Artificial Intelligence Approaches to Quantum Materials: Quantifying Entanglement

Artificial intelligence has gained much interest over the last years - broadly, but also in the field of physics specifically. Here we present a numerical study of machine learning (ML) methods applied to data analysis for strongly correlated quantum systems. The main goal of this work is to establish a framework for entanglement learning: an application of neural networks (NNs) - a supervised ML technique - to detect the entanglement of many-body magnetic systems at finite temperatures. As a preliminary result, we applied an unsupervised ML technique called principal component analysis (PCA) to detect superconducting phase transitions in exotic magnetic materials from real-life muon spectroscopy data. We find that the PCA can identify phase transition points correctly, even when the data vary in a subtle manner, which proves to be difficult for more traditional methods. Moreover, the joint PCA of data from materials with vastly different magnetic properties and different kinds of phase transitions enhances rather than diminishes the performance of the method. In order to demonstrate that the entanglement can in principle be learned, we first examine to which extent the two-point correlators constrain entanglement measures. To this end, we conduct a study of the fitness landscape of neutron-scattering functions obtained from Heisenberg-like Hamiltonians. We discover that the fitness landscape shows a linear correlation as a function of distances between quantum states at zero temperature, and a sub-linear correlation for finite temperatures. This makes the scattering functions - or closely related two-point correlators and structure factors - perfect candidates as input for entanglement learning. We find that entanglement entropy can be correctly predicted by NNs from previously mentioned observables, even when trained on a fraction of the full data set, or when trained on data from a different system than the one for which the prediction is made. Specifically, when trained on observables from an anisotropic transverse-field XY model, in order to obtain an accurate prediction, we only require 3% (6%) of the data to train the network if using the dynamic two-point correlators (structure factors) for learning.