Counting Majorana bound states using complex momenta

Abstract

Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov-de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a formula has been proposed to count the number (n) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symmetry, these solutions are the so-called "exceptional points", where two or more eigenvalues of the complexified Hamiltonian coalesce.

Majorana fermions in finite-size strips with in-plane magnetic fields

“…Second we use the singular points (SP) technique introduced in Refs. [1][2][3][4], based on the momentum values where the determinant of the Hamiltonian vanishes, and we show that it yields results consistent with the numerical ones.…”

“…Since the chiral symmetry (a combination of the PHS and the TRS) is absent, we cannot easily use the counting formula introduced in Ref. [3] to obtain the number of Majorana modes using this method. The counting formula is in principle also applicable in the absence of the chiral symmetry, but the broken TRS case is very cumbersome and much harder to implement numerically.…”

Signatures of two- and three-dimensional semimetals from circular dichroism

“…This chirality is in analogy with the chiral Majorana quasiparticles that appear at the edges of topological nanowires[26][27][28][29][30][31]. In the context of the semimetals, the sign of the topological charge (or in turn, the chirality) can be related to the chiral edge states observed in various experiments[32].…”