Stability of zero modes in parafermion chains

TL;DR

This paper investigates the stability of zero-energy modes in parafermion chains, revealing that while ground states show near-degeneracy, higher-energy states exhibit power-law splitting, challenging the existence of exact localized zero modes.

Contribution

The study provides an analytical and numerical analysis of zero mode stability in parafermion chains, highlighting the transition from exponential to power-law splitting in excited states.

Findings

Ground states maintain exponential degeneracy

Higher-energy states show power-law splitting

Critical behavior appears in excited states

Abstract

One-dimensional topological phases can host localized zero-energy modes that enable high-fidelity storage and manipulation of quantum information. Majorana fermion chains support a classic example of such a phase, having zero modes that guarantee two-fold degeneracy in all eigenstates up to exponentially small finite-size corrections. Chains of `parafermions'---generalized Majorana fermions---also support localized zero modes, but, curiously, only under much more restricted circumstances. We shed light on the enigmatic zero mode stability in parafermion chains by analytically and numerically studying the spectrum and developing an intuitive physical picture in terms of domain-wall dynamics. Specifically, we show that even if the system resides in a gapped topological phase with an exponentially accurate ground-state degeneracy, higher-energy states can exhibit a splitting that scales as a power law with system size---categorically ruling out exact localized zero modes. The transition to power-law behavior is described by critical behavior appearing exclusively within excited states.