✎ Topological Defects [The Physics Travel Guide]

You've loaded an old revision of the document! If you save it, you will create a new version with this data.

====== Topological Defects ======

<blockquote>
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 [[advanced_tools:group_theory:quotient_group|coset space]] $G/H$. The [[advanced_tools:topology:homotopy|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.

<cite>[[https://arxiv.org/abs/0905.1720|Topological Dark Matter]] by Hitoshi Murayama, Jing Shu</cite>
</blockquote>

----

**Important Topological Defects:**

* [[advanced_notions:topological_defects:magnetic_monopoles|Magnetic Monopoles]]
  * [[advanced_notions:topological_defects:vortices]]
  * [[advanced_notions:topological_defects:notions:kinks]]
  * [[advanced_notions:topological_defects:strings]]

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student>

Topological defects are most easily understood by considering the scalar potential. A non-trivial vev breaks a given group $G$ to some subgroup $H$. Depending on the [[advanced_tools:topology:homotopy|homotopy class]] of the vacuum manifold $G/H$ (speak [[advanced_tools:group_theory:quotient_group|G mod H]]), we get different topological defects.

* If the [[advanced_tools:topology:homotopy|homotopy class]] of $G/H$ is non-trivial this tells us that the vacuum manifold is not connected and thus there are **domain walls** between the different sectors. A domain wall is a **two-dimensional** object with non-zero field energy. An example, is a scalar potential with $Z_2$ symmetry that breaks to the trivial subgroup $1$. The vacuum manifold is $Z_2/1=Z_2$ and is therefore disconnected. For a surface, or domain wall, what we have said so far leads us to expect a map $S^0 \to M$ (for a d-dimensional singularity has led to a map $S^{2-d} \to M$). So, the unit sphere $S^0$ in R, consists of the two points $\pm 1$. One of the points, +1, corresponds to a point on one side of the domain wall and the other, -1, to a point on the other side."
  * If the first homotopy class of $G/H$ is non-trivial, we get **one-dimensional** topological defects that are called **strings**. An example is when a scalar potential with $U(1)$ symmetry breaks to the trivial subgroup $1$. The vacuum manifold is $U(1)/1=U(1) \simeq S^1$, which is connected, but not simply connected. This makes it possible that strings show up. (For an explanation with $S^1$ is not simply connected, see section 1.3.1 [[http://people.physics.tamu.edu/pope/geom-group.pdf|here]])
  * If the second homotopy class of $G/H$ is non-trivial, we get a **zero-dimensional** topological defect, a "a pointlike singularity" that is called a **[[advanced_notions:topological_defects:magnetic_monopoles|monopole]]**.  An example is when a scalar potential with $SU(2)$ symmetry breaks to $U(1)$. The vacuum manifold is $SU(2)/U(1)\simeq S^2$.
  * If the third homotopy class of $G/H$ is non-trivial, we get so called “**textures**”. In fact the notion "textures" is more popular among condensed matter physicists and particle physicists call this kind of topological defect **Skyrmions**.  "If space is a three sphere, we can have a texture wrapped around the entire three sphere and this would give a static solution in the model." ([[https://arxiv.org/pdf/hep-ph/9710292.pdf|source]]). An example is when a scalar potential with $SU(2)$ symmetry breaks to the trivial subgroup $1$. The vacuum manifold is $SU(2)/1 =SU(2) \simeq S^3$.

----

** 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

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Domain Wall#

<blockquote>
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

<cite>https://arxiv.org/pdf/hep-ph/9710292.pdf</cite>
</blockquote>

<--

--> What topological defects are present in a given model?#
<blockquote>
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.

<cite>https://arxiv.org/pdf/hep-ph/9710292.pdf</cite>
</blockquote>

<--

--> Why are Topological Defects Stable?#

<blockquote>
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.

<cite>[[http://iopscience.iop.org/article/10.1088/0034-4885/49/7/001/pdf|Topological structures in field theories]] by Goddard and Mansfield </cite>
</blockquote>

<--

--> Example2:#

<--

--> Example2:#

<--

--> Example2:#

<--
  
<tabbox History>

</tabbox>

Please solve the following equation to prove you're human. 163 -9 = Please keep this field empty:

Edit summary

Note: By editing this page you agree to license your content under the following license: CC Attribution-Share Alike 4.0 International

User Tools

Site Tools

Page Tools