Machine learning offers an unprecedented perspective for the problem of classifying phases in condensed matter physics. We employ neural-network machine learning techniques to distinguish finite-temperature phases of the strongly correlated fermions on cubic lattices. We show that a three dimensional convolutional network trained on auxiliary field configurations produced by quantum Monte Carlo simulations of the Hubbard model can correctly predict the magnetic phase diagram of the model at the average density of one (half filling). We then use the network, trained at half filling, to explore the trend in the transition temperature as the system is doped away from half filling. This transfer learning approach predicts that the instability to the magnetic phase extends to at least 5% doping in this region. Our results pave the way for other machine learning applications in correlated quantum many-body systems.Comment: 9 pages, 8 figure

Essentials of the scientific discovery process have remained largely unchanged for centuries 1 : systematic human observation of natural phenomena is used to form hypotheses that, when validated through experimentation, are generalized into established scientific theory. Today, however, we face major challenges because automated instrumentation and large-scale data acquisition are generating data sets of such volume and complexity as to defy human analysis. Radically different scientific approaches are needed, with machine learning (ML) showing great promise, not least for materials science research 2-5 . Hence, given recent advances in ML analysis of synthetic data representing electronic quantum matter (EQM) 6-16 , the next challenge is for ML to engage equivalently with experimental data. For example, atomic-scale visualization of EQM yields arrays of complex electronic structure images 17 , that frequently elude effective analyses. Here we report development and training of an array of artificial neural networks (ANN) designed to recognize different types of hypothesized order hidden in EQM imagearrays. These ANNs are used to analyze an experimentally-derived EQM image archive from carrier-doped cuprate Mott insulators. Throughout these noisy and complex data, the ANNs discover the existence of a lattice-commensurate, four-unitcell periodic, translational-symmetry-breaking EQM state. Further, the ANNs find these phenomena to be unidirectional, revealing a coincident nematic EQM state. Strong-coupling theories of electronic liquid crystals 18,19 are congruent with all these observations. 1Frontier research in EQM concentrates on exotic electronic phases that emerge when electrons interact so strongly that they lack a definite momentum. These electrons often self-organize into complex new states of EQM including, for example, electronic liquid crystals 18,19 , high temperature superconductors 20,21 , fractionalized electronic fluids and quantum spin liquids. In this field, vast experimental data sets have emerged, for example from real space (r-space) visualization of EQM using spectroscopic imaging scanning tunneling microscopy 17 (SISTM), from momentum space (k-space) visualization of EQM using angle resolved photoemission (ARPES), or from modern X-ray 22 and neutron scattering. The challenge is to develop ML strategies capable of scientific discovery using such large and complex experimental data structures from EQM experiments. 2An excellent example is the electronic structure of the CuO2 plane in the cuprate compounds supporting high temperature superconductivity 20 (Fig. 1a). With one electron per Cu site, strong Coulomb interactions produce charge localization in an antiferromagnetic Mott insulator (MI) state. Removing p electrons (adding p 'holes') perCuO2 plaquette generates the 'pseudogap' (PG) phase 20 . It exhibits strongly depleted density-of-electronic states ( ) for energies |E| < Δ ! , where Δ ! is the characteristic pseudogap energy scale that emerges for < * ( ) (Fig. 1a). Although the PG pha...

We employ several unsupervised machine learning techniques, including autoencoders, random trees embedding, and t-distributed stochastic neighboring ensemble (t-SNE), to reduce the dimensionality of, and therefore classify, raw (auxiliary) spin configurations generated, through Monte Carlo simulations of small clusters, for the Ising and Fermi-Hubbard models at finite temperatures. Results from a convolutional autoencoder for the three-dimensional Ising model can be shown to produce the magnetization and the susceptibility as a function of temperature with a high degree of accuracy. Quantum fluctuations distort this picture and prevent us from making such connections between the output of the autoencoder and physical observables for the Hubbard model. However, we are able to define an indicator based on the output of the t-SNE algorithm that shows a near perfect agreement with the antiferromagnetic structure factor of the model in two and three spatial dimensions in the weak-coupling regime. t-SNE also predicts a transition to the canted antiferromagnetic phase for the three-dimensional model when a strong magnetic field is present. We show that these techniques cannot be expected to work away from half filling when the "sign problem" in quantum Monte Carlo simulations is present.