Topological phases of the compass ladder model

TL;DR

This paper characterizes the phases of the compass ladder model, revealing topological and symmetry-breaking phases, and introduces effective models and numerical methods to identify quantum phase transitions and universality classes.

Contribution

It provides a comprehensive analysis of the compass ladder model's phases using perturbation theory, symmetry fractionalization, and numerical techniques, highlighting the topological and symmetry-breaking properties.

Findings

Cluster phase exhibits symmetry-protected topological order.

Ising phase characterized by a local order parameter and magnetization exponent.

Partial symmetry breaking occurs at the quantum phase transition.

Abstract

We characterize phases of the compass ladder model by using degenerate perturbation theory, symmetry fractionalization, and numerical techniques. Through degenerate perturbation theory we obtain an effective Hamiltonian for each phase of the model, and show that a cluster model and the Ising model encapsulate the nature of all phases. In particular, the cluster phase has a symmetry-protected topological order, protected by a specific $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry, and the Ising phase has a $\mathbb{Z}_2$-symmetry-breaking order characterized by a local order parameter expressed by the magnetization exponent $0.12\pm0.01$. The symmetry-protected topological phases inherit all properties of the cluster phases, although we show analytically and numerically that they belong to different classes. In addition, we study the one-dimensional quantum compass model, which naturally emerges from the compass ladder, and show that a partial symmetry breaking occurs upon quantum phase transition. We numerically demonstrate that a local order parameter accurately determines the quantum critical point and its corresponding universality class.