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Majorana Fermions in Condensed Matter

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Abstract

Fermi’s opinion that Ettore Majorana ranked within the Olympus of geniuses finds its best demonstration in the ninth Majorana’s paper [1], the last being published before his misterious disappearance on the 26th March 1938 at the age of 31. The paper, perfectly translated from the original Italian by Luciano Maiani [2] and entitled “A symmetric theory of electrons and positron”, aimed at modifying Dirac equation through a new quantization process, so as to avoid the solutions of negative energy. While for charged fermions such as electrons and positrons the new symmetric theory necessarily implies complex fields, Majorana shows that for neutral fermions the theory leads to the identity of particles with their own antiparticles, thus arguing “that there is no reason now to infer the existence of antineutrons and antineutrinos.” [2]. The process consists in replacing Dirac’s anticommuting gamma matrices γ μ with a new set of fully imaginary Majorana’s anticommuting matrices \( {\tilde{\gamma}}^{\mu }=-\left({\tilde{\gamma}}^{\mu}\right)^{\ast} \) which fulfil the Clifford algebra. In this way the operator \( i\;{\tilde{\gamma}}^{\mu }{\partial}_{\mu } \) is real, thus allowing for real eigenfuction. The reality of field operators is a prerequisite for a particle being its own eigenfuction, as it implies the identity between particle and antiparticle creator operators, ψ*  ≡ ψ. Among the numerous contributions made by Majorana in his short life, that of “Majorana fermions” is perhaps the most profound and far-reaching one. As Frank Wilczek remarks in his fashinating 2009 Nature Physics perspective “Majorana returns” [3], “for 70 years his modified equation remained a rather obscure footnote in theoretical physics. Now suddenly …. Majorana’s concept is ubiquitous, and his equation is central to recent work not only in neutrino physics, super symmetry and dark matter, but also on some exotic states of ordinary matter.”

Just the day this note was completed I received the sad news that Shoucheng Zhang passed away on December 1st, 2018, at only 55. This note is dedicated to his memory in consideration of his fundamental theoretical and experimental contribution to the discovery of Majorana fermions in condensed matter.

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  1. University of Milano-Bicocca, Milan, Italy

    Giorgio Benedek

  2. Donostia International Physics Center, Donostia/San Sebastian, Spain

    Giorgio Benedek

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  1. Department of Physics, University of Bologna, Bologna, Italy

    Luisa Cifarelli

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Benedek, G. (2020). Majorana Fermions in Condensed Matter. In: Cifarelli, L. (eds) Scientific Papers of Ettore Majorana. Springer, Cham. https://doi.org/10.1007/978-3-030-23509-3_14

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