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advanced_notions:topological_defects [2017/12/20 11:09]
jakobadmin [Layman]
advanced_notions:topological_defects [2017/12/20 11:11] (current)
jakobadmin [FAQ]
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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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   * 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 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 $SU(2)/1 =SU(2) \simeq S^3$.   * 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:​**
 +
  
   ​   ​
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---> ​Example2:#+--> ​What classes of topological defects exist?# 
 + 
 +<​blockquote>​ 
 +The field theories discussed above fall into two classes from the point of view of their topology. Suppose we are working in d space dimensions. In the first place we have theories, like the Abelian Higgs model of §2.1, where we have a potential function 
 +$V(\varphi)$ and $\varphi$ must tend to a zero (i.e. minimum) of V as we approach spatial infinity. In this case, at any given time, $\varphi$ defines a map 
 +$$ \varphi_\infty (\hat n) = \lim_{r\to\infty} \varphi(r \hat n)$$ 
 +which takes its values in the set of values which minimises V, 
 +$$ M=\{ \varphi: V(\varphi)=0 $$ 
 +The directions $\hat n$ in which one can approach infinity constitute a (d - 1)-dimensional 
 +sphere, the unit sphere in $R^d$. Thus $\varphi_\infty$ defines a map $S^{d-1} \to M$.  
 + 
 +**The second class of possibilities is not the result of non-trivial boundary conditions.** 
 +Here we have a field which is always constrained to take its values in some manifold 
 +$M$, which is not simply a linear space. This time the boundary conditions are actually 
 +supposed to be trivial in the sense that $\varphi$ tends to a limit $\varphi_\infty \in M$ of course, independently 
 +of the direction in which we approach spatial infinity. In this case we can compactify 
 +space, $R^d$, by adding a point at infinity, to obtain what is topologically a sphere $S^d$ 
 +(cf stereographic projection),​ with $\varphi$ being assigned the value $\varphi_N$ at the point of $S^d$ 
 +corresponding to infinity. In this way we obtain a map $\varphi$ : $S^d \to M$.  
 + 
 +<​cite>​Topological structures in field theories by Goddard and Mansfield</​cite>​ 
 +</​blockquote>​
  
    
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---> ​Example2:#+--> ​What's the experimental status of topological defects?# 
 + 
 + 
 + 
 +<​blockquote>​ 
 +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 can therefore be largely ruled out. 
 + 
 +<​cite>​https://​arxiv.org/​pdf/​1206.1294.pdf</​cite>​ 
 +</​blockquote>​
  
    
advanced_notions/topological_defects.1513764574.txt.gz · Last modified: 2017/12/20 10:09 (external edit)