Nonuniform Parafermion Chains: Low-Energy Physics and Finite-Size Effects

Abstract

The nonuniform ${\mathbb{Z}}_2$ symmetric Kitaev chain, comprising alternating topological and normal regions, hosts localized states known as edge-zero modes (EZMs) at its interfaces. These EZMs can pair to form qubits that are resilient to quantum decoherence, a feature expected to extend to higher symmetric chains, i.e., parafermion chains. However, finite-size effects may impact this ideal picture. Diagnosing these effects requires first a thorough understanding of the low-energy physics where EZMs may emerge. Previous studies have largely focused on uniform chains, with nonuniform cases inferred from these results. While recent work [Narozhny, Sci. Rep. 7, 1447 (2017)] provides an insightful analytical solution for a nonuniform ${\mathbb{Z}}_2$ chain with two topological regions separated by a normal one, its complexity limits its applicability to chains with more regions or higher symmetries. Here, we present a new approach based on decimating the highest-energy terms, facilitating the scalable analysis of ${\mathbb{Z}}_n$ chains with any number of regions. We provide analytical results for both ${\mathbb{Z}}_2$ and${\mathbb{Z}}_3$ chains, supported by numerical findings, and identify the critical lengths necessary to preserve well-separated EZMs.

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