Counting Majorana bound states using complex momenta

Abstract

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, $\text{110}$, 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.

21 pages, 10 figures

References in corpus (20)

• Non-Abelian Anyons and Topological Quantum Computation
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Classification of topological insulators and superconductors in three spatial dimensions
• The physics of exceptional points
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Reversing the Pump-Dependence of a Laser at an Exceptional Point
• Non-Abelian quantum order in spin-orbit-coupled semiconductors: The search for topological Majorana particles in solid state systems
• $p_x+i{p}_y$ superfluid from s-wave interactions of fermionic cold atoms
• A Majorana smoking gun for the superconductor-semiconductor hybrid topological system
• Transition from fractional to Majorana fermions in Rashba nanowires
• Zero modes and the edge states of the honeycomb lattice
• Kramers Pairs of Majorana Fermions and Parafermions in Fractional Topological Insulators
• Tunneling of anyonic Majorana excitations in topological superconductors
• Zero modes of two-dimensional chiral p-wave superconductors
• An Index Theorem for the Majorana Zero Modes in Chiral P-Wave Superconductors
• Majorana Fermions in Chiral Topological Ferromagnetic Nanowires
• Topological indices, defects and Majorana fermion in chiral superconductors
• Armchair graphene nanoribbons: PT-symmetry breaking and exceptional points without dissipation
• Exceptional point description of one-dimensional chiral topological superconductors/superfluids in BDI class

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• No-go theorem for a time-reversal invariant topological phase in noninteracting systems coupled to conventional superconductors
• Interaction-driven topological superconductivity in one dimension
• Signatures of two- and three-dimensional semimetals from circular dichroism
• Majorana fermions in finite-size strips with in-plane magnetic fields
• Symmetry-breaking signatures of multiple Majorana zero modes in one-dimensional spin-triplet superconductors
• Nested Defects on the Boundary of Topological Superconductors