Zero energy modes with Gaussian, exponential, or polynomial decay: Exact solutions in hermitian and nonhermitian regimes

Abstract

Topological zero modes in topological insulators or superconductors are exponentially localized at the phase transition between a topologically trivial and nontrivial phase. These modes are solutions of a Jackiw-Rebbi equation modified with an additional term which is quadratic in the momentum. Moreover, localized fermionic modes can also be induced by harmonic potentials in superfluids and superconductors or in atomic nuclei. Here, by using inverse methods, we consider in the same framework exponentially-localized zero modes, as well as Gaussian modes induced by harmonic potentials (with superexponential decay) and polynomially decaying modes (with subexponential decay), and derive the explicit and analytical form of the modified Jackiw-Rebbi equation (and of the Schrödinger equation) which admits these modes as solutions. We find that the asymptotic behavior of the mass term is crucial in determining the decay properties of the modes. Furthermore, these considerations naturally extend to the nonhermitian regime. These findings allow us to classify and understand topological and nontopological boundary modes in topological insulators and superconductors.

18 pages, 6 figures, published on PTEP

References in corpus (53)

• Topological Insulators
• Topological insulators and superconductors
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Helical liquids and Majorana bound states in quantum wires
• New directions in the pursuit of Majorana fermions in solid state systems
• Search for Majorana fermions in superconductors
• Non-Hermitian Physics
• Topological superconductors: a review
• Majorana fermions in a tunable semiconductor device
• Majorana Zero Modes and Topological Quantum Computation
• Dirac materials
• Colloquium: Majorana Fermions in nuclear, particle and solid-state physics
• Introduction to topological superconductivity and Majorana fermions
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Non-Abelian Topological Order in S-Wave Superfluids of Ultracold Fermionic Atoms
• Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems
• Symmetry and Topology in Superconductors - Odd-frequency pairing and edge states -
• Majorana Fermions in Semiconductor Nanowires: Fundamentals, Modeling, and Experiment
• From Andreev to Majorana bound states in hybrid superconductor-semiconductor nanowires
• Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p_x+ip_y paired superfluids
• Andreev bound states versus Majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaks
• Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement
• Non-Hermitian topological phenomena: A review
• Majorana Fermions and Topology in Superconductors
• Probing Atomic Structure and Majorana Wavefunctions in Mono-Atomic Fe-chains on Superconducting Pb-Surface
• Classification of Topological Insulators and Superconductors
• Non-Abelian Topological Orders and Majorana Fermions in Spin-Singlet Superconductors
• Reproducing topological properties with quasi-Majorana states
• Physical mechanisms for zero-bias conductance peaks in Majorana nanowires
• Non-Hermitian topology: a unifying framework for the Andreev versus Majorana states controversy
• Pseudospin symmetry and its approximation in real nuclei
• Topological insulator and the Dirac equation
• Tensor coupling and pseudospin symmetry in nuclei
• Majorana nanowires for topological quantum computation
• Distinguishing topological Majorana bound states from trivial Andreev bound states: Proposed tests through differential tunneling conductance spectroscopy
• Quantifying wave-function overlaps in inhomogeneous Majorana nanowires
• Theory of Majorana zero modes in unconventional superconductors
• Distinguishing Majorana bound states from localized Andreev bound states by interferometry
• Majorana bound states in semiconducting nanostructures
• Distinguishing trivial and topological zero energy states in long nanowire junctions
• Majorana modes in emergent-wire phases of helical and cycloidal magnet-superconductor hybrids
• Disentangling Majorana fermions from conventional zero energy states in semiconductor quantum wires
• Majorana corner flat bands in two-dimensional second-order topological superconductors
• Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires
• Chain of Majorana States from Superconducting Dirac Fermions at a Magnetic Domain Wall
• Crossover between trivial zero modes in Majorana nanowires
• Differentiating Majorana from Andreev Bound States in a Superconducting Circuit
• 1D Majorana Goldstinos and partial supersymmetry breaking in quantum wires
• Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors
• Revealing Majorana Fermion states in a superfluid of cold atoms subject to a harmonic potential
• Dispersive 1D Majorana modes with emergent supersymmetry in 1D proximitized superconductors via spatially-modulated potentials and magnetic fields
• Majorana modes in striped two-dimensional inhomogeneous topological superconductors