advanced_notions:topological_defects

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+ | ====== Topological Defects ====== | ||

+ | |||

+ | <tabbox Why is it interesting?> | ||

+ | |||

+ | <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</cite> | ||

+ | </blockquote> | ||

+ | |||

+ | |||

+ | ---- | ||

+ | |||

+ | **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]] | ||

+ | |||

+ | |||

+ | |||

+ | |||

+ | <tabbox Layman> | ||

+ | |||

+ | |||

+ | * 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 | ||

+ | <tabbox Student> | ||

+ | |||

+ | |||

+ | |||

+ | |||

+ | |||

+ | ---- | ||

+ | |||

+ | ** 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 | ||

+ | |||

+ | |||

+ | |||

+ | <tabbox Researcher> | ||

+ | |||

+ | 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:** | ||

+ | |||

+ | |||

+ | | ||

+ | <tabbox Examples> | ||

+ | |||

+ | --> 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> | ||

+ | |||

+ | |||

+ | |||

+ | |||

+ | |||

+ | <-- | ||

+ | |||

+ | <tabbox FAQ> | ||

+ | |||

+ | --> 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</cite> | ||

+ | </blockquote> | ||

+ | |||

+ | |||

+ | <-- | ||

+ | |||

+ | --> 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 </cite> | ||

+ | </blockquote> | ||

+ | |||

+ | |||

+ | <-- | ||

+ | |||

+ | |||

+ | --> 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> | ||

+ | |||

+ | |||

+ | <-- | ||

+ | |||

+ | |||

+ | --> 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> | ||

+ | |||

+ | |||

+ | <-- | ||

+ | |||

+ | |||

+ | --> Example2:# | ||

+ | |||

+ | |||

+ | <-- | ||

+ | | ||

+ | <tabbox History> | ||

+ | |||

+ | </tabbox> | ||

+ | |||

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