Unsupervised Learning of Non-Hermitian Topological Phases

Non-Hermitian topological phases bear a number of exotic properties, such as the non-Hermitian skin effect and the breakdown of conventional bulk-boundary correspondence. In this Letter, we introduce an unsupervised machine learning approach to classify non-Hermitian topological phases based on diffusion maps, which are widely used in manifold learning. We find that the non-Hermitian skin effect will pose a notable obstacle, rendering the straightforward extension of unsupervised learning approaches to topological phases for Hermitian systems ineffective in clustering non-Hermitian topological phases. Through theoretical analysis and numerical simulations of two prototypical models, we show that this difficulty can be circumvented by choosing the “on-site” elements of the projective matrix as the input data. Our results provide a valuable guidance for future studies on learning non-Hermitian topological phases in an unsupervised fashion, both in theory and experiment.

Figure 1

A schematic illustration of the 1D non-Hermitian Su-Schrieffer-Heeger (NH SSH) model, and the unsupervised learning of its harboring topological phases based on the diffusion map. Theoretically, with open boundary conditions (OBCs) this model entails two distinct phases with a phase diagram shown in the lower-left subfigure. Samples from the same phase correspond to the same diagonal blocks of the Gaussian kernel matrix and the boundary between two blocks indicates the topological phase transition point.

Figure 2

Numerical results of unsupervised learning without the NHSE for the 1D non-Hermitian SSH [(a)–(c)] and 2D QWZ model [(d)]. (a) Heat map for Gaussian kernel value distribution between samples with varying $t_1$. (b) Eigenvalues of the one-step diffusion matrix $\mathcal{P}$. (c) Scatter diagram of eigenvectors $\{{\psi }_1,{\psi }_2\}$ with the corresponding eigenvalues ${\lambda }_{1,2}\approx 1$, where the samples are clustered into three topological phases. (d) For the QWZ model, the input samples are classified into seven categories [110].

Figure 3

Numerical results of unsupervised learning with the NHSE. (a)–(c) show respectively the heat map of the Gaussian kernel, the eigenvalues of the one-step diffusion matrix, and the scatter diagram of eigenvectors for the non-Hermitian SSH model. The red arrow in (a) indicates the phase transition point $t_1\approx 0.6993$. The input data samples are clustered into two (rather than three) topological categories, in sharp contrast to the case of learning without the NHSE shown in Fig. 2. (d) The eigenvalues of the one-step diffusion matrix for the non-Hermitian QWZ model, with the inset showing the scatter diagram of eigenvectors corresponding to the largest two eigenvalues. The input samples are classified into two (rather than seven) categories [110].