# Haldane Topological Orders in Motzkin Spin Chains

## TL;DR

This paper investigates Haldane topological orders in Motzkin spin chains, revealing phase transitions, topological properties, and their dependence on spin value, thus classifying various hidden antiferromagnetic regimes.

## Contribution

It introduces a detailed analysis of topological phases in Motzkin spin chains, highlighting their dependence on deformation parameters and spin values, and classifying different Haldane regimes.

## Key findings

- Identification of Berezinskii-Kosterlitz-Thouless transition at t=1
- Gapped Haldane topological orders for t<1
- Dependence of topological properties on spin value

## Abstract

Motzkin spin chains are frustration-free models whose ground-state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter t. As a function of the latter these models, beside the formation of domain wall structures, exhibit a Berezinskii-Kosterlitz-Thouless phase transition for t=1 and gapped Haldane topological orders with constant decay of the string order parameters for t < 1. By means of numerical calculations we show that the topological properties of the Haldane phases depend on the spin value. This allows to classify different kinds of hidden antiferromagnetic Haldane gapped regimes associated to nontrivial features like symmetry-protected topological order. Our results from one side allow to clarify the physical properties of Motzkin frustration-free chains and from the other suggest them as a new interesting and paradigmatic class of local spin Hamiltonians.