Engineering topological phases guided by statistical and machine learning methods

The search for materials with topological properties is an ongoing effort. In this article we propose a systematic statistical method, supported by machine learning techniques, that is capable of constructing topological models for a generic lattice without prior knowledge of the phase diagram. By sampling tight-binding parameter vectors from a random distribution, we obtain data sets that we label with the corresponding topological index. This labeled data is then analyzed to extract those parameters most relevant for the topological classification and to find their most likely values. We find that the marginal distributions of the parameters already define a topological model. Additional information is hidden in correlations between parameters. Here we present as a proof of concept the prediction of the Haldane model as the prototypical topological insulator for the honeycomb lattice in Altland-Zirnbauer (AZ) class A. The algorithm is straightforwardly applicable to any other AZ class or lattice, and could be generalized to interacting systems.

Figure 1

Features are uniformly distributed over a circular region with radius $r_i=\alpha \left|x_{\mathrm{ref}}^i\right|$ around the reference point $x_{\mathrm{ref}}$. The spread in the real parameter $x_j$ is given by $r_j=\alpha \left|x_{\mathrm{ref}}^j\right|$.

Figure 2

Illustration of a probability distribution function for two features $x_0$ and $x_1$. When restricted to the data subset with class label $L$, the distribution of feature $x_1$ deviates significantly from the base distribution, i.e., the feature is more important for the classification.

Figure 3

Illustration of the redundancy $R$ [Eq. (4)] (top row) and the Pearson correlation coefficient $r$ [Eq. (7)] (bottom row). In (a) a joint probability density function for two dependent random variables is shown. The redundancy is nonzero. The product of the corresponding marginal distributions is shown in (b) and clearly differs from the true joint distribution. The redundancy between independent variables vanishes. (c) and (d) are examples for joint distribution functions for positively and negatively correlated variables. Note the respective sign of the PCC.

Figure 4

Phase diagram of the Haldane model for $\varphi =\pi /2$ in terms of next-nearest neighbor hopping $t_2$ and mass $m$. Starting out from the trivial phase (0), one can reach a nontrivial phase by changing either $m$, $t_2$, or both. The reference point $x_{\mathrm{ref}}$ is marked by $\times$.

Figure 5

(a) Nearest and next-nearest neighbor hopping terms accounted for in the honeycomb lattice. We draw independent parameters in different colors. (b) Percentage of samples categorized by the topological class label (outer ring) and the corresponding fraction of insulators/metals (inner ring). Here, we find only $y=0,1,-1$ in the surveyed region. The total sample size is $n_s={10}^7$.

Figure 6

Importance scores in terms of the Bhattacharyya distance for all (real) features. Here, we take into account only the topological class with Chern index $C=1$. Most relevant are apparently the mass $m$, the real part and phase of $t_1$, and the phase of $t_2$. We plot a separate bar for every individual hopping vector. Equal colors indicate equal lengths.

Figure 7

Relative frequency (approximate probability density function) for four important features. We chose here the mass $m$, the real part of a nearest neighbor hopping $t_1$, and the phases of two next-nearest neighbor hoppings connecting $A$ and $B$ sites, respectively. For all terms we observe a clear distinction of the PDF of the nontrivial phase ($C=1,-1$) from that describing the trivial phase ($C=0$).

Figure 8

Top: (a) Redundancy $R$, Eq. (4), shown here for the phases of all parameters for the unbiased (left) and biased (right) topological data set. The nearest neighbor hoppings show a small redundancy in the unbiased data. This is not the case for the next-nearest neighbor hoppings. Imposing the bias on the data reveals a redundancy between $t_{2A}$ and $t_{2B}$. (b) Joint probability density function $p[\phi \left(t_{2A}\right),\phi \left(t_{2B}\right)]$ for the next-nearest neighbor hoppings for unbiased (left) and biased (right) data sets. Apparently, noise due to the large number of degrees of freedom for the six next-nearest neighbor hoppings reduces the contrast in the PDF, and therefore reduces the measured redundancy. Bottom: Pearson correlation coefficient [Eq. (7)] for the (c) $C=1$ and (d) $C=-1$ phase. We observe positive and negative correlations between the nearest-neighbor hoppings, respectively.

Figure 9

(a) Hopping parameters taken into account. Terms related by symmetry are colored equally. (b) Percentage of samples from the improved model categorized by the topological class label (outer ring) and the fraction of insulators and metals therein (inner ring). In the surveyed region we find almost exclusively insulators with labels $y=0,1$; the number of $y=-1$ samples is statistically irrelevant. The total sample size is ${10}^6$.

Figure 10

Relative frequency (approximate probability density function) for the mass $m$, a nearest neighbor term $t_1$ and two next-nearest neighbor terms $t_2$ for the biased data. Compared to the unbiased data (see Fig. 7), the nearest neighbor term is suddenly completely indistinguishable between different phases, while the contrast of the next-nearest neighbor terms is increased. The $y=-1$ label can apparently only appear for a specific sign of the next-nearest neighbor phases.

Figure 11

Results for the honeycomb lattice. Top row: Fully unbiased model. Bottom row: Symmetrized model. (a) The fraction of topological samples is comparable to the Haldane case in the unbiased data, but much smaller in the symmetric data because the reference point is far away from a topological phase. While the majority of samples are metallic, all samples have separable bands. (b) PDFs of features with highest importance score $D_B$, separated from the rest by an order of magnitude. For the unbiased model we use all data points, while in the biased case we restrict to insulators only. (c) and (d) are the same representation of the data for the biased calculation. The phases of the next-nearest neighbor hoppings are now a strong indicator for the topological phase. In (e) we show the overlap of the features with the parameters of a generic Haldane model (grey) for the $C=1$ (orange) and $C=-1$ (green) phase. The effective model contains the characteristics of the Haldane model.