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The discovery of topological features of quantum states plays an importantrole in modern condensed matter physics and various artificial systems. Due tothe absence of local order parameters, the detection of topological quantumphase transitions remains a challenge. Machine learning may provide effectivemethods for identifying topological features. In this work, we show that theunsupervised manifold learning can successfully retrieve topological quantumphase transitions in momentum and real space. Our results show that theChebyshev distance between two data points sharpens the characteristic featuresof topological quantum phase transitions in momentum space, while the widelyused Euclidean distance is in general suboptimal. Then a diffusion map orisometric map can be applied to implement the dimensionality reduction, and tolearn about topological quantum phase transitions in an unsupervised manner. Wedemonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model,the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, andfurther provide implications and demonstrations for learning in real space,where the topological invariants could be unknown or hard to compute. Theinterpretable good performance of our approach shows the capability of manifoldlearning, when equipped with a suitable distance metric, in exploringtopological quantum phase transitions.

Topological quantum phase transitions retrieved through unsupervised machine learning