# Constructing edge zero modes through domain wall angle conservation

## TL;DR

This paper explores the construction and properties of edge zero modes in topologically ordered spin chains, focusing on a $ ext{Z}_3$ generalization and emphasizing the role of domain wall angle symmetry for locality and iterative construction.

## Contribution

It introduces a general algorithm for perturbative construction of zero modes, applies it to various models, and provides analytical formulas for zero modes with model-dependent coefficients.

## Key findings

- Preservation of total domain wall angle symmetry guarantees locality of zero modes.
- A perturbative algorithm for constructing zero modes is proposed and tested.
- Analytical formulas for zero modes are derived for multiple models.

## Abstract

We investigate the existence, normalization and explicit construction of edge zero modes in topologically ordered spin chains. In particular we give a detailed treatment of zero modes in a $\mathbb{Z}_3$ generalization of the Ising/Kitaev chain, which can also be described in terms of parafermions. We analyze when it is possible to iteratively construct strong zero modes, working completely in the spin picture. An important role is played by the so called total domain wall angle, a symmetry which appears in all models with strong zero modes that we are aware of. We show that preservation of this symmetry guarantees locality of the iterative construction, that is, it imposes locality conditions on the successive terms appearing in the zero mode's perturbative expansion. The method outlined here summarizes and generalizes some of the existing techniques used to construct zero modes in spin chains and sheds light on some surprising common features of all these types of methods. We conjecture a general algorithm for the perturbative construction of zero mode operators and test this on a variety of models, to the highest order we can manage. We also present analytical formulas for the zero modes which apply to all models investigated, but which feature a number of model dependent coefficients.