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 advanced_notions:topological_defects [2017/12/20 10:11] advanced_notions:topological_defects [2017/12/20 11:11] (current)jakobadmin [FAQ] Both sides previous revision Previous revision 2017/12/20 11:11 jakobadmin [FAQ] 2017/12/20 11:10 jakobadmin [Student] 2017/12/20 11:10 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Layman] 2017/12/20 11:08 jakobadmin [Researcher] 2017/12/20 11:08 jakobadmin [Student] 2017/12/20 11:07 jakobadmin [Examples] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:04 jakobadmin [Why is it interesting?] 2017/12/20 11:03 jakobadmin ↷ Page moved from basic_notions:topological_defects to advanced_notions:topological_defects2017/12/20 11:03 jakobadmin [Student] 2017/12/20 11:02 jakobadmin created Previous revision 2017/12/20 11:11 jakobadmin [FAQ] 2017/12/20 11:10 jakobadmin [Student] 2017/12/20 11:10 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Layman] 2017/12/20 11:08 jakobadmin [Researcher] 2017/12/20 11:08 jakobadmin [Student] 2017/12/20 11:07 jakobadmin [Examples] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:04 jakobadmin [Why is it interesting?] 2017/12/20 11:03 jakobadmin ↷ Page moved from basic_notions:topological_defects to advanced_notions:topological_defects2017/12/20 11:03 jakobadmin [Student] 2017/12/20 11:02 jakobadmin created Line 1: Line 1: + ====== 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​ + ​ + + + ---- + + **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]] + + + + +  ​ + + + * A great laymen introduction to topoltogical defects can be found at https://​skepticsplay.blogspot.de/​2013/​02/​what-are-topological-defects.html + * 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 +  ​ + + + + + + ---- + + ** 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 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 + + + +  ​ + + 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:​** + + + ​ +  ​ + + --> 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​ + ​ + + + + + + <-- + +  ​ + + --> 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​ + ​ + + + <-- + + --> 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 ​ + ​ + + + <-- + + + --> 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​ + ​ + + + <-- + + + --> 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​ + ​ + + + <-- + + + --> Example2:# + + + <-- + ​ +  ​ + + ​ +