Exchangeless braiding of Majorana zero modes in weakly coupled Kitaev chains

Abstract

Exchangeless braiding of Majorana modes is studied in minimal networks of weakly hybridized Kitaev chains of finite length using a rigorous many-body framework. In particular, for two coupled chains it is shown that exchangeless braiding is achieved by $2\pi$ rotations of the phase $\varphi$ of the superconducting order parameter of one of the chains. This braiding protocol is verified by the numerical calculation of the non-Abelian Wilczek-Zee phase of the low-energy many-body subspaces ${\mathcal{H}}_0(\varphi )$ based on the Bertsch-Robledo ground-state overlap formula. In the parameter space spanned by the total chain length, the strength of the weak hybridization connecting the chains, and the on-site potential, we identify two regions with different braiding outcomes, i.e., a projective ${\sigma }_x$-gate and a projective ${\sigma }_z$-gate phase. The transition between these phases is a continuous crossover, the location of which is reliably given by a simple four-Majorana mode model. This demonstrates the resilience of the anyonic properties of the Majorana modes against finite-size effects and weak links between Kitaev chains.

16 pages, 6 figures, v2 with minor changes, as published

References in corpus (26)

• Non-Abelian Anyons and Topological Quantum Computation
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Epitaxy of Semiconductor-Superconductor nanowires
• Hard gap in epitaxial semiconductor-superconductor nanowires
• Measurement-Only Topological Quantum Computation
• Distinguishing a Majorana zero mode using spin resolved measurements
• Quantum Computation Using Vortices and Majorana Zero Modes of a $p_x$ + $i{p}_y$ Superfluid of Fermionic Cold Atoms
• Spin-selective Peierls transition in interacting one-dimensional conductors with spin-orbit interaction
• Topological Superconductivity induced by Ferromagnetic Metal Chains
• The sign of the overlap of HFB wave functions
• Measurement-Only Topological Quantum Computation via Anyonic Interferometry
• Shortcuts to nonabelian braiding
• Majorana Braiding Dynamics on Nanowires
• Many-body computations by stochastic sampling in Hartree-Fock-Bogoliubov space
• Technical aspects of the evaluation of the overlap of Hartree- Fock- Bogoliubov wave functions
• Braiding of non-Abelian anyons using pairwise interactions
• Many-body Majorana braiding without an exponential Hilbert space
• Optimizing the transport of Majorana zero modes in one-dimensional topological superconductors
• Manipulation of Majorana Kramers Qubit and its Tolerance in Time-Reversal-Invariant Topological Superconductor
• Majorana phase-gate based on the geometric phase
• Long-time relaxation dynamics of a spin coupled to a Chern insulator
• Emergent Non-Abelian Gauge Theory in Coupled Spin-Electron Dynamics
• Fock space representations of Bogolyubov transformations as spin representations
• Controlling the real-time dynamics of a spin coupled to the helical edge states of the Kane-Mele model

Cited by in corpus (2)

• Revisiting the Poor Man's Majoranas: The Spin-Exchange Induced Spillover Effect
• Dynamics of Majorana zero modes across hybrid Kitaev chain