Machine learning non-Hermitian topological phases

Non-Hermitian topological phases have gained widespread interest due to their unconventional properties, which have no Hermitian counterparts. In this work, we propose to use machine learning to identify and predict non-Hermitian topological phases, based on their winding number. We consider two examples—non-Hermitian Su-Schrieffer-Heeger model and its generalized version in one dimension and non-Hermitian nodal line semimetal in three dimensions—to demonstrate the use of neural networks to accurately characterize the topological phases. We show that for the one-dimensional model, a fully connected neural network gives an accuracy greater than 99.9% and is robust to the introduction of disorder. For the three-dimensional model, we find that a convolutional neural network accurately predicts the different topological phases.

Figure 1

Fully connected neural network for the non-Hermitian Su-Schrieffer-Heeger model. A fully connected neural network with two hidden layers was constructed. The hidden layers were comprised of 100 and 32 neurons, respectively. The training set consisted of ${10}^5$ samples. (a) The loss or cost with each epoch of training. The network was tested on a set of ${10}^4$ samples, not seen by the network during the training. (b) The predicted winding number $W_p$ (in blue) and its rounded off value (in pink) for the test set. The network was able to predict with an accuracy of 99.98%. The dashed vertical lines show the analytical value of $t_1$ at the phase transition. Here we have chosen $\gamma =4/3$ and $t_2=1$.

Figure 2

Effect of noise on the predicted winding number. Randomly sampled Hamiltonian from a computation lacks noise. Data collected from experiments would invariably show noise. To simulate this, noise was artificially introduced in the training data. This was achieved by randomly adding a value between $[-0.5t_2,0.5t_2]$ to $h_x$ or $h_y$. The robustness of the network was checked by training the network on training sets with 1%, 5%, and 10% noise and testing it on a separate test set. Predicted winding numbers and their rounded off value, for training with 1% (in magenta), 5% (in red), and 10% (in green) noise, are shown in (a) and (b), respectively. We obtained very high accuracy of 99.92%, 99.90%, and 99.70%, respectively. The vertical dashed lines are the analytically obtained values of $t_1$ at the phase transition.

Figure 3

Learning the non-Bloch winding number for generalized SSH model. The predicted winding number (in blue) and the rounded off value (in pink) for the generalized non-Hermitian SSH model. A total of ${10}^5$ training samples were generated with $t_1\in [-3,3]$, $t_2=1$, $t_3=1/5$, ${\gamma }_1=4/3$, and ${\gamma }_2=0$. The network was tested on ${10}^4$ samples, not seen by the network during training. An accuracy of 99.8% was obtained. The inset shows the generalized Brillouin zone $C_{\beta }$ for $t_1=1.1$.

Figure 4

Convolutional neural network for non-Hermitian nodal line semimetal model. (a) Computed and (b) predicted winding number as a function of $k_x$ and $k_y$, when $m=0.4$, ${\gamma }_z=0.2$, $v_z=B=1$, and $k_z=0$. Areas with winding numbers 0, $-1/2$, and $-1$ are shown in green, blue, and red, respectively. We obtained an accuracy of 99.95%. In (b) the incorrect predictions are marked in magenta. These occur predominantly near the phase boundaries. (c) Schematic of our convolutional neural network with two convolutional layers, with 80 and 64 filters and kernel size of $2\times 2$ and $1\times 1$, followed by a fully connected layer with 20 neurons before the output layer, which was used to predict the winding numbers.