advanced_notions:topological_defects
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advanced_notions:topological_defects [2017/12/20 11:08] jakobadmin [Student] |
advanced_notions:topological_defects [2017/12/20 11:10] jakobadmin [Student] |
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| - | <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. | + | * A great laymen introduction to topoltogical defects can be found at https://skepticsplay.blogspot.de/2013/02/what-are-topological-defects.html |
| - | </note> | + | * See also http://web.mit.edu/8.334/www/grades/projects/projects14/TrungPhan_8334WP/foundation-5.2.2/index.html for some very nice illustration of topological defects |
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| * A great introduction is http://www.dartmouth.edu/~dbr/topdefects.pdf and | * A great introduction is http://www.dartmouth.edu/~dbr/topdefects.pdf and | ||
| - | * see also http://www.lassp.cornell.edu/sethna/pubPDF/OrderParameters.pdf | + | * see also http://www.lassp.cornell.edu/sethna/pubPDF/OrderParameters.pdf and |
| + | * [[https://www.scribd.com/document/85012149/From-Monopoles-to-Textures-A-Survey-of-Topological-Defects-in-Cosmological-Quantum-Field-Theory|From Monopoles to Textures]] by Damian Sowinski | ||
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| <tabbox Researcher> | <tabbox Researcher> | ||
| - | <note tip> | + | 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. |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | * 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 $SU(2)/1 =SU(2) \simeq S^3$. | ||
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| + | ---- | ||
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| + | **Recommended Resources:** | ||
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advanced_notions/topological_defects.txt · Last modified: 2017/12/20 11:11 by jakobadmin
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