  Despite rapidly growing interest in harnessing machine learning in the study
of quantum many-body systems, training neural networks to identify quantum
phases is a nontrivial challenge. The key challenge is in efficiently
extracting essential information from the many-body Hamiltonian or wave
function and turning the information into an image that can be fed into a
neural network. When targeting topological phases, this task becomes
particularly challenging as topological phases are defined in terms of
non-local properties. Here we introduce quantum loop topography (QLT): a
procedure of constructing a multi-dimensional image from the "sample"
Hamiltonian or wave function by evaluating two-point operators that form loops
at independent Monte Carlo steps. The loop configuration is guided by
characteristic response for defining the phase, which is Hall conductivity for
the cases at hand. Feeding QLT to a fully-connected neural network with a
single hidden layer, we demonstrate that the architecture can be effectively
trained to distinguish Chern insulator and fractional Chern insulator from
trivial insulators with high fidelity. In addition to establishing the first
case of obtaining a phase diagram with topological quantum phase transition
with machine learning, the perspective of bridging traditional condensed matter
theory with machine learning will be broadly valuable.
