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

TL;DR

This paper develops a scalable analytical approach to study low-energy physics and finite-size effects in nonuniform parafermion chains, providing insights into edge-zero modes and their robustness in complex chain configurations.

Contribution

It introduces a new decimation-based method for analyzing nonuniform $ ext{Z}_n$ chains with multiple regions, extending beyond previous uniform or simpler models.

Findings

Analytical results for $ ext{Z}_2$ and $ ext{Z}_3$ chains

Identification of critical lengths for preserving EZMs

Numerical validation of the analytical approach

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.