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.

Some entries in bib file fixed. Companion paper to arXiv:1503.06804

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Cited by in corpus (17)

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• Signatures of two- and three-dimensional semimetals from circular dichroism
• Enhanced eigenvector sensitivity and algebraic classification of sublattice-symmetric exceptional points
• Exceptional point description of one-dimensional chiral topological superconductors/superfluids in BDI class
• Acceleration toward polarization singularity inspired by relativistic ExB drift
• Majorana fermions in finite-size strips with in-plane magnetic fields
• Identifying gap-closings in open non-Hermitian systems by Biorthogonal Polarization
• Symmetry-breaking signatures of multiple Majorana zero modes in one-dimensional spin-triplet superconductors
• Anomalous Josephson effect of s-wave pairing states in chiral double layers
• Non-Hermitian generalizations of the Yao-Lee model augmented by SO(3)-symmetry-breaking terms
• Nested Defects on the Boundary of Topological Superconductors
• Characterizing topological pumping of charges in exactly solvable Rice-Mele chains of the non-Hermitian variety
• Chiral Josephson effect in double layers: the role of particle-hole duality