  The Landau description of phase transitions relies on the identification of a
local order parameter that indicates the onset of a symmetry-breaking phase. In
contrast, topological phase transitions evade this paradigm and, as a result,
are harder to identify. Recently, machine learning techniques have been shown
to be capable of characterizing topological order in the presence of human
supervision. Here, we propose an unsupervised approach based on diffusion maps
that learns topological phase transitions from raw data without the need of
manual feature engineering. Using bare spin configurations as input, the
approach is shown to be capable of classifying samples of the two-dimensional
XY model by winding number and capture the Berezinskii-Kosterlitz-Thouless
transition. We also demonstrate the success of the approach on the Ising gauge
theory, another paradigmatic model with topological order. In addition, a
connection between the output of diffusion maps and the eigenstates of a
quantum-well Hamiltonian is derived. Topological classification via diffusion
maps can therefore enable fully unsupervised studies of exotic phases of
matter.
