# Composite symmetry protected topological order and effective models

## TL;DR

This paper investigates how effective models can accurately represent symmetry-protected topological phases in one-dimensional spin systems, using tensor network methods to classify phases and validate model applicability.

## Contribution

It introduces a methodology combining tensor network classification and cohomology to determine when effective models preserve the original phase in 1D spin systems.

## Key findings

- Effective spin-1 models capture phases under strong ferromagnetic coupling.
- Weak coupling leads to symmetry-broken phases differing from effective models.
- Tensor network methods effectively classify and compare phases in complex quantum systems.

## Abstract

Strongly correlated quantum many-body systems at low dimension exhibit a wealth of phenomena, ranging from features of geometric frustration to signatures of symmetry-protected topological order. In suitable descriptions of such systems, it can be helpful to resort to effective models which focus on the essential degrees of freedom of the given model. In this work, we analyze how to determine the validity of an effective model by demanding it to be in the same phase as the original model. We focus our study on one-dimensional spin-1/2 systems and explain how non-trivial symmetry protected topologically ordered (SPT) phases of an effective spin 1 model can arise depending on the couplings in the original Hamiltonian. In this analysis, tensor network methods feature in two ways: On the one hand, we make use of recent techniques for the classification of SPT phases using matrix product states in order to identify the phases in the effective model with those in the underlying physical system, employing Kuenneth's theorem for cohomology. As an intuitive paradigmatic model we exemplify the developed methodology by investigating the bi-layered delta-chain. For strong ferromagnetic inter-layer couplings, we find the system to transit into exactly the same phase as an effective spin 1 model. However, for weak but finite coupling strength, we identify a symmetry broken phase differing from this effective spin-1 description. On the other hand, we underpin our argument with a numerical analysis making use of matrix product states.