Pasquale Marra

Abstract Topological zero modes in topological insulators or superconductors are exponentially localized at the phase transition between a topologically trivial and a topologically 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 non-Hermitian regime. These findings allow us to classify and understand topological and nontopological boundary modes in topological insulators and superconductors.

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.

Majorana zero modes have gained significant interest due to their potential applications in topological quantum computing and in the realization of exotic quantum phases. These zero-energy quasiparticle excitations localize at the vortex cores of two-dimensional topological superconductors or at the ends of one-dimensional topological superconductors. Here we describe an alternative platform: a two-dimensional topological superconductor with inhomogeneous superconductivity, where Majorana modes localize at the ends of topologically nontrivial one-dimensional stripes induced by the spatial variations of the order parameter phase. In certain regimes, these Majorana modes hybridize into a single highly nonlocal state delocalized over spatially separated points, with exactly zero energy at finite system sizes and with emergent quantum-mechanical supersymmetry. We then present detailed descriptions of braiding and fusion protocols and showcase the versatility of our proposal by suggesting possible setups which can potentially lead to the realization Yang-Lee anyons and the Sachdev-Ye-Kitaev model.

The Hofstadter model allows to describe and understand several phenomena in condensed matter such as the quantum Hall effect, Anderson localization, charge pumping, and flat-bands in quasiperiodic structures, and is a rare example of fractality in the quantum world. An apparently unrelated system, the relativistic Toda lattice, has been extensively studied in the context of complex nonlinear dynamics, and more recently for its connection to supersymmetric Yang-Mills theories and topological string theories on Calabi-Yau manifolds in high-energy physics. Here we discuss a recently discovered spectral relationship between the Hofstadter model and the relativistic Toda lattice which has been later conjectured to be related to the Langlands duality of quantum groups. Moreover, by employing similarity transformations compatible with the quantum group structure, we establish a formula parametrizing the energy spectrum of the Hofstadter model in terms of elementary symmetric polynomials and Chebyshev polynomials. The main tools used are the spectral duality of tridiagonal matrices and the representation theory of the elementary quantum group.

Majorana bound states are quasiparticle excitations localized at the boundaries of a topologically nontrivial superconductor. They are zero-energy, charge-neutral, particle–hole symmetric, and spatially-separated end modes which are topologically protected by the particle–hole symmetry of the superconducting state. Due to their topological nature, they are robust against local perturbations and, in an ideal environment, free from decoherence. Furthermore, unlike ordinary fermions and bosons, the adiabatic exchange of Majorana modes is noncommutative, i.e., the outcome of exchanging two or more Majorana modes depends on the order in which exchanges are performed. These properties make them ideal candidates for the realization of topological quantum computers. In this tutorial, I will present a pedagogical review of 1D topological superconductors and Majorana modes in quantum nanowires. I will give an overview of the Kitaev model and the more realistic Oreg–Lutchyn model, discuss the experimental signatures of Majorana modes, and highlight their relevance in the field of topological quantum computation. This tutorial may serve as a pedagogical and relatively self-contained introduction for graduate students and researchers new to the field, as well as an overview of the current state-of-the-art of the field and a reference guide to specialists.

Abstract Realizing Majorana modes in topological superconductors, i.e., the condensed-matter counterpart of Majorana fermions in particle physics, may lead to a major advance in the field of topologically-protected quantum computation. Here, we introduce one-dimensional, counterpropagating, and dispersive Majorana modes as bulk excitations of a periodic chain of partially-overlapping, zero-dimensional Majorana modes in proximitized nanowires via periodically-modulated fields. This system realizes centrally-extended quantum-mechanical supersymmetry with spontaneous partial supersymmetry breaking. The massless Majorana modes are the Nambu-Goldstone fermions (Goldstinos) associated with the spontaneously broken supersymmetry. Their experimental fingerprint is a dip-to-peak transition in the zero-bias conductance, which is generally not expected for Majorana modes overlapping at a finite distance. Moreover, the Majorana modes can slide along the wire by applying a rotating magnetic field, realizing a “Majorana pump”. This may suggest new braiding protocols and implementations of topological qubits.