Rapid Exploration of Topological Band Structures Using Deep Learning

The design of periodic nanostructures allows to tailor the transport of photons, phonons, and matter waves for specific applications. Recent years have seen a further expansion of this field by engineering topological properties. However, what is missing currently are efficient ways to rapidly explore and optimize band structures and to classify their topological characteristics for arbitrary unit-cell geometries. In this work, we show how deep learning can address this challenge. We introduce an approach where a neural network first maps the geometry to a tight-binding model. The tight-binding model encodes not only the band structure but also the symmetry properties of the Bloch waves. This allows us to rapidly categorize a large set of geometries in terms of their band representations, identifying designs for fragile topologies. We demonstrate that our method is also suitable to calculate strong topological invariants, even when (like the Chern number) they are not symmetry indicated. Engineering of domain walls and optimization are accelerated by orders of magnitude. Our method directly applies to any passive linear material, irrespective of the symmetry class and space group. It is general enough to be extended to active and nonlinear metamaterials.

Popular Summary

The fabrication of nanostructures offers an avenue to tailor the transport of waves, be they electromagnetic, sound, or matter waves. The way in which a periodic structure modifies the propagation of waves—such as which light colors can propagate and at what speed—is encoded in a single mathematical object, known as the band structure. The task of optimizing the fabrication pattern to modify the propagation of waves in a desired manner requires the calculation of the band structure for a huge number of configurations within the infinitely large space of all possible patterns. Here, we show how an artificial neural network can map arbitrary fabrication patterns to the resulting band structures.

Instead of naive direct mapping, we introduce an approach where our neural network learns to predict the parameters of an auxiliary “tight-binding” model that can yield the band structure using an inexpensive calculation. Beyond the dramatic acceleration of band-structure predictions, our auxiliary model also encodes the symmetries of the underlying normal modes and thus can give access to topological properties.

The rapid exploration and optimization of band structures made possible by our neural network can be a powerful tool to aid in the physical discovery of future nanostructures.

Figure 1

Neural-network-based prediction of band structures from unit-cell geometries or potentials. (a) The geometry is fed into a multilayer convolutional network (Appendix pp1) producing the coefficients of a symmetry-enhanced tight-binding model, which is then diagonalized to obtain the band structure. (b) Cut along the $\mathbf{k}$ path indicated in (a), for the same band structure (now including also the first two bands), comparing the Schrödinger equation (dark thin lines) and NN (thick light lines). The two types of predictions are difficult to distinguish with the bare eye. The symmetry labels are also indicated ($0/1$ indicate the quasiangular momentum; $+$ and $-$ label the even and odd states, respectively). (c) Fraction of correctly predicted symmetry labels at each high-symmetry point ($\mathrm{\Gamma }$, $K$, and $M$) and for each band on validation geometries unknown to the NN. (d) Comparing the $\mathbf{k}$-resolved band gap ${\omega }_{n+1}(\mathbf{k})-{\omega }_n(\mathbf{k})$ for five validation geometries (NN vs Schrödinger). For the last potential, the band-gap zeros are marked by white dots. The crossings of the second and third band define a sextuplet of Dirac cones that are not pinned to any high-symmetry point. In all plots, the height of the potential is $V_{\max }=(16\hslash {)}^2/(2m{a}^2)$.

Figure 2

Designing a band inversion for topological transport using rapid band-structure evaluation and symmetry predictions provided by a neural network. (a) Geometry of the potential: Six circular holes of fixed radius are placed at a varying distance $R$ from the ${\mathcal{C}}_6$ center. (b) Energy spectrum at the $\mathrm{\Gamma }$ point as a function of $R$. The energies of the $p$ and $d$ bands cross for $R=a/3$. (c) Band structure for three different values of $R$ [marked in panel (b) by the horizontal lines] before, at, and after the band-inversion transition (thick lines, NN; thin lines, SEq). The corresponding potentials are shown as insets. At the band-inversion transition, the Wigner-Seitz primitive cell becomes smaller. The resulting folded band structure supports a pair of degenerate Dirac cones at the $\mathrm{\Gamma }$ point. Moreover, two pairs of bands become degenerate along the $\mathbf{k}$ path from $M$ to $K$. These are essential degeneracies enforced by the rotational symmetry and the smaller unit cell. The NN is able to reproduce these features, although it has not been trained on potentials with a smaller unit cell.

Figure 3

Using the NN to explore the topological properties of large sets of potentials. (a) Examples of randomly generated potentials and the band structure predicted by the NN. The tables indicate the irreps predicted by the NN for each of the first six bands. (b) Possible pathway leading to the annihilation of two (one) sextuplets of Dirac cones. For one sextuplet, this process is possible only at $\mathrm{\Gamma }$ or $M$. (c) The EBRs for the wallpaper group $p_6$. The sketches depict the “Wyckoff” positions and point symmetry of the underlying orbitals. The “occupations” $n_i$ in the “symmetry fingerprint” $(n_s,n_p,n_d,n_f,n_0,n_{+})$ count the number of (pairs of) Bloch waves for the corresponding irreps; e.g., $n_p=1$ means that there is one pair of $p$ waves at $\mathrm{\Gamma }$. (d) Distribution of (quasi-) BRs for the first (upper chart) and second (lower chart) set of bands. The light gray slice represents sets of bands that cannot be classified based on the first six bands only but are likely to be composite BRs; see Appendix pp9. All other composite BRs are grouped in the dark gray slice. (e) t-SNE visualization [36], with each point representing a random potential (left) or the corresponding output (TB coefficients) of the NN (right); the latter visualization allows for improved clustering (colors indicate EBRs). (f) Phase map indicating EBR and fragile topology for the first (inset) and the second sets of bands. The parameter space interpolates between the potentials “1,” “2,” and “3” marked in panel (a), in terms of their underlying Fourier coefficients.

Figure 4

Neural-network-based prediction of 3D band structures for potential distributions in the $p{4}_{2}22$ nonsymmorphic space group. (a) Sketch of the symmetry-enhanced TB model. Pairs of orbitals localized about different sublattices are mapped onto each other via screw rotations. Symmetry-related nearest-neighbors transitions are highlighted in the same color. The different orbital types (bottom) transform according to the different representations of the point group ${\mathcal{D}}_2$ (twofold rotations about the axes $x$, $y$, and $z$); see the Appendix. (b) Example of steplike potential with $p{4}_{2}22$ space group. (c) BZ and high-symmetry path for the primitive tetragonal lattice. (d) Band structure along the high-symmetry path. (e) Comparison of NN and Schrödinger equation predictions for the first three band gaps on two different 2D cuts of the BZ. [The results in (d) and (e) refer to the validation potential shown in (b).] (f) Fraction of correctly predicted symmetry labels ordered by band and high-symmetry point.

Figure 5

Using the NN to investigate the spin Chern numbers in a family of 2D topological insulator metamaterials. (a) Sketch of the spin-polarized excitation spectrum in a semi-infinite-plane geometry for a trivial (top) and a topological (bottom) material close to the $\mathrm{\Gamma }$ point, close to the Fermi energy $E_F=0$. The topological material supports gapless edge states (red line). (b) Sketch of a metamaterial geometry. Its unit cell (blue) is subdivided into trivial (gray) and topological (red) domains according to a randomly drawn fourfold rotationally symmetric pattern. (c) Probability field for a metamaterial’s Bloch wave with energy within the single-domain band gap. (d) Metamaterial band structure. Shown are the first (last) four positive (negative) bands together with the symmetry labels and the Chern numbers (in red). Exact results and NN predictions are not distinguishable with bare eyes. (e) Exact and NN-predicted Berry fluxes for the first four bands. [(c)–(e) refer to the validation geometry shown in (b).] (f) Stacked bar chart of the Chern number distribution for the first four positive bands. (g) Scatter plot of the Chern and fingerprint accuracies for 13 trained NNs. Each NN is represented by a red and a blue circle. For the red circles, the rms band-structure deviation (radius of the circles), the fraction of correct fingerprints ($x$ coordinate), and Chern numbers ($y$ coordinate) are obtained, averaging over 2500 validation samples and the eight central bands. The blue circles take into account only bands separated from neighboring bands by a minimal splitting larger than $0.01m$.

Figure 6

Using the NN for gradient-based optimization. (a) Pipeline for finding a physical implementation of a given TB model (unit cell shown). The band structure of the model is provided as a target. A randomly initialized geometry is evolved until the NN-predicted band structure (dark lines) approaches the target (bright lines); see also the animation in the Supplemental Video [60]. We select those random trials that end up at a loss comparable to the NN accuracy itself (blue shaded region). (b) Physically accessible regions of the TB model parameter space: Contours show the minimal loss achieved after 50 trials as a function the parameters (on-site energy ${\omega }_0$, average hopping $\overline{J}=(J+J^{\prime })/2$, hopping difference $\delta J=J^{\prime }-J$). In the blue region, the minimal loss is lower than the network accuracy. (c) Solutions for a more general model, including also next-nearest-neighbor hopping supporting fragile topological phases; see Appendix pp13 for more details. The parameters are $\hslash {\omega }_0=0.6V_{\max }$, $\hslash J^{\prime }=0.038V_{\max }$, $\hslash J=0.035V_{\max }$, $\hslash L^{\prime }=0.002V_{\max }$, $\hslash L=-0.002V_{\max }$.

Figure 7

Details of the NN layout. (a) Sketch of the layout of the neural network. The input image is the unit cell of the potential (the region within the blue contour). The numbers indicate the dimensions of the layers, where for the convolutional the last value is the number of filters. The image is flattened before it is processed by the dense layers. The final dense layer has 245 output neurons encoding the independent coefficients of the TB model. (b) Detailed list of the parameters for the network. “Layer 0” labels the input image.

Figure 8

rms deviation of the individual bands for the NN used to produce Figs. 1, 2, 3 and 6.

Figure 9

(Quasi-) BR distribution calculated using three different methods to determine the bands connectivity. The methods are (from left to right for each entry) (i) standard method of topological quantum chemistry which neglects connections away from high-symmetry points, (ii) taking into account the connections away from high-symmetry points which are predicted by the parity compatibility relation, and (iii) looking for $\pi$ defects in the Berry flux. This method is able to detect all connections (if a fine enough grid is used). The column top1 and top2 refer to the two fragile topological phases discovered by our NN; cf. Fig. 3 of the main text. The standard method would predict other types of topological bands (top. el.). These sets of bands turn out not to be isolated once the connections away from the high-symmetry points are taken into account with the other two methods. All results are calculated using the first six bands. The set of bands that is connected to the seventh band can not be assigned to any (quasi-) BR based on the available information and is grouped in the last entry [tba (to be assigned)].

Figure 10

Comparison of the (quasi-) BR distribution calculated using the NN and the Schrödinger equation. The probability bars for a given (quasi-) BR but different methods are shown side by side. The results obtained using the Schrödinger equation are plotted in slightly darker colors.

Figure 11

(a) Comparison of the (quasi-) BR distributions for two different distributions of potentials. The probability bars for a given (quasi-) BR but different distributions are shown side by side. (b) Number of samples as a function of the filling factor [portion of the unit cell where $V(\mathbf{x})=V_{\max }$] for the two distributions of interest. In all panels, the results referring to the training distribution are plotted in slightly darker colors.

Figure 12

Sketch of the NN layout for the 3D nonsymmorphic case study.

Figure 13

Sketch of the NN layout for the two-component topological insulator case study.

Figure 14

Three different phases for the graphene TB model with next-nearest-neighbor hopping. For $\delta J>0$, $|\delta L|=0$ the folded graphene band structure is split into two kagomelike band structures (central panel). For $\delta L<-\delta J/\sqrt{12}$, after band-inversion transitions at both the $M$ and $K$ points (the orbitals that are exchanged are marked by dashed lines), the lower kagome bands split into a triangularlike $s$ band and a pair of topological bands (left panel). For $\delta L>\delta J/\sqrt{12}$, similar band inversions lead to the splitting of the higher kagome bands into a topological pair of bands and triangularlike $f$ band (right panel).