Exceptional points for chiral Majorana fermions in arbitrary dimensions

Abstract: 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 changes of signs at a topological phase transition point signal the appearance or disappearance of chiral Majorana zero modes. 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, capture 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, although one can use our formula for counting $n$, using complex $k$ solutions where the determinant of the corresponding BdG Hamiltonian vanishes.