Machine learning electron correlation in a disordered medium

Learning from data has led to a paradigm shift in computational materials science. In particular, it has been shown that neural networks can learn the potential energy surface and interatomic forces through examples, thus bypassing the computationally expensive density functional theory calculations. Combining many-body techniques with a deep-learning approach, we demonstrate that a fully connected neural network is able to learn the complex collective behavior of electrons in strongly correlated systems. Specifically, we consider the Anderson-Hubbard (AH) model, which is a canonical system for studying the interplay between electron correlation and strong localization. The ground states of the AH model on a square lattice are obtained using the real-space Gutzwiller method. The obtained solutions are used to train a multitask multilayer neural network, which subsequently can accurately predict quantities such as the local probability of double occupation and the quasiparticle weight, given the disorder potential in the neighborhood as the input.

Figure 1

Summary of the GA solution for the AH model on a $30\times 30$ square lattice. The panels show the scatter diagram of (a) local energy correction ${\mu }_i$, (b) site electron density $n_i$, (c) probability of double occupation $D_i=d_i^2$, and (d) local quasiparticle weight $Z_i={\mathcal{R}}_i^2$ vs the random site energy ${\varepsilon }_i$. The data points were obtained from calculations with random strength $W/t=6,10,14,18$ and three different $U=4t$, $10t$, and $16t$. The smooth curves showing the underlying overall trend for a given $U$ were obtained using polynomial regression with up to 14th-order polynomials. The red, blue, and green curves correspond to $U=4t$, $10t$, and $16t$, respectively.

Figure 2

(a) Schematic showing the target site at ${\mathbf{R}}_0$ on the square lattice. Random potentials ${\varepsilon }_j$ of neighbors up to some cutoff radius $r_c$ are used as the input to the neural network (NN). (b) Basic invariant subgroups of neighbors include two types of squares and a octagons. (c) Architecture of the fully connected NN for the disordered correlated systems. For input, we use all the random-distributed on-site energies in the certain circle. The ReLU activation function is used in the five hidden-feature extraction layers with $512\times 256\times 256\times 128\times 64$ nodes. The linear activation function is used to predict local quantities including $\mu$, $n$, $D$, and $Z$.

Figure 3

Comparison of the ML predictions with references obtained from the GA solvers, for (a) the local potential renormalization ${\mu }_i$, (b) site electron density $n_i$, (c) probability of double occupation $D_i$, and (d) local quasiparticle weight $Z_i$. The blue and orange data points denote predictions for training and test data sets, respectively. The insets show the normalized count of the error $\delta$ defined as the difference between prediction and reference values.