  The discovery of topological features of quantum states plays an important
role in modern condensed matter physics and various artificial systems. Due to
the absence of local order parameters, the detection of topological quantum
phase transitions remains a challenge. Machine learning may provide effective
methods for identifying topological features. In this work, we show that the
unsupervised manifold learning can successfully retrieve topological quantum
phase transitions in momentum and real space. Our results show that the
Chebyshev distance between two data points sharpens the characteristic features
of topological quantum phase transitions in momentum space, while the widely
used Euclidean distance is in general suboptimal. Then a diffusion map or
isometric map can be applied to implement the dimensionality reduction, and to
learn about topological quantum phase transitions in an unsupervised manner. We
demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model,
the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and
further provide implications and demonstrations for learning in real space,
where the topological invariants could be unknown or hard to compute. The
interpretable good performance of our approach shows the capability of manifold
learning, when equipped with a suitable distance metric, in exploring
topological quantum phase transitions.
