Identifying quantum phase transitions with adversarial neural networks

The identification of phases of matter is a challenging task, especially in quantum mechanics, where the complexity of the ground state appears to grow exponentially with the size of the system. Traditionally, physicists have to identify the relevant order parameters for the classification of the different phases. We here follow a radically different approach: we address this problem with a state-of-the-art deep learning technique, adversarial domain adaptation. We derive the phase diagram of the whole parameter space starting from a fixed and known subspace using unsupervised learning. This method has the advantage that the input of the algorithm can be directly the ground state without any ad hoc feature engineering. Furthermore, the dimension of the parameter space is unrestricted. More specifically, the input data set contains both labeled and unlabeled data instances. The first kind is a system that admits an accurate analytical or numerical solution, and one can recover its phase diagram. The second type is the physical system with an unknown phase diagram. Adversarial domain adaptation uses both types of data to create invariant feature extracting layers in a deep learning architecture. Once these layers are trained, we can attach an unsupervised learner to the network to find phase transitions. We show the success of this technique by applying it on several paradigmatic models: the Ising model with different temperatures, the Bose-Hubbard model, and the Su-Schrieffer-Heeger model with disorder. The method finds unknown transitions successfully and predicts transition points in close agreement with standard methods. This study opens the door to the classification of physical systems where the phase boundaries are complex such as the many-body localization problem or the Bose glass phase.

Figure 1

Schematic representation of our architecture. (a) Given a parametric Hamiltonian, we find the ground states of two different distributions. For one of them—the source—we know the labels. For the other one—the target—we do not. A convolutional neural network is used as a feature extractor. The final layer of the representation is fed into a domain and a label classifier to find the correct phase labeling and to identify which domain the data comes from, respectively. The gradient reverse layer adds a negative constant to the back propagation of the domain classifier, which makes the feature distributions of the two domains similar. (b) We send the unlabeled examples across the trained feature extractor, and feed the high-dimensional representation to unsupervised learning methods to identify the phase transition.

Figure 2

Sketches of the Bose-Hubbard model (a), the SSH model (b), and the SSH model with long-range hopping (c).

Figure 3

Phase diagram of the Bose-Hubbard model predicted by the label classifier of the DANN. The dashed line represents the phase transition directly calculated via the compressibility $\kappa$. The predictions of the DANN and the exact value are in good agreement.

Figure 4

SSH with open boundary conditions. (a) Neuron output of the SSH model for different disorder strength $W$ and for a system of 64 sites with open boundary conditions. The results are averaged over 1000 disorder realizations. (b) Comparison of the phase predictions of a convolutional neural network without domain adaptation (CNN), with the domain adversarial approach (DANN), and with the winding number for different values of $j_1$ and fixed $W=2$. While the transfer learning fails, the phase transitions predicted by the DANN are in good agreement with the winding number.

Figure 5

(a) k-means classification applied on the feature space of the DANN trained on SSH model with periodic boundary conditions compared to the DANN label classifier output. For periodic boundaries. The classifier of the trained DANN cannot distinguish the phases of states with disorder. (b) Clustering of the two phases with t-SNE. The shapes indicate the correct labeling; the colors show the labeling found by k-means. If we apply k-means clustering on the low-dimensional embedding provided by t-SNE on the feature space, the labeling works well and the phase boundary can be found with an accuracy of $j_1/j_2=1\pm 0.01$. The plot shows the SSH model with periodic boundary conditions with disorder strength $W=0.2$.

Figure 6

SSH long-range label prediction of the classifier (solid line). If we fix $j_1=j_2=1$ and $j_3=0$ we can find phase transitions $\nu =1\to 0$ at $j_4=0$ and $\nu =0\to 2$ at $j_4=1$ [48]. The dashed line shows the labeling found by k-means directly on the feature space. We can see that there is a mislabeling at the boundary between $\nu =1$ and $\nu =0$. In color are the effective winding numbers. $j_1$ and $j_2$ are the nearest neighbor hopping terms; $j_3$ and $j_4$ are the third nearest neighbor hopping terms. In the right panel is the SSH long-range feature space classification via k-means and graphical embedding by t-SNE. The shapes indicate the correct labeling; the colors show the labeling found by k-means.

Figure 7

CNN phase prediction of Ising configurations with and without random magnetic field. Triangular markers show the training set with $T_{\mathrm{crit}}\approx 2.27$ and the cross markers show the test set with $T_{\mathrm{crit}}\approx 1.3$.

Figure 8

(a) DANN label prediction of the classifier for the Kitaev model. The learned phase transition of the SSH model can be transferred via domain adaptation. The prediction of the transition point gets better for bigger system sizes. The inaccuracy of the predictions comes from the fact that the DANN classifier was not trained on the Kitaev inputs, only on SSH. (b) k-means applied to the feature space directly after the convolutional layers shows an accurate labeling. The shapes indicate the correct labeling; the colors show the labeling found by k-means. The effective phase transition occurs at $\mu /t=\left|2\right|$. The k-means algorithm finds almost all labels correctly (colors). We obtain an error of $\mathrm{\Delta }\mu /t=-0.01$. The spatial clustering has been done by the t-SNE method.