Quantum vortex

In physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was predicted by Lars Onsager in 1947 in connection with superfluid helium. Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions. These ideas of Onsager were further developed by Richard Feynman in 1955^[1] and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov.^[2]

Quantum vortices are observed experimentally in Type-II superconductors, liquid helium, and atomic gases (see Bose–Einstein condensate).

In a superfluid, a quantum vortex “carries” quantized angular momentum, thus allowing the superfluid to rotate; in a superconductor, the vortex carries quantized magnetic flux.

Vortex-quantisation in a superfluid

In a superfluid, a quantum vortex is a hole with the superfluid circulating around the vortex axis; the inside of the vortex may contain excited particles, air, vacuum, etc. The thickness of the vortex depends on a variety of factors; in liquid helium, the thickness is of the order of a few Angstroms.

A superfluid has the special property of having phase, given by the wavefunction, and the velocity of the superfluid is proportional to the gradient of the phase. The circulation around any closed loop in the superfluid is zero if the region enclosed is simply connected. The superfluid is deemed irrotational, however, if the enclosed region actually contains a smaller region with an absence of superfluid, for example a rod through the superfluid or a vortex, then the circulation is:

$\oint_{C} \mathbf{v}\cdot\,d\mathbf{l} = \frac{\hbar}{m}\oint_{C}\nabla\phi_v\cdot\,d\mathbf{l} = \frac{\hbar}{m}\Delta^{tot}\phi_v,$

where $\hbar$ is Planck's constant divided by $2\pi$, m is the mass of the superfluid particle, and $\Delta^{tot}\phi_v$ is the total phase difference around the vortex. Because the wavefunction must return to its same value after an integer number of turns around the vortex (similar to what is described in the Bohr model), then $\Delta^{tot}\phi_v= 2\pi n$, where n is an integer. Thus, the circulation is quantized:

$\oint_{C} \mathbf{v}\cdot\,d\mathbf{l} \equiv \frac{2\pi\hbar}{m}n.$

Vortex-quantization in a superconductor

A principal property of superconductors is that they expel magnetic fields; this is called the Meissner effect. If the magnetic field becomes sufficiently strong, one scenario is for the superconductive state to be “quenched”. However, in some cases, it may be energetically favorable for the superconductor to form a lattice of quantum vortices, which carry quantized magnetic flux through the superconductor. A superconductor that is capable of supporting vortex lattices is called a type-II superconductor, vortex-quantization in superconductors is general.

Over some enclosed area S, the magnetic flux is

$\Phi = \iint_S\mathbf{B}\cdot\mathbf{\hat{n}}\,d^2x = \oint_{\partial S}\mathbf{A}\cdot d\mathbf{l},$ where $\mathbf A$ is the vector potential of the magnetic induction $\mathbf B.$

Substituting a result of London's equation: $\mathbf{j}_s = -\frac{n_se_s^2}{m}\mathbf{A} + \frac{n_se_s\hbar}{m}\mathbf{\nabla}\phi$, we find (with $\mathbf B=\mathrm{curl}\,\, \mathbf A$):

$\Phi =-\frac{m}{n_s e^2}\oint_{\partial S}\mathbf{j}_s\cdot d\mathbf{l} +\frac{\hbar}{e_s}\oint_{\partial S}\mathbf{\nabla}\phi\cdot d\mathbf{l}$,

where n_s, m, and e_s are, respectively, number density, mass, and charge of the Cooper pairs.

If the region, S, is large enough so that $\mathbf{j}_s = 0$ along $\partial S$, then

$\Phi = \frac{\hbar}{e_s}\oint_{\partial S}\mathbf{\nabla}\phi\cdot d\mathbf{l} = \frac{\hbar}{e_s}\Delta^{tot}\phi = \frac{2\pi\hbar}{e_s}n.$

The flow of current can cause vortices in a superconductor to move, causing the electric field due to the phenomenon of electromagnetic induction. This leads to energy dissipation and causes the material to display a small amount of electrical resistance while in the superconducting state.^[3]

Constrained vortices in ferromagnets and antiferomagnets

The vortex states in ferromagnetic or antiferromagnetic material are also important, mainly for information technology^[4] They are exceptional, since in contrast to superfluids or superconducting material one has a more subtle mathematics: instead of the ususal equation of the type $\operatorname{curl} \ \vec v (x,y,z,t)\propto\vec \Omega (\mathrm r,t)\cdot\delta (x,y),$ where $\vec \Omega (\mathrm r,t)$ is the vorticity at the spatial and temporal coordinates, and where $\delta (x,y)$ is the Dirac function, one has:

$\operatorname{curl} \, \vec v (x,y,z,t)\propto\vec m_\mathrm{eff} (\mathrm r,t)\cdot\delta(x,y) \ \ (eqn. *),$

where now at any point and at any time there is the constraint $m_x^2(\mathrm r, t)+m_y^2(\mathrm r,t)+m_z^2(\mathrm r,t)\equiv M_0^2.$ Here $M_0$ is constant, the constant magnitude of the non-constant magnetization vector $\vec m(x,y,z,t).$ As a consequence the vector $\vec m$ in eqn. (*) has been modified to a more complex entity $\vec m_\mathrm{eff}.$ This leads, among other points, to the following fact:

In ferromagnetic or antiferromagnetic material a vortex can be moved to generate bits for information storage and recognition, corresponding, e.g., to changes of the quantum number n. ^[4] But although the magnetization has the usual azimutal direction, and although one has vorticity quantization as in superfluids, as long as the circular integration lines surround the central axis at far enough perpendicular distance, this apparent vortex magnetization will change with the distance from an azimutal direction to an upward or downward one, as soon as the vortex center is approached.

Thus, for each directional element $\mathrm d\varphi \,\mathrm d\vartheta$ there are now not two, but four bits to be stored by a change of vorticity: The first two bits concern the sense of rotation, clockwise or counterclockwise; the remaining bits three and four concern the polarization of the central singular line, which may be polarized up- or downwards. The change of rotation and/or polarization involves subtle topology. ^[5]

Statistical mechanics of vortex lines

As first discussed by Onsager and Feynman, If the temperature in a superfluid or a superconductor is raised, the vortex loops undergo a second-order phase transition. This happens when the configurational entropy overcomes the Boltzmann factor which suppresses the thermal or heat generation of vortex lines. The lines form a condensate. Since the center of the lines, the vortex cores, are normal liquid or normal conductors, respectively, the condensation transforms the superfluid or superconductor into the normal state. The ensembles of vortex lines and their phase transitions can be described efficiently by a gauge theory.

See also

• Macroscopic quantum phenomena
• Vortex
• Abrikosov vortex
• Josephson vortex
• Fractional vortices
• Superfluid helium-4
• Superfluid film
• Superconductor
• Type-II superconductor
• Type-1.5 superconductor
• Quantum turbulence
• Bose–Einstein condensate

References

1. Feynman, R. P. (1955). "Application of quantum mechanics to liquid helium". Progress in Low Temperature Physics. Progress in Low Temperature Physics 1: 17–53. doi:10.1016/S0079-6417(08)60077-3. ISBN 978-0-444-53307-4.
2. • Abrikosov, A. A. (1957) "On the Magnetic properties of superconductors of the second group", Sov.Phys.JETP 5:1174–1182 and Zh.Eksp.Teor.Fiz.32:1442-1452.
3. "First vortex 'chains' observed in engineered superconductor". Physorg.com. Retrieved 2011-03-23.
4. Magnetic vortices in nanodisks reveal information (with a drawing), physics.org, march 03, 2015.
5. O.V. Pylipovskyi et al.: Polarity Switching in Magnets with Surface Anisotropy, arxiv.org/pdf/1501.06548, January 2015