Topological zero modes and bounded modes at smooth domain walls: Exact solutions and dualities

Abstract

Topology describes global quantities invariant under continuous deformations, such as the number of elementary excitations at a phase boundary, without detailing specifics. Conversely, differential laws are needed to understand the physical properties of these excitations, such as their localization and spatial behavior. For instance, topology mandates the existence of solitonic zero-energy modes at the domain walls between topologically inequivalent phases in topological insulators and superconductors. However, the spatial dependence of these modes is only known in the idealized (and unrealistic) case of a sharp domain wall. Here, we find the analytical solutions of these zero-modes by assuming a smooth and exponentially-confined domain wall. This allows us to characterize the zero-modes using a few length scales: the domain wall width, the exponential decay length, and oscillation wavelength. These quantities define distinct regimes: featureless modes with "no hair" at sharp domain walls, and nonfeatureless modes at smooth domain walls, respectively, with "short hair", i.e., featureless at long distances, and "long hair", i.e., nonfeatureless at all length scales. We thus establish a universal relation between the bulk excitation gap, decay rate, and oscillation momentum of the zero modes, which quantifies the bulk-boundary correspondence in terms of experimentally measurable physical quantities. Additionally, we reveal an unexpected duality between topological zero modes and Shockley modes, unifying the understanding of topologically-protected and nontopological boundary modes. These findings shed some new light on the localization properties of edge modes in topological insulators and Majorana zero modes in topological superconductors and on the differences and similarities between topological and nontopological zero modes in these systems.

7 pages, 4 figures, 5 tables, published in PTEP

References in corpus (29)

• Topological Insulators
• Topological insulators and superconductors
• New directions in the pursuit of Majorana fermions in solid state systems
• Majorana Zero Modes and Topological Quantum Computation
• Dirac materials
• Introduction to topological superconductivity and Majorana fermions
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems
• 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
• Majorana Fermions and Topology in Superconductors
• Classification of Topological Insulators and 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
• Topological insulator and the Dirac equation
• 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
• Disentangling Majorana fermions from conventional zero energy states in semiconductor quantum wires
• Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires
• Crossover between trivial zero modes in Majorana nanowires
• Differentiating Majorana from Andreev Bound States in a Superconducting Circuit