Deep learning topological invariants of band insulators

Abstract

In this work we design and train deep neural networks to predict topological invariants for onedimensional four-band insulators in AIII class whose topological invariant is the winding number, and two-dimensional two-band insulators in A class whose topological invariant is the Chern number. Given Hamiltonians in the momentum space as the input, neural networks can predict topological invariants for both classes with accuracy close to or higher than 90%, even for Hamiltonians whose invariants are beyond the training data set. Despite the complexity of the neural network, we find that the output of certain intermediate hidden layers resembles either the winding angle for models in AIII class or the solid angle (Berry curvature) for models in A class, indicating that neural networks essentially capture the mathematical formula of topological invariants. Our work demonstrates the ability of neural networks to predict topological invariants for complicated models with local Hamiltonians as the only input, and offers an example that even a deep neural network is understandable.Learning topological invariants of these two models is significantly harder than that in Ref. [15], as the mathematical formula of topological invariants in these models are intrinsically more complicated (see Eq.(2) and Eq. (7)) and the sizes of the input data are much larger. Consequently, to guarantee a good performance, neural networks used in this work are much deeper than the one used in Ref. [15]. As shown in Fig. 1, there are more than nine hidden layers in each neural network. Because the neural network becomes more complicated, it becomes more difficult to analyze how the neural network works. Nevertheless, we show that the intermediate output of certain hidden layer is, for case (i) the local winding angle, and for case (ii) the local Berry curvature -both are the integrands in the mathematical formula of the corresponding topological invariant. In this way, we demonstrate that the complicated function fitted by the neural network is essentially the same as the mathematical formula for the topological invariant.The paper is organized as follows. In Section II we train a neural network to learn the winding number of one-dimensional four-band models in AIII class. After introducing the model Hamiltonian and the mathematical formula of the winding number, we present our neural network in detail and report its performance. We then analyze the mechanism of why the neural network works. arXiv:1805.10503v2 [cond-mat.str-el] 9 Jun 2018 * They contribute equally to this work.

Unsupervised learning using topological data augmentation

“…Fig. 10 (right), compared to the explicitly calculated Berry curvature in discretized momentum space [21] (left). These maps are summed up to give the network output and the exact Chern number respectively.…”

“…Once successfully trained on trivial and nontrivial ensembles, we find that these networks can accurately classify any band insulator in the relevant symmetry class by its topological index, as exemplified in The algorithm is bona fide unsupervised learning, relying only on having a formalism for performing continuous, topology-preserving, deformations on a discretized representation of the object. This is the crucial main distinction from the important work by Zhang et al [20] and Sun et al [21] that also use neural networks to calculate the topological index of band insulators. The latter rely on having access to an auxiliary-independent calculator of the index in order to generate the ensembles of labeled training data.…”

“…This is in contrast to the neural networks used for supervised learning of topological indices in Refs. [20,21] where the networks had to learn to integrate the local features. Our network structure provides efficiency by limiting the number of trainable parameters and by learning to calculate the relevant local quantities (see Fig.…”

“…Artificial neural networks (NNs), one of the most efficient and widely used tools of ML [8], are currently at the frontier of research activity. It has been shown that they are capable of predicting phase transitions and critical temperatures [9][10][11][12][13][14][15][16][17][18][19], learning topological indices of quantum phases [20][21][22][23][24][25][26], efficiently representing many-body states [27][28][29][30][31][32][33][34], improving known numerical computational methods [35][36][37], and decoding topological quantum correcting codes [38][39][40][41][42].…”

Multifaceted machine learning of competing orders in disordered interacting systems

“…Topological phases form a central class of such phases. Though there has been recent progress in using machine learning for topological phases 3,4,9,12,15,19,30,31 , these early efforts naturally centered around benchmarking the neural network based approaches to the conventional approaches on established problems by suppressing either disorder or interaction.…”