Non-Abelian holonomy in a chaotic Majorana billiard

A ``Majorana billiard'' consists of a chaotic cavity coupled to a topological superconductor via tunneling contacts. In the limit of zero transmission, each of these contacts hosts a Majorana zero mode. Close to a cavity resonance and at a finite contact transparency, the resonant mode couples the Majorana modes, but a ground state degeneracy per fermion parity subspace remains if the number of Majorana modes coupled to the cavity exceeds five. Upon varying shape-defining gate voltages while remaining close to resonance, a nontrivial evolution within the degenerate ground-state manifold can be achieved. We characterize the corresponding non-Abelian holonomy using random matrix theory and discuss measurable signatures of the non-Abelian time-evolution.