Majorana fermion

In elementary particle physics, Majorana fermions are particles with half-integer spin ( fermions ) whose antiparticles have the same properties as the particles themselves. They were named after Ettore Majorana .

Majorana fermions in particular do not carry any electrical charge , because otherwise particles and antiparticles would have opposite charges and would thus be clearly distinguishable, e.g. B. electrons and positrons . Such fermions that can carry a charge are called Dirac fermions .

A distinction must also be made between the majorana fermions and the hypothetical majorons, which are also named after Ettore Majorana, but as Goldstone bosons carry integer spin.

Occur

In the standard model: unexplained position of the neutrinos

In the standard model of particle physics (SM) none of the elementary particles is a Majorana fermion. Instead, all fermions are described here by Dirac spinors , including the neutrinos , which can be distinguished from antineutrinos by default. However, the neutrinos in the Standard Model are massless, in contradiction to experimental results. A popular explanation for the neutrino masses assumed because of the observed neutrino oscillation, the see-saw mechanism, on the other hand , requires the description of the neutrinos by Majorana spinors and thus the equality of neutrinos and antineutrinos. This in turn would imply a violation of lepton number conservation.

Whether a distinction can be made between neutrinos and antineutrinos is currently still open. One possibility for experimental clarification is offered by a questionable decay mode, the neutrino-free double beta decay, which is only possible if neutrinos are Majorana spinors and not Dirac spinors. This mode of decay is sought in experiments such as the Enriched Xenon Observatory (EXO200).

In the MSSM

In supersymmetric extensions of the standard model such as the minimal supersymmetric standard model (MSSM), both the gluinos and the neutralinos are described by Majorana spinors. Neutralinos are candidates for WIMPs and dark matter .

Solid state physics

In solid state physics, the particle-hole identity takes the place of the particle-antiparticle identity. In superconductors, excitations ( quasiparticles ) are formed, which are composed of particle (electron) and hole states and, if there is a suitable spin-orbit coupling (although this does not exist in ordinary superconductors), are equal to their antiparticles, so-called bound majorana -States or Majorana zero modes. The name is the name preferred "Majorana fermions" here as they take a more complex Fermi statistics anyons have statistic. They therefore play a role in topological, fault-tolerant quantum computers ( Alexei Kitaev ). The name zero mode comes from the fact that they have vanishing excitation energy as a particle-hole superposition. Since the Majorana states are topologically protected, one also speaks of topological superconductors.

It had long been known that Majorana states can occur in superconductors with triplet pairing (p-wave superconductors in one dimension, pairing in two dimensions), but this had not yet been observed in experiments. A more realistic way of creating such Majorana states was proposed by Charles L. Kane and Liang Fu in 2008, the coupling of a superconductor to a topological insulator , where a pairing occurs at the surface of the contact of the topological insulator with the ordinary s-wave superconductor . In 2010, Sankar Das Sarma (University of Maryland) and Felix von Oppen (FU Berlin) and their colleagues independently suggested that heterostructures made of ordinary superconductors and suitable semiconductors could be used: a wire made of a semiconductor material with strong spin-orbit coupling of the electrons is brought into contact with a superconductor and is thus partially superconducting itself. In addition, a magnetic field is applied that creates a band gap between the bands for both spin directions. Then a robust Majorana state (topologically protected against interference) should develop at both ends of the wire. Other implementations of Majorana states have also been discussed. In 2015, Arnab Banerjee and co-workers detected Majorana fermions in a quantum spin fluid made from layers of α-ruthenium (III) chloride. ${\displaystyle p_{x}\pm ip_{y}}$ ${\displaystyle p_{x}\pm ip_{y}}$

A group led by Leo P. Kouwenhoven and Sergey Frolov (University of Delft) found evidence of Majorana states following this suggestion in 2012. They used a nanowire made of indium-antimony semiconductor material with a normal gold contact and a superconductor contact. Normally, a current would only be measurable in the wire with an external voltage, but with Majorana zero modes this would also be possible without external voltage, which was also found in the experiment.

Further confirmations of Majorana states in solids were then provided by other groups, such as a Princeton University group using scanning tunneling microscopy.

There are also other applications in statistical mechanics: In 1964 Elliott Lieb , Daniel C. Mattis and Theodore D. Schultz introduced a description of the two-dimensional Ising model with Majorana fermions.

Mathematical description

Similar to the massless Weyl fermions , for which the Dirac equation decouples (see Weyl equation ), Majorana fermions are 2-component particles, but with Majorana mass.

The Lagrangian of a Majorana particle is ${\displaystyle \psi _{M}}$

${\displaystyle {\mathcal {L}}={\frac {1}{2}}{\bar {\psi }}_{M}{\Bigl (}\mathrm {i} \gamma ^{\mu }\partial _{\mu }-m{\Bigr )}\,\psi _{M},}$

where, as is usual in relativistic quantum mechanics , applies. ${\displaystyle {\bar {\psi }}_{M}=\psi _{M}^{\dagger }\gamma ^{0}}$

The corresponding Dirac equation for is: ${\displaystyle \psi _{M}}$

${\displaystyle {\Bigl (}\mathrm {i} \gamma ^{\mu }\partial _{\mu }-m{\Bigr )}\,\psi _{M}=0}$

Set as with the Weyl fermions and note that under a Lorentz transformation ${\displaystyle \psi _{M}={\begin{pmatrix}\psi _{L}\\\psi _{R}\end{pmatrix}}}$

${\displaystyle \psi _{R}=i\sigma ^{2}\psi _{L}^{*}}$

applies, so you can

${\displaystyle \psi _{M}={\begin{pmatrix}\chi \\i\sigma ^{2}\chi ^{*}\end{pmatrix}}}$

and the Majorana equation for the 2-component field results : ${\displaystyle \chi }$

${\displaystyle i{\bar {\sigma }}^{\mu }\partial _{\mu }\chi -im\sigma ^{2}\chi ^{*}=0}$

Here, according to contains the three Pauli spin matrices , and is the second Pauli matrix. ${\displaystyle {\bar {\sigma }}}$ ${\displaystyle {\bar {\sigma }}=(1,-{\vec {\sigma }})}$ ${\displaystyle \sigma ^{2}}$

The Majorana equation is Lorentz invariant and implies the Klein-Gordon equation , which defines the energy-momentum relationship .

literature

• Andreas Aste: A direct road to Majorana fields. 2008. arxiv : 0806.1690 . A compact introduction to the Majorana formalism.
• Martin Leijnse, Karsten Flensberg: Introduction to topological superconductivity and Majorana fermions. Semic. Sci. Techn., Volume 27, 2012, p. 124003.

References and footnotes

1. ↑ Enriched Xenon Observatory
2. ^ Liang Fu, Charles L. Kane : Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett., Vol. 100, 2008, p. 096407, Arxiv.
3. ↑ RM Lutchyn, JD Sau, S. Das Sarma: Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures. Phys. Rev. Lett., Vol. 105, 2010, p. 077001, Arxiv.
4. ↑ Y. Oreg, G. Refael, F. von Oppen: Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett., Vol. 105, 2010, p. 177002, Arxiv.
5. ↑ Jason Alicea: New directions in the pursuit of Majorana fermions in solid state systems. Reports on Progress in Physics, Volume 75, 2012, p. 076501, Arxiv.
6. ↑ A. Banerjee, CA Bridges, J.-Q. Yan, AA Aczel, L. Li, MB Stone, GE Granroth, MD Lumsden, Y. Yiu, J. Knolle, S. Bhattacharjee, DL Kovrizhin, R. Moessner, DA Tennant, DG Mandrus, SE Nagler: Proximate Kitaev quantum spin liquid behavior in a honeycomb magnet. In: Nature Materials. 2016, doi: 10.1038 / nmat4604 .
7. ^ V. Mourik, K. Zuo, SM Frolov, SR Plissard, EPAM Bakkers, LP Kouwenhoven: Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science, Volume 336, 2012, pp. 1003-1007, Arxiv.
8. ^ Rainer Scharf: Majorana's footsteps. Pro-Physik, April 13, 2012.
9. ↑ Majorana fermion: Physicists observe elusive particle that is its own antiparticle. Phys.org, October 2014.
10. ↑ Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen, Sangjun Jeon, Jungpil Seo, Allan H. MacDonald, Andrei Bernevig , Ali Yazdani: Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science, Volume 346, 2014, pp. 602-607.
11. ↑ Jian Li, Hua Chen, Ilya K. Drozdov, A. Yazdani, B. Andrei Bernevig, AH MacDonald: Topological Superconductivity induced by Ferromagnetic Metal Chains. Phys. Rev. B, Volume 90, 2014, p. 235433, Arxiv, for the theoretical treatment.
12. ↑ John Kogut : An introduction to lattice gauge theory and spin systems. In: Reviews of Modern Physics. Vol. 51, 1979, pp. 659-713. Abstract.
13. ↑ TD Schultz, DC Mattis, EH Lieb: Two-Dimensional Ising Model as a Soluble Problem of Many Fermions. In: Reviews of Modern Physics. Vol. 36, 1964, p. 856.