Abstract
Majorana fermions feature non-Abelian exchange statistics and promise fascinating applications in topological quantum computation. Recently, second-order topological superconductors (SOTSs) have been proposed to host Majorana fermions as localized quasiparticles with zero excitation energy, pointing out a new avenue to facilitate topological quantum computation. We provide a minimal model for SOTSs and systematically analyze the features of Majorana zero modes with analytical and numerical methods. We further construct the fundamental fusion principles of zero modes stemming from a single or multiple SOTS islands. Finally, we propose concrete schemes in different setups formed by SOTSs, enabling us to exchange and fuse the zero modes for non-Abelian braiding and holonomic quantum gate operations.
Highlights
Majorana fermions are self-conjugate fermions [1]
We show that the desired non-Abelian braiding operations of Majorana zero modes (MZMs) as well as a full set of holonomic gates can be achieved in the second-order topological superconductors (SOTSs) platform
We provide a minimal model for SOTSs with inversion symmetry and discuss the features and behaviors of individual MZMs in a disk geometry, both analytically and numerically
Summary
Majorana fermions are self-conjugate fermions [1]. They can arise as zero-energy Bogoliubov quasiparticles in condensed matter [2,3,4,5], such as vortex bound states in p-wave superconductors [6,7], Majorana bound states in Josephson junctions [8,9], and end states of nanowires with Rashba spinorbit coupling or of ferromagnetic atomic chains [10,11,12,13,14,15]. When more than two MZMs are present, the braiding (exchange) operations on them correspond to nonAbelian rotations in the ground-state manifold spanned by them They can serve as basic building blocks for topological quantum computation [3,7,16,17,18]. We put forward a number of setups formed by the SOTSs, as sketched, and present in detail corresponding schemes for braiding the MZMs based on variations of chemical potential, applied magnetic field, and geometry engineering. We propose a shamrock-like trijunction setup constituted by three incomplete disks This trijunction hosts three MZMs that fuse exclusively with a fourth one.
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