Automatic learning of topological phase boundaries

Topological phase transitions, which do not adhere to Landau's phenomenological model (i.e., a spontaneous symmetry breaking process and vanishing local order parameters), have been actively researched in condensed matter physics. Machine learning of topological phase transitions has generally proved difficult due to the global nature of the topological indices. Only recently has the method of diffusion maps been shown to be effective at identifying changes in topological order. However, previous diffusion map results required adjustments of two hyperparameters: a data length scale and the number of phase boundaries. In this article we introduce a heuristic that requires no such tuning. This heuristic allows computer programs to locate appropriate hyperparameters without user input. We demonstrate this method's efficacy by drawing remarkably accurate phase diagrams in three physical models: the Haldane model of graphene, a generalization of the Su-Schreiffer-Haeger model, and a model for a quantum ring with tunnel junctions. These diagrams are drawn, without human intervention, from a supplied range of model parameters.

Figure 1

(a) A data set is composed of classical XY spins with different winding numbers. Points on a single manifold in parameter space can have a large pairwise separation, but a small diffusion separation as defined in Eq. (4). (b) The XY spins projected to two dimensions. The data are plotted as a function of the angles in the first and last lattice sites. The data are stratified because of the presence of spin winding. However, $k$-means clustering (color code) does not recognize the topological order of the full-dimensional data set. (c) The 20 largest eigenvalues of the transition matrix $P_{ij}$ for the data in (a), as a function of the resolution parameter $\epsilon$ in Eq. (2) (note the semilog scale). The timescale here was chosen to be $t=500$. $P_{ij}$ is a probability matrix, and thus its eigenvalues (${\lambda }_k$) vary between 0 and 1. The number of eigenvalues of order unity roughly correspond to the number of manifolds in the data. Three eigenvalues survive for a large range of $\epsilon$, indicating there are truly three sets of topologically distinct data. The inset shows the eigenvalues for $\epsilon =0.8$. (d) The similarity matrix of the data for $\epsilon =0.8$, derived from Eq. (2) using the ${\mathbb{L}}^{\infty }$ norm. Much can be learned from visual inspection: (1) there are three manifolds in the data; (2) the manifolds are well separated with weak interconnections and strong intraconnections; (3) each manifold is the same size; (4) the manifolds have the same density. (e) The data set transformed to two dimensions by diffusion maps. The distribution of the data along the zeroth and first diffusion modes is shown. Clearly, the spin data in diffusion space are clustered according to their winding numbers, and $k$-means analysis automatically relates states of the same topological index. The intercluster distance is much larger than the intracluster distance.

Figure 2

(a) The augmented MSE (black, solid line) from Eq. (6) of the spin data in Fig. 1 as a function of resolution given three clusters. Also plotted are the five largest transition probability eigenvalues (red, dashed line) [Eq. (3)]. The minimum of the MSE can be found with standard computer routines, and allows a program to automatically select $\epsilon$ without an experimenter choosing the hyperparameter. (b) The similarity matrix when the MSE is minimized. The resolution parameter is much larger here than in Fig. 1, resulting in a more uniform matrix. (c) The “ideal” similarity which the heuristic tries to match. (d) The error for different numbers of $k$-means centroids on a log-log scale. The error reaches its smallest possible value for $n=3$ and $\epsilon =18.2$. Based on the heuristic, those are the values with which diffusion maps are performed. The lack of discontinuities in the $n=3$ curve may also be evidence for the proper number of cluster centers. The locations of the centroids smoothly change for $n=3$, suggesting the data are inherently triply clustered in diffusion space.

Figure 3

The optimized resolution hyperparameter as a function of XY spin chain length where three different winding numbers are present (black, solid line). This value changes rapidly for small lattices but gradually increases once the system is large enough to find the topological features. Similar behavior is exhibited by the MSE itself (green, dotted line).

Figure 4

The learned phase boundaries (black dots) match that of the known phase boundaries (black line) for the Haldane model. The learned boundaries are in locations where the $k$-means labels of the Bloch vectors transformed by diffusion maps change. Matter phases are identified by their Chern number and are color coded here for extra visibility.

Figure 5

SSH model with next-next-nearest-neighbor hopping enabled, corresponding to the Hamiltonian of Eq. (13).

Figure 6

(a) Learned phase diagram of the standard SSH model ($t_3=0$), matching theory to high precision. (b) When next-next-nearest hopping is allowed ($t_3\ne 0,t_1=1$), the heuristic is markedly less accurate but still close to the ground truth diagram. Topological phases are identified even when they occupy a small volume of parameter space. The ability of the heuristic to draw several boundaries simultaneously is also demonstrated.

Figure 7

A schematic of the triple-ring junction, which can be understood as a single ring wrapped into the “trefoil” pattern. At points where the wires cross, there is a $\delta$-function coupling that allows an electron to tunnel from one loop to the next.

Figure 8

The phase diagram of the single-electron wave function across a triple-ring junction in Fig. 7. As one varies the real and imaginary parts of the $\delta$-function coupling at the junctions, the wave function features stark shifts of its winding numbers as defined in Eq. (15). The machine learning method (dotted line) is faithful to the explicit calculation of the winding number (solid line).