as providing the initial 'seed'-gravity around which the large-scale structure of the cosmos of matter has condensed. Textures are similarly benign.[ clarification needed ] In late 2007, a cold spot in the cosmic microwave background provided evidence of a possible texture. [4]

Classes of stable defects in biaxial nematics Biaxial.png
Classes of stable defects in biaxial nematics

Condensed matter

In condensed matter physics, the theory of homotopy groups provides a natural setting for description and classification of defects in ordered systems. [1] Topological methods have been used in several problems of condensed matter theory. Poénaru and Toulouse used topological methods to obtain a condition for line (string) defects in liquid crystals that can cross each other without entanglement. It was a non-trivial application of topology that first led to the discovery of peculiar hydrodynamic behavior in the A-phase of superfluid helium-3. [1]

Stable defects

Homotopy theory is deeply related to the stability of topological defects. In the case of line defect, if the closed path can be continuously deformed into one point, the defect is not stable, and otherwise, it is stable.

Unlike in cosmology and field theory, topological defects in condensed matter have been experimentally observed. [5] Ferromagnetic materials have regions of magnetic alignment separated by domain walls. Nematic and bi-axial nematic liquid crystals display a variety of defects including monopoles, strings, textures etc. [1]

Images

A static solution to
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m
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{\displaystyle {\mathcal {L}}=\partial _{\mu }\phi \partial ^{\mu }\phi -\left(\phi ^{2}-1\right)^{2}}
in (1 + 1)-dimensional spacetime. DoubleWellSoliton.png
A static solution to in (1 + 1)-dimensional spacetime.
A soliton and an antisoliton colliding with velocities +-sinh(0.05) and annihilating. DoubleWellSolitonAntisoliton.gif
A soliton and an antisoliton colliding with velocities ±sinh(0.05) and annihilating.

See also

Related Research Articles

Condensed matter physics Branch of physics

Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the subject deals with "condensed" phases of matter: systems of many constituents with strong interactions between them. More exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, and the Bose–Einstein condensate found in ultracold atomic systems. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other theories to develop mathematical models.

Soliton a self-reinforcing single wave packet that maintains its shape while it propagates at a constant velocity

In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

Magnetic monopole Hypothetical particle with one magnetic pole

In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole. A magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence.

Homotopy Continuous deformation between two continuous maps

In topology, a branch of mathematics, 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. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

An instanton is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime.

Spontaneous symmetry breaking Physical process

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry-breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking was not simply connected. It is expected that at least one string per Hubble volume is formed. Their existence was first contemplated by the theoretical physicist Tom Kibble in the 1970s.

In gauge theory and mathematical physics, a topological quantum field theory is a quantum field theory which computes topological invariants.

Linking number

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.

In quantum mechanics, superselection extends the concept of selection rules.

In theoretical physics, the 't Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without any singularities. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field which spontaneously breaks it down to a smaller group H via the Higgs mechanism. It was first found independently by Gerard 't Hooft and Alexander Polyakov.

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 1961. 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.

Montonen–Olive duality

Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or S-duality according to current terminology. It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite", can in fact be viewed as "elementary" quantized particles with electrons playing the reverse role of "composite" topological solitons; the viewpoints are equivalent and the situation dependent on the duality. It was later proven to hold true when dealing with a N = 4 supersymmetric Yang–Mills theory. It is named after Finnish physicist Claus Montonen and British physicist David Olive after they proposed the idea in their academic paper Magnetic monopoles as gauge particles? where they state:

There should be two "dual equivalent" field formulations of the same theory in which electric (Noether) and magnetic (topological) quantum numbers exchange roles.

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. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary, on which the boundary conditions are specified, has a non-trivial homotopy group that is preserved by the differential equations. The topological quantum number of a solution is sometimes called the winding number of the solution, or, more precisely, it is the degree of a continuous mapping.

Topological order Type of order at absolute zero

In physics, topological order is a kind of order in the zero-temperature phase of matter. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders cannot change into each other without a phase transition.

Erick J. Weinberg is a theoretical physicist and professor of physics at Columbia University.

The Kibble–Zurek mechanism (KZM) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate. It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.

The index of physics articles is split into multiple pages due to its size.

In theoretical physics, the curvature renormalization group (CRG) method is an analytical approach to determine the phase boundaries and the critical behavior of topological systems. Topological phases are phases of matter that appear in certain quantum mechanical systems at zero temperature because of a robust degeneracy in the ground-state wave function. They are called topological because they can be described by different (discrete) values of a nonlocal topological invariant. This is to contrast with non-topological phases of matter that can be described by different values of a local order parameter. States with different values of the topological invariant cannot change into each other without a phase transition. The topological invariant is constructed from a curvature function that can be calculated from the bulk Hamiltonian of the system. At the phase transition, the curvature function diverges, and the topological invariant correspondingly jumps abruptly from one value to another. The CRG method works by detecting the divergence in the curvature function, and thus determining the boundaries between different topological phases. Furthermore, from the divergence of the curvature function, it extracts scaling laws that describe the critical behavior, ie how different quantities behave as the topological phase transition is approached. The CRG method has been successfully applied to a variety of static, periodically driven, weakly and strongly interacting systems to classify the nature of the corresponding topological phase transitions.

References

  1. 1 2 3 4 5 Mermin, N. D. (1979). "The topological theory of defects in ordered media". Reviews of Modern Physics. 51 (3): 591–648. Bibcode:1979RvMP...51..591M. doi:10.1103/RevModPhys.51.591.
  2. 1 2 Nakahara, Mikio (2003). Geometry, Topology and Physics. Taylor & Francis. ISBN   978-0-7503-0606-5.
  3. Poénaru, V.; Toulouse, G. (1977). "The crossing of defects in ordered media and the topology of 3-manifolds". Le Journal de Physique. 38 (8): 887–895. CiteSeerX   10.1.1.466.9916 . doi:10.1051/jphys:01977003808088700.
  4. Cruz, M.; Turok, N.; Vielva, P.; Martínez-González, E.; Hobson, M. (2007). "A Cosmic Microwave Background Feature Consistent with a Cosmic Texture". Science. 318 (5856): 1612–1614. arXiv: 0710.5737 . Bibcode:2007Sci...318.1612C. doi:10.1126/science.1148694. PMID   17962521. S2CID   12735226.
  5. "Topological defects". Cambridge cosmology.