Why is it interesting?

Topological defects are of common interest to condensed matter physics, atomic physics, astrophysics and cosmology, as well as algebraic topology. When the symmetry group $G$ spontaneously breaks down to its subgroup $H$, there are continuously connected ground states parametrized by the coset space $G/H$. The homotopy groups of the coset space then tell us what kinds of topological effects are possible. In most cases, non-trivial $\pi_d(G/H)$ implies the existence of $(2-d)$-dimensional topological defect. If the coset space has disconnected pieces ($\pi_0 (G/H) \neq 0$), we expect domain walls. For multiply-connected space ($\pi_1 (G/H)\neq 0$), there are strings/ vortices. If the boundary of space can map non-trivially to the coset space ($\pi_2(G/H)\neq 0$), we expect point-like defects such as monopoles. An exception to the rule is when the whole space is mapped non-trivially to the coset space ($\pi_3(G/H) \neq 0$), where skyrmions are stabilized by non-renormalizable terms in the low-energy effective theory. In this case, it is not the boundary condition that is topologically non-trivial, but the configuration in the bulk.

Topological Dark Matter by Hitoshi Murayama, Jing Shu

Important Topological Defects:

• Magnetic Monopoles
• Vortices
• kinks
• strings

Layman

Student

Recommended Resources:

• A great introduction is http://www.dartmouth.edu/~dbr/topdefects.pdf and
• see also http://www.lassp.cornell.edu/sethna/pubPDF/OrderParameters.pdf

Researcher

Examples

Domain Wall

Consider the possibility that φ = +η at x = +∞ and φ = −η at x = −∞. In this case, the continuous function φ(x) has to go from −η to +η as x is taken from −∞ to +∞ and so must necessarily pass through φ = 0. But then there is energy in this field configuration since the potential is non-zero when φ = 0. Also, this configuration cannot relax to either of the two vacuum configurations, say φ(x) = +η, since that involves changing the field over an infinite volume from −η to +η, which would cost an infinite amount of energy

https://arxiv.org/pdf/hep-ph/9710292.pdf

FAQ

What topological defects are present in a given model?

The problem of finding the types of topological defects present in a given model reduces to finding the homotopy groups for a certain symmetry breaking G → H. That is, we need to find πn(G/H) (n = 0, 1, 2, 3) given the groups G and H. In general, this can be quite complicated but there is an immensely useful theorem which is often applicable and simplifies matters.

https://arxiv.org/pdf/hep-ph/9710292.pdf

Why are Topological Defects Stable?

Suppose we consider a point defect and enclose it by an imaginary sphere S2. If we avoid other defects, all spheres about the point will be equivalent for our purposes, because they can be continuously deformed into one another inside V. Restricted to the sphere, cp defines a map S2+ A. If there is some topologically non-trivial aspect of this map, such as a winding number, the point defect will be topologically stable. It will not be able to just disappear leaving the medium in a uniform state with cp constant because, once it had gone, we could continuously deform the sphere to a point. The winding number, or whatever, would have had to have abruptly vanished, contradicting the assumed continuity. Similarly for a line defect, we take a circle S' about the line and obtain a map S' + A whose topological characteristics could ensure the stability of the defect.

Topological structures in field theories by Goddard and Mansfield

Example2:

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History