Topological Protection and Non-Abelian Braiding of Majorana Zero Modes — E8 Intelligence Research

Abstract

FINDING: Majorana zero modes (MZMs) realize non-Abelian braid statistics via the Ising anyon model, with topological protection scaling exponentially in wire length. | MATH: Braid group \( B_n \) generators \( \sigma_i \) act on a 2D degenerate ground space (for \( 2n \) MZMs) via \( \gamma_i \to \gamma_j = U^\dagger \gamma_i U \), where \( U = \exp(\frac{\pi}{4}\gamma_i\gamma_j) \). The non-Abelian exchange yields \( \sigma_i^2 = -1 \) (fermion parity flip), and the topological gap \( \Delta_{\rm top} \propto e^{-L/\xi} \) (exponential suppression of splitting with wire length \( L \), coherence length \( \xi \)). Kitaev chain Hamiltonian: \( H = -\mu\sum c_i^\dagger c_i - t\sum(c_{i+1}^\dagger c_i + {\rm h.c.}) + \Delta\sum(c_{i+1}^\dagger c_i^\dagger + {\rm h.c.}) \), with MZM operators \( \gamma_1 = c_1 + c_1^\dagger \), \( \gamma_2 = -i(c_N - c_N^\dagger) \) at \( \mu=0, t=\Delta \). | CONNECTION: The braid group \( B_n \) is the fundamental group of configuration space of \( n \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com