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advanced_notions:topological_defects [2017/12/20 11:07]
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 <tabbox Layman> ​ <tabbox Layman> ​
  
-<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
-  ​+
 <tabbox Student> ​ <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 $SU(2)/1 =SU(2) \simeq S^3$. 
  
  
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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-trivialwe 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.1513764435.txt.gz · Last modified: 2017/12/20 10:07 (external edit)