advanced_notions:topological_defects
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advanced_notions:topological_defects [2017/12/20 11:03] jakobadmin ↷ Page moved from basic_notions:topological_defects to advanced_notions:topological_defects |
advanced_notions:topological_defects [2017/12/20 11:11] (current) jakobadmin [FAQ] |
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| - | 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 [[group_theory:notions:quotient_group|coset space]] $G/H$. The [[topology:notions: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. | + | 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> | <cite>[[https://arxiv.org/abs/0905.1720|Topological Dark Matter]] by Hitoshi Murayama, Jing Shu</cite> | ||
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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. <q>"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." </q> | ||
| - | * 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 **[[topology:notions: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**. <q>"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."</q> ([[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 $SU(2)/1 =SU(2) \simeq S^3$. | ||
| - | <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> | ||
| ---- | ---- | ||
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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 $SU(2)/1 =SU(2) \simeq S^3$. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | **Recommended Resources:** | ||
| | | ||
| <tabbox Examples> | <tabbox Examples> | ||
| - | --> Example1# | + | --> 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> | ||
| - | <-- | ||
| - | --> Example2:# | ||
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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.1513764227.txt.gz · Last modified: 2017/12/20 10:03 (external edit)
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