# Detecting composite orders in layered models via machine learning

## TL;DR

This paper demonstrates how machine learning, specifically convolutional neural networks, can effectively identify phases and composite order parameters in layered spin models, including hidden orders, without prior phase knowledge.

## Contribution

The authors introduce a machine learning approach that accurately characterizes phases and detects composite orders in layered models directly from Monte Carlo data, without preprocessing.

## Key findings

- Successfully identified all phases in the Ashkin-Teller model, including hidden composite orders.
- Correctly distinguished ferromagnetic and paramagnetic phases in bilayer and trilayer Ising models.
- Method is versatile and applicable to various layered systems without prior phase information.

## Abstract

Determining the phase diagram of systems consisting of smaller subsystems 'connected' via a tunable coupling is a challenging task relevant for a variety of physical settings. A general question is whether new phases, not present in the uncoupled limit, may arise. We use machine learning to study layered spin models, in which the spin variables constituting each of the uncoupled systems (to which we refer as layers) are coupled to each other via an interlayer coupling. In such systems, in general, composite order parameters involving spins of different layers may emerge as a consequence of the interlayer coupling. We focus on the layered Ising and Ashkin-Teller models as a paradigmatic case study, determining their phase diagram via the application of a machine learning algorithm to the Monte Carlo data. Remarkably our technique is able to correctly characterize all the system phases also in the case of hidden order parameters, i.e., order parameters whose expression in terms of the microscopic configurations would require additional preprocessing of the data fed to the algorithm. We correctly retrieve the three known phases of the Ashkin-Teller model with ferromagnetic couplings, including the phase described by a composite order parameter. For the bilayer and trilayer Ising models the phases we find are only the ferromagnetic and the paramagnetic ones. Within the approach we introduce, owing to the construction of convolutional neural networks, naturally suitable for layered image-like data with arbitrary number of layers, no preprocessing of the Monte Carlo data is needed, also with regard to its spatial structure. The physical meaning of our results is discussed and compared with analytical data, where available. Yet, the method can be used without any a priori knowledge of the phases one seeks to find and can be applied to other models and structures.