Quantizing Majorana fermions in a superconductor

Abstract

A Dirac-type matrix equation governs surface excitations in a topological insulator in contact with an s-wave superconductor. The order parameter can be homogenous or vortex valued. In the homogenous case a winding number can be defined whose nonvanishing value signals topological effects. A vortex leads to a static, isolated, zero-energy solution. Its mode function is real and has been called "Majorana." Here we demonstrate that the reality/Majorana feature is not confined to the zero-energy mode but characterizes the full quantum field. In a four-component description a change in basis for the relevant matrices renders the Hamiltonian imaginary and the full, space-time-dependent field is real, as is the case for the relativistic Majorana equation in the Majorana matrix representation. More broadly, we show that the Majorana quantization procedure is generic to superconductors, with or without the Dirac structure, and follows from the constraints of fermionic statistics on the symmetries of Bogoliubov-de Gennes Hamiltonians. The Hamiltonian can always be brought to an imaginary form, leading to equations of motion that are real with quantized real-field solutions. Also we examine the Fock space realization of the zero-mode algebra for the Dirac-type systems. We show that a two-dimensional representation is natural, in which fermion parity is preserved.

Evolution of Density of States and a Spin-Resolved Checkerboard-Type Pattern Associated with the Majorana Bound State

“…The detailed wave functions of MBS are Fig. 4 with spin up and down separately, where the amplitudes oscillate with distance from vortex center in terms of Bessel functions [25,26]. According to our numerical calculations, the excited state with energy E = 0.2∆ 0 and total angular momentum j = −1 exhibits a spatial distribution almost the same as the MBS [see Fig.…”

“…As shown in Fig. 3(e), the MBSs distribute over a larger area in the vortex core when the chemical potential is lowered, since their wave function is given by the Bessel function J 0 (rk F ) [25,26], where k F , the intercept of Dirac dispersion with Fermi level, decreases with decreasing µ upon lowering the Fermi level [see Fig. 1(b)].…”

“…The spatial distributions of the two MBSs on the top and bottom surfaces are the same since in the present simplified model the effect of SC substrate is taken into account by a uniform proximity-induced SC gap. Because the two MBSs are localized at the two surfaces of TI slab, they are well described by the 2D model [13], and the analytical solution [25,26].…”

“…MBS is a unique quasiparticle excitation of topological SC carrying zero energy and zero total angular momentum, which appears at ends of one-dimensional systems or at vortex cores, where the SC gap is closed [13][14][15][16][20][21][22][23][24][25][26][27][28]. The unique property of equivalence between particle and antiparticle, which renders MBS noble for advanced applications, makes them difficult to be identified [9,[29][30][31].…”

Annihilation of Colliding Bogoliubov Quasiparticles Reveals their Majorana Nature

“…A real wave equation implies the linear relation Ψ † ðr; tÞ ¼ UΨðr; tÞ between the particle and antiparticle field operators, which is the hallmark of a Majorana fermion. As argued forcefully by Chamon et al [14], fermionic statistics plus superconductivity by itself produces Majorana fermions, irrespective of considerations of dimensionality, topology, or broken symmetries.Here we propose an experiment to probe the Majorana nature of Bogoliubov quasiparticles in conventional, nontopological, superconductors. Existing proposals apply to topological superconductors [15][16][17][18][19][20][21][22][23][24][25], where Majorana fermions appear as charge-neutral edge states with a distinct signature in DC transport experiments.…”