Unsupervised machine learning account of magnetic transitions in the Hubbard model

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.

Figure 1

Values of neurons in the coding layer of a trained autoencoder that takes Monte Carlo spin configurations of a 3D Ising model on a $N=8\times 8\times 8$ lattice as the input. The architecture of our 3D convolutional autoencoder is depicted on top with neurons in the coding layer as filled (orange) circles. The encoder (decoder) part consists of four convolutional (upsampling) layers with 32, 8, 4, and 4 feature maps. Different symbol colors (red middle cluster and blue clusters on the sides) in the scatter plot correspond to different temperatures. Top inset: The output of a similar autoencoder, in which the coding layer consists of only a single neuron, as a function of temperature. The dashed line is a fit to $A{(B-T)}^{\beta }+C$ with $A=0.38, B=4.55, \beta =0.34$, and $C$ kept fixed at $-0.25$, which is the average output over all $T$. Bottom inset: Temperature dependence of the spread of the data as measured through $k$-means. The vertical dashed line marks the location of $T_c$.

Figure 2

The output of the 3D convolutional autoencoder for the 3D Hubbard model with (a) $U=4$, (2) 9, and (3) $U=14$ at half filling. Each of the $\mathcal{L}=200$ imaginary time slices is treated as a different “color” channel. The input is a four-dimensional array of size $\mathcal{L}\times N$ with $N=4^3$. Similarly to the output for the Ising model, the character of the output in the space of the neuron outputs of the coding layer changes as a function of temperature. However, we do not find a formation of distinct clusters below the expected Néel temperatures. In (d), we show $\mathcal{D}$ for the data in (a) and (b), normalized to fit the antiferromagnetic structure factors (shown as red solid lines) in each case, as a function of temperature. For $U=14$, this indicator is dominated by noise, and therefore, not shown.

Figure 3

The output of random trees embedding algorithm trained on the four latent variables of a fully-connected autoencoder for the 3D Hubbard model with (a) $U=4$, (b) $U=9$, and (c) $U=14$ at half filling. The input layer of the autoencoder (shown on top) is a 1D array of size $\mathcal{L}\times N$. The hidden layers in the encoder and decoder parts have 80, 30, and 10 neurons. The random trees embedding algorithm further reduces the dimension of the data from four to two. In this case, the outputs clearly separate data points from different temperatures. The dashed lines are line fits to data at the estimated Néel temperature for each $U$. (d)–(f) Same as in (a)–(c), except that the inputs are the latent variables of the convolutional autoencoder used for Fig. 2, modified to have four neurons in the coding layer. In this case, the temperature gradient is larger along output 1 and clustering of points at low temperatures is visible for $U=4$ and $U=9$.

Figure 4

Output of the $t$-SNE algorithm in two dimensions for the 3D Hubbard model at half filling for three different values of the interaction strength, (a) $U=4$, (b) $U=9$, and (c) $U=14$. We use the raw auxiliary field configurations from DQMC simulations of the model at 51 temperatures on a uniform grid that extends from $T=0.10$ to $T=0.60$ as input. Different symbol colors (densities in the greyscale) in (a)–(c) correspond to different temperatures. (d) Same indicators as in Fig. 2 as a function of temperature calculated for the $t$-SNE outputs. For $U=4$, the indicator follows the AFM structure factor very closely.

Figure 5

(a) The convolutional autoencoder (CAE) and (b) $t$-SNE outputs for the 3D Hubbard model with $U=4$ at half filling in the presence of a magnetic field $h=1.0$. (c) The corresponding indicators as a function of temperature. (d) The AFM structure factors of the model for the $z$ and $xy$ components of the spin as a function of temperature.

Figure 6

Output of the $t$-SNE algorithm for the 2D Hubbard model with $N={10}^2$ and $U=4$ at half filing. We use 800 raw auxiliary field configurations for each temperature in the uniform grid of $T$ between 0.1 and 0.6. We use the same $t$-SNE parameters as in the training for the 3D Hubbard model in Fig. 4. (b) The indicator $\mathcal{D}$ (green symbols) and the AFM structure factor (red line) as a function of temperature.

Figure 7

(a)–(h) The $t$-SNE output for the 3D Hubbard model with $U=9$ away from half-filing. We have used 1500 auxiliary field configurations per chemical potential $\mu$, generated deep in the AFM phase ($T=0.16$) for $\mu$ ranging from $\mu =-0.2$ to $\mu =-4.0$ in steps of 0.2. We show snapshots of the output at select $\mu$, separated into configurations with negative sign ($\mathcal{S}=-1$) in the top row and those with a positive sign ($\mathcal{S}=+1$) in the bottom row. We use the same $t$-SNE parameters as in the training for the 3D Hubbard model at half filling. (i) The expectation value of the AFM structure factor obtained using configurations with positive or negative signs separately as a function of the chemical potential. (j) The average sign as a function of $\mu$.

Figure 8

Same as Fig. 7, except that here, $t$-SNE calculations have been performed separately on configurations with the same sign. We also use 2000 auxiliary field configurations per $\mu$.