# Multi-faceted machine learning of competing orders in disordered interacting systems

## TL;DR

This paper introduces a neural network framework to classify and analyze phases in disordered, interacting quantum systems, successfully mapping phase boundaries and identifying complex phases like fractional quantum Hall and charge density wave states.

## Contribution

The study presents a novel multi-input neural network approach to detect and distinguish topological and symmetry-broken phases in disordered quantum systems, including a comprehensive phase diagram analysis.

## Key findings

- The neural network accurately identifies phase boundaries consistent with traditional methods.
- It reveals the robustness of the fractional quantum Hall state under disorder and interactions.
- Maps the emergence of charge density wave micro-emulsions before full disorder.

## Abstract

While the non-perturbative interaction effects in the fractional quantum Hall regime can be readily simulated through exact diagonalization, it has been challenging to establish a suitable diagnostic that can label different phases in the presence of competing interactions and disorder. Here we introduce a multi-faceted framework using a simple artificial neural network (ANN) to detect defining features of a fractional quantum Hall state, a charge density wave state and a localized state using the entanglement spectra and charge density as independent input. We consider the competing effects of a perturbing interaction ($l = 1$ pseudopotential $\Delta V_1$), a disorder potential $W$, and the Coulomb interaction to the system at filling fraction ${\nu} = 1/3$. Our phase diagram benchmarks well against previous estimates of the phase boundary using conventional measures along the $\Delta V_1 = 0$ and $W = 0$ axes, the only regions where conventional approaches have been explored. Moreover, exploring the entire two-dimensional phase diagram for the first time, we establish the robustness of the fractional quantum Hall state and map out the charge density wave micro-emulsion phase wherein droplets of charge density wave region appear before the charge density wave is completely disordered. Hence we establish that the ANN can access and learn the defining traits of topological as well as broken symmetry phases using multi-faceted inputs of entanglement spectra and charge density.