Longer modeled chains of proximitised quantum dots show less overlap between their end modes and a smaller low-energy splitting, while a robust gap appears above a critical chemical potential. The comparison covers modeled chains with two, four and six dots.
The study asks whether topological properties develop as a quantum-dot chain grows. Its simplest finite example is an open four-dot structure whose two end dots are coupled to two external metallic leads.
A deliberately narrow model
The proposed platform places a quantum-dot chain between an s-wave superconductor and a strong spin-orbit semiconductor under a Zeeman field. The simplifying Hamiltonian neglects spin-conserving hopping and local on-dot pairing, allowing it to separate into two equivalent independent Hilbert-space parts.
At the paper’s “sweet spot”, the modeled hopping and pairing parameters are both set to 1, while the chemical potential and Zeeman field are both 0; the lowest modeled eigen-energies are zero. For even chain lengths, the reported scaling shows slow energy growth away from that setting, while protection increases in longer chains.
A phase map for four dots
To examine the short-chain signatures, the authors calculate a zero-energy retarded Green function in a chiral-Majorana representation, a mathematical way of tracing propagation through the chain. For the open four-dot case, the phase map uses modeled hopping of 1, pairing of 0.5 and equal lead couplings of 0.05.
Those Green-function structures agree with characteristic boundaries from complementary theoretical diagnostics and distinguish oscillatory standing-wave modes, strongly hybridised finite-size states and localized edge-Majorana regimes.
A transfer-matrix calculation separates complex-root solutions that oscillate from real-root solutions that decay exponentially and localize. This provides a separate way to classify the spatial character of the modeled zero modes.
In the finite four-dot model, exact zero-energy parity crossings occur when the determinant of its Hamiltonian vanishes, at the reported golden-ratio critical points.
The bulk check
Those crossings are finite-size features. The paper treats a strict topological phase transition as a property of the long-chain limit, not of the finite four-dot chain.
In that limit, a Pfaffian, a mathematical test for the bulk phase, labels the nontrivial region with ν = −1. When the pairing strength squared is smaller than the hopping strength squared, the modeled oscillatory sector lies inside that topological region.
Lead coupling moves Green-function poles off the real axis and broadens resonant peaks; increasing dissipation widens the phase-map arcs.
What the model leaves open
The detailed phase map is specific to the selected parameters of the open four-dot model, and the simplifying reduction omits spin-conserving hopping and local on-dot pairing. The reported protection and phase boundaries are therefore conditional on those assumptions.
The document is an arXiv version 1 preprint dated 20 August 2026 and presents theoretical diagnostics for short chains in the proposed geometry. Its agreement among the calculations is an internal check of the model, not an experimental validation.
Paper data and sources
Original title: Search for Majorana Bound States in Short Chains of Proxmitised Quantum Dots
Authors: Bogdan R. Bułka, Karol I. Wysokiński
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text