Exceptional points for chiral Majorana fermions in arbitrary dimensions

Certain real parameters of a Hamiltonian, when continued to complex values, can give rise to singular points called exceptional points ($EP$'s), where two or more eigenvalues coincide and the complexified Hamiltonian becomes non-diagonalizable. We show that for a generic $d$-dimensional topological superconductor/superfluid with a chiral symmetry, one can find $EP$'s associated with the chiral zero energy Majorana fermions bound to a topological defect/edge. Exploiting the chiral symmetry, we propose a formula for counting the number ($n$) of such chiral zero modes. We also establish the connection of these solutions to the Majorana fermion wavefunctions in the position space. The imaginary parts of these momenta are related to the exponential decay of the wavefunctions localized at the defect/edge, and hence their change of sign at a topological phase transition point signals the appearance or disappearance of a chiral Majorana zero mode. Our analysis thus explains why topological invariants like the winding number, defined for the corresponding Hamiltonian in the momentum space for a defectless system with periodic boundary conditions, captures the number of admissible Majorana fermion solutions for the position space Hamiltonian with defect(s). Finally, we conclude that $EP$'s cannot be associated with the Majorana fermion wavefunctions for systems with no chiral symmetry, though one can use our formula for counting $n$, using complex $k$ solutions where the determinant of the corresponding BdG Hamiltonian vanishes.Comment: 5 pages; published versio

Counting Majorana bound states using complex momenta

Recently, the connection between Majorana fermions bound to 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, $\textbf{110}$, 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.Comment: 21 pages, 10 figure