Tunable Splitting of the Ground-State Degeneracy in 1D Parafermionic Wires

TL;DR

This paper analyzes the splitting of topological ground-state degeneracy in 1D parafermionic wires, revealing oscillatory behavior due to Berry phases and providing a detailed evaluation of tunneling effects relevant for quantum computing.

Contribution

It introduces an instanton approach to evaluate ground-state degeneracy splitting in parafermionic systems, linking Berry phases to interference effects and mapping them to quantum clock models.

Findings

Ground-state splitting oscillates with chemical potential.

Berry phases cause interference in tunneling events.

Mapping to quantum clock models elucidates phase effects.

Abstract

Systems with topologically protected ground-state degeneracies are currently of great interest due to their potential applications in quantum computing. In practise this degeneracy is never exact, and the magnitude of the ground-state degeneracy splitting imposes constraints on the timescales over which information is topologically protected. In this Letter we use an instanton approach to evaluate the splitting of topological ground-state degeneracy in quasi-1D systems with parafermion zero modes, in the specific case where parafermions are realized by inducing a superconducting gap in pairs of fractional quantum Hall edges. We show that, like 1D topological superconducting wires, this splitting has an oscillatory dependence on the chemical potential, which arises from an intrinsic Berry phase that produces interference between distinct instanton tunneling events. These Berry phases can be mapped to chiral phases in a (dual) quantum clock model using a Fradkin-Kadanoff transformation. Comparing our low-energy spectrum to that of phenomenological parafermion models allows us to evaluate the real and imaginary parts of the hopping integral between adjacent parafermionic zero modes as a function of the chemical potential.