Topological Defects

Prerequisites: Symmetry Breaking, Topology. Full path (14 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract
• Examples
• FAQ

Why is it interesting?

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 coset space $G/H$. The 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 Dark Matter by Hitoshi Murayama, Jing Shu

Important Topological Defects:

• Magnetic Monopoles
• Vortices
• kinks
• strings

Intuitive

When a lake freezes, ice starts to form at many places at once. Each patch of ice grows with its own crystal orientation, and where two patches with different orientations meet, they cannot fit together perfectly. A seam of misaligned molecules remains between them. Such seams are defects, and they persist because removing one would mean rearranging a whole patch of ice.

The same thing happens whenever a system settles into one of several equivalent ordered states. Think of a field of compass needles that all want to point in the same direction but have no preference for which direction. In one region they agree on north, in a distant region on east. In between, the direction has to turn smoothly. Sometimes this works out with no trouble. In other cases the arrangement contains a twist that cannot be smoothed out anywhere: walk in a circle around a certain point and the needles turn once around the full compass. Near that point the needles cannot decide on any direction, so the order breaks down there, and this costs energy.

A topological defect is such a place where the order must break down because of the way the surrounding field is arranged. The word "topological" says that the reason is a counting property, like the number of times the needles wind around, which cannot change under small smooth deformations. So the defect cannot simply decay. It can only disappear by meeting a defect of the opposite kind, or by moving out of the system.

Depending on the kind of order, defects are walls (between two regions in different states), lines (like the swirl in a whirlpool, or a vortex in a superfluid), or points (like a hedgehog of needles all pointing outward). They are seen in liquid crystals, magnets, superfluids and superconductors. In cosmology one expects them to have formed when the early universe cooled through symmetry breaking phase transitions, for the same reason as the seams in the ice.

• A great laymen introduction to topoltogical defects can be found at Skeptic's Play: What are topological defects?
• 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

Concrete

The three standard examples below follow the same pattern. A field theory has a potential with a set of degenerate minima, the vacuum manifold. Far from the defect the field sits in this manifold, and the way it does so on a large sphere or circle around the defect cannot be undone continuously. Somewhere inside, the field must leave the vacuum manifold, and the potential energy stored there is the defect's energy. We use a static real or complex scalar field with energy density $\frac{1}{2}(\nabla\phi)^2 + V(\phi)$ (or $|\nabla\phi|^2 + V$ for complex $\phi$).

Domain wall (kink). Take a real scalar with $$V(\phi) = \frac{\lambda}{4}\left(\phi^2 - \eta^2\right)^2 ,$$ with two minima $\phi = \pm\eta$, related by the $\mathbb{Z}_2$ symmetry $\phi\to-\phi$. A configuration with $\phi \to -\eta$ for $x \to -\infty$ and $\phi\to+\eta$ for $x\to+\infty$ must pass through $\phi = 0$. The static field equation $\phi'' = V'(\phi) = \lambda\phi(\phi^2 - \eta^2)$ is solved by $$\phi(x) = \eta\tanh\left(\sqrt{\frac{\lambda}{2}}\,\eta\, x\right),$$ as one checks by differentiating twice. The wall has thickness $\delta \sim (\sqrt\lambda\,\eta)^{-1}$ and energy per unit area $$\sigma = \int_{-\infty}^{\infty} dx\left[\frac{1}{2}\phi'^2 + V(\phi)\right] = \frac{2\sqrt 2}{3}\sqrt\lambda\,\eta^3 .$$ For this solution the two terms in the integrand are equal pointwise, which makes the integral easy. In three dimensions the kink extended in $y$ and $z$ is a two-dimensional wall.

Vortex (string). Take a complex scalar with $V = \frac{\lambda}{4}(|\phi|^2 - \eta^2)^2$, whose minima form a circle $\phi = \eta e^{i\alpha}$. In the plane, the ansatz $$\phi(r,\theta) = f(r)\, e^{in\theta}, \qquad f(0) = 0,\; f(\infty) = \eta ,$$ has winding number $$n = \frac{1}{2\pi}\oint_C \nabla\alpha\cdot d\vec l \in \mathbb{Z}$$ around any large circle $C$, where $\alpha$ is the phase of $\phi$. Single-valuedness of $\phi$ forces $n$ to be an integer, and an integer cannot change under continuous deformations, so a configuration with $n \neq 0$ cannot be deformed into the uniform vacuum. Far from the core $|\nabla\phi|^2 \approx \eta^2 n^2/r^2$, so the energy per unit length within radius $R$ grows as $2\pi n^2\eta^2\ln(R/\delta)$: a global vortex has a logarithmically divergent energy. If the $U(1)$ is gauged, the gauge field cancels the gradient at large $r$ and the vortex has finite energy per length and carries a quantised magnetic flux (see vortices). In three dimensions the vortex extends along a line, a string.

Monopole (hedgehog). Take a real triplet $\vec\phi$ with $V = \frac{\lambda}{4}(\vec\phi^{\,2} - \eta^2)^2$, whose minima form a sphere $S^2$. The hedgehog $$\vec\phi(\vec x) = \eta\, h(r)\,\frac{\vec x}{r}, \qquad h(0) = 0,\; h(\infty) = 1 ,$$ maps a large sphere around the origin once onto the vacuum sphere and cannot be unwound. Coupling to an $SO(3)$ gauge field gives the 't Hooft-Polyakov magnetic monopole.

Formation. In a phase transition the field picks its vacuum value independently in regions farther apart than a correlation length, since there is no reason for distant regions to agree. Where several regions meet, the vacuum values can wind, and defects are left behind. This is the Kibble mechanism, and it predicts a defect density set by the correlation length at the transition.

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
• From Monopoles to Textures by Damian Sowinski

Abstract

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 homotopy groups of the vacuum manifold $G/H$ (speak G mod H), we get different topological defects.

• If $\pi_0(G/H)$ (the set of connected components) 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 group $\pi_1(G/H)$ 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 here)
• If the second homotopy group $\pi_2(G/H)$ is non-trivial, we get a zero-dimensional topological defect, a "a pointlike singularity" that is called a 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 group $\pi_3(G/H)$ is non-trivial, we get so called “textures”. The name "textures" is common in cosmology; closely related solitons in particle physics are called 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." (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:

Examples

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Functional
4. Group Theory
5. Newton's Second Law
6. Product Spaces
7. Vector Calculus
8. Newtonian Mechanics
9. Symmetry
10. Topology
11. Variational Calculus
12. Euler-Lagrange Equations
13. Lagrangian Formalism
14. Symmetry Breaking

Leads to: Magnetic Monopoles, Vortices.