Vortices

Prerequisites: Topological Defects. Full path (15 pages)

• Intuitive
• Concrete
• Abstract
• History

Intuitive

An example of a vortex is universally encountered by people taking baths or washing dishes. As the water flows down the drain, it circulates. We cannot interpolate the circulating velocity field all the way to the center of the vortex since it would have to become multi-valued. Instead the fluid density in the central region of the vortex vanishes.

https://arxiv.org/pdf/hep-ph/9710292.pdf

Concrete

Consider the Lagrangian: $$\begin{equation} L = |\partial _\mu \phi |^2 - {\lambda\over 4}(|\phi |^2 -\eta^2 )^2 \end{equation}$$ where $\phi$ is now taken to be a complex scalar field. The Lagrangian is invariant under $$\phi \rightarrow \phi ' = e^{i\alpha} \phi$$ and hence the model has a $U(1)$ (global) symmetry. The vacuum expectation value of $\phi$ is $\eta e^{i\alpha}$ where $\alpha$ can take any value. So the ground state of the model has continuous degeneracy. The degeneracy is labelled by the phase angle $\alpha$ and hence the vacuum manifold is a circle.

Vortices are formed if we consider the model in two spatial dimensions and let $\alpha$ be such as to wrap around the vacuum manifold. For example, we could take $\alpha =\theta$, the polar angle. Then, since the field is single valued everywhere, there must be at least one point at which $\phi =0$. The field carries energy at this point since $\phi =0$ is not on the vacuum manifold. The location of this point may be defined as the location of a vortex.

https://arxiv.org/pdf/hep-ph/9710292.pdf

Abstract

Topological classification. A vortex in two spatial dimensions (a string in three) is a defect of codimension two. Far from the core the field lies in the vacuum manifold $\mathcal M_0 \cong G/H$, so a large circle around the core is mapped to a loop in $G/H$. Free homotopy classes of such loops classify vortices, and for a connected $G/H$ they correspond to conjugacy classes of $$\pi_1(G/H).$$ For the model in the Concrete layer, $G = U(1)$, $H = 1$ and $\pi_1(S^1) = \mathbb{Z}$, the winding number. If $G$ is connected and simply connected, the exact sequence of homotopy groups gives $\pi_1(G/H) \cong \pi_0(H)$, so vortices exist when the unbroken group is disconnected. The group $\pi_1(G/H)$ can be non-Abelian. Then in three dimensions two strings whose classes do not commute cannot pass through each other without leaving a connecting string. Examples from condensed matter are uniaxial nematics with $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$ and biaxial nematics, whose line defects are classified by the quaternion group.

Abelian Higgs model. Gauging the $U(1)$ of the Concrete layer (units $\hbar = c = 1$, $D_\mu = \partial_\mu - ieA_\mu$) gives the static energy $$E = \int d^2x \left[\frac{1}{2}B^2 + |D_i\phi|^2 + \frac{\lambda}{4}\left(|\phi|^2 - \eta^2\right)^2\right], \qquad B = \partial_1A_2 - \partial_2A_1 .$$ The global vortex has energy diverging as $\ln R$. Here the gauge field can make $D_i\phi \to 0$ at infinity, so the energy is finite. With $\phi \to \eta\, e^{in\theta}$ at large $r$, $D_i\phi \to 0$ requires $A_\theta \to n/(er)$, and the magnetic flux is quantised: $$\Phi = \int d^2x\, B = \oint A\cdot dl = \frac{2\pi n}{e} .$$ These are the Nielsen-Olesen vortices. In a superconductor, where $e \to 2e$ for Cooper pairs, this is the flux quantum $h/2e$ carried by an Abrikosov vortex.

Bogomolny bound. The vector mass is $m_A^2 = 2e^2\eta^2$ and the scalar mass is $m_\phi^2 = \lambda\eta^2$. At critical coupling $\lambda = 2e^2$ (so $m_\phi = m_A$), completing squares gives, up to a boundary term that vanishes for finite-energy configurations, $$E = \int d^2x\left[\frac{1}{2}\left(B \mp e\left(|\phi|^2 - \eta^2\right)\right)^2 + \left|(D_1 \pm iD_2)\phi\right|^2\right] \pm e\eta^2\,\Phi \;\geq\; 2\pi\eta^2|n|,$$ with the sign chosen to match the sign of $n$ and with sign conventions for $B$ that vary between references. The bound is saturated by solutions of the first-order Bogomolny equations $(D_1 \pm iD_2)\phi = 0$, $B = \pm e(|\phi|^2 - \eta^2)$. At critical coupling static vortices exert no net force on each other, and the $n$-vortex solutions form a moduli space of real dimension $2n$, parametrised by the positions of the $n$ zeros of $\phi$ in the plane. Away from critical coupling, vortices attract for $\lambda < 2e^2$ (type I) and repel for $\lambda > 2e^2$ (type II).

Geometric view. On a compact surface, or on $\mathbb{R}^2$ with the boundary condition above, $A$ is a connection on a $U(1)$ bundle $L$ and $\phi$ is a section of $L$. The vortex number is the first Chern number, $n = \frac{e}{2\pi}\int B$, which equals the number of zeros of $\phi$, each counted with its local winding number.

History

Lord Kelvin famously tried to explain the different atoms as vortices.

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
15. Topological Defects