# Three-Phase Confusion Learning

## TL;DR

This paper introduces a new machine learning method to study quantum systems with three phases, expanding on previous work with binary classification.

## Contribution

The paper generalizes the Learning by Confusion technique to systems with three phases using a ternary classifier.

## Key findings

- The ternary classifier successfully identifies three phases in quantum systems.
- Results on Kitaev chains and the Extended Hubbard model match previous studies.
- The method shows improved accuracy when data are correctly partitioned.

## Abstract

The use of Neural Networks in quantum many-body theory has undergone a formidable rise in recent years. Among the many possible applications, their pattern recognition power can be utilized when dealing with the study of equilibrium phase diagrams. Learning by Confusion has emerged as an interesting and unbiased scheme within this context. This technique involves systematically reassigning labels to the data in various ways, followed by training and testing the Neural Network. While random labeling results in low accuracy, the method reveals a peak in accuracy when the data are correctly and meaningfully partitioned, even if the correct labeling is initially unknown. Here, we propose a generalization of this confusion scheme for systems with more than two phases, for which it was originally proposed. Our construction relies on the use of a slightly different Neural Network: from a binary classifier, we move to a ternary one, which is more suitable to detect systems exhibiting three phases. After introducing this construction, we test it on free and interacting Kitaev chains and on the one-dimensional Extended Hubbard model, consistently achieving results that are compatible with previous works. Our work opens the way to wider use of Learning by Confusion, demonstrating once more the usefulness of Machine Learning to address quantum many-body problems.