Colloquium: Majorana fermions in nuclear, particle, and solid-state physics

Ettore Majorana (1906–1938) disappeared while traveling by ship from Palermo to Naples in 1938. His fate has never been fully resolved and several articles have been written that explore the mystery itself. His demise intrigues us still today because of his seminal work, published the previous year, that established symmetric solutions to the Dirac equation that describe a fermionic particle that is its own antiparticle. This work has long had a significant impact in neutrino physics, where this fundamental question regarding the particle remains unanswered. But the formalism he developed has found many uses as there are now a number of candidate spin-$1/2$ neutral particles that may be truly neutral with no quantum number to distinguish them from their antiparticles. If such particles exist, they will influence many areas of nuclear and particle physics. Most notably the process of neutrinoless double beta decay can exist only if neutrinos are massive Majorana particles. Hence, many efforts to search for this process are underway. Majorana’s influence does not stop with particle physics, however, even though that was his original consideration. The equations he derived also arise in solid-state physics where they describe electronic states in materials with superconducting order. Of special interest here is the class of solutions of the Majorana equation in one and two spatial dimensions at exactly zero energy. These Majorana zero modes are endowed with some remarkable physical properties that may lead to advances in quantum computing and, in fact, there is evidence that they have been experimentally observed. This Colloquium first summarizes the basics of Majorana’s theory and its implications. It then provides an overview of the rich experimental programs trying to find a fermion that is its own antiparticle in nuclear, particle, and solid-state physics.

Figure 1

Generic diagram showing a $\mathrm{\Delta }L=2$ process involving the exchange of a Majorana neutrino. $f$ is a generic fermion, $W^{\pm }$ is the weak-interaction intermediate vector boson, and $l^{\pm }$ is an outgoing lepton. The two $W$ and $l$ have the same charge for this $\mathrm{\Delta }L=2$ process.

Figure 2

Diagrams showing the $\beta \beta (2\nu )$ (top) and $\beta \beta (0\nu )$ (bottom) processes. Within the group of nucleons inside a nucleus, two neutrons simultaneously emit $\beta$ particles while producing protons.

Figure 3

The spectrum of the sum of the energies of the two electrons from $\beta \beta (2\nu )$ (dotted) and $\beta \beta (0\nu )$ (solid). The resolution used for $\beta \beta (0\nu )$ is arbitrary and the relative strength of $\beta \beta (0\nu )$ compared to that of $\beta \beta (2\nu )$ is exaggerated for clarity. In reality, if $\beta \beta (0\nu )$ exists, the peak would be very weak.

Figure 4

$⟨m_{\beta \beta }⟩$ as a function of the smallest of the three mass eigenvalues. The plot is done for the best fit oscillation parameters. From [77].

Figure 5

A history of the effective Majorana neutrino mass limit from $\beta \beta (0\nu )$. The shaded band indicates the range of masses one would deduce from the reported $T_{1/2}^{0\nu }$ as a result of the spread of matrix element calculations. The straight line is an extrapolation drawn by the eye for a mass value improvement of a factor of 2 every 6 years. The lower shaded region is the target derived from neutrino oscillation experiments.

Figure 6

Two phases of the Kitaev chain. (a) In the trivial phase Majorana fermions on each lattice site can be thought of as bound into ordinary fermions. (b) In the topological phase Majoranas on neighboring sites are bound leaving two unpaired Majorana fermions at the ends of the chain. (c) The phase diagram of the Kitaev chain in the $\mu \text{−}2t$ plane, showing the topological phase (TSC) and the normal phase. The dots mark the special points in the parameter space considered in the text.

Figure 7

(a) 2D topological insulator interfaced with a superconductor (SC) and two ferromagnetic (FM) insulators. MZMs are expected to occur at the SC and FM boundaries. (b) Schematic spectrum of the surface state in a 2D TI. The shaded regions represent the bulk conduction and valence bands. (c) SC and magnetic order parameter profiles near the SC and FM boundary assumed in the calculation. The dashed line shows the resulting Majorana wave function amplitude.

Figure 8

(a) A 3D topological insulator wire in longitudinal magnetic field $\mathit{B}$. (b) Spectrum of the surface state excitations when the total flux piercing the wire $\eta =\mathrm{\Phi }/{\mathrm{\Phi }}_0=SB/{\mathrm{\Phi }}_0$ is half integral. The dashed lines indicate nondegenerate bands while those represented by the solid lines are doubly degenerate. (c) The topological phase diagram in the limit of small $\mathrm{\Delta }$. Shaded regions represent the topological phase and the numerals indicate the number of Fermi points in the right half of the Brillouin zone.

Figure 9

Excitation spectra of a single channel semiconductor quantum wire Eq. (78). The three panels show representative cases with different Zeeman coupling: (a) $V_Z/E_{\mathrm{SO}}=0.0$, (b) 0.4, and (c) 1.2. The arrows indicate the spin direction at the Fermi points.

Figure 10

(a) Schematic setup for the Fu-Kane model with a 3D TI covered with a thin layer of SC film. A vortex is depicted in the surface layer with a small core, phase winding $2\pi$ and a sketch of a magnetic field profile $\mathit{B}(\mathit{r})$. (b) A vortex with Majorana zero mode ${\gamma }_2$ encircles a vortex placed at the origin with Majorana ${\gamma }_1$ in its core.

Figure 11

(a) Upon exchange of two Majoranas one of them must cross the branch cut implied by the wave function Eq. (90). When the branch cuts are chosen as indicated by the dashed lines then counterclockwise exchange of ${\gamma }_i$ and ${\gamma }_j$ results in rules summarized in Eq. (95). (b) Setup used for the simplest topologically protected operations on the internal Hilbert space spanned by four MZMs.

Figure 12

The Delft experiment. (a) Tunneling conductance $g(V)$ as a function of the voltage bias showing a SC gap at low magnetic field and the emergence of a zero-bias peak attributed to the MZM at higher fields. (b) Scanning electron microscope image of the device with normal ($N$) and superconducting ($S$) contacts attached to an InSb nanowire. Adapted from [127].