  We propose a method to directly probe bulk topological quantum numbers in
topological matter by measuring the momentum-space single-particle spectral
function (SPSF). Angle-resolved photoemission spectroscopy (ARPES) can detect
SPSF and is often used to determine the bulk band structure of quantum
materials. Here, we show that while one part of the momentum-space SPSF gives
band structure, it also contains the knowledge of winding and Chern numbers of
various topological materials. For this, we derive SPSF in three different
models of topological systems, such as the Kitaev model of topological
superconductors, the long-range Su-Schrieffer-Heeger model, and the Qi-Wu-Zhang
model and explain how to extract the winding or Chern numbers in different
topological phases from the SPSF. Such information from SPSF seems
experimentally accessible due to the recent advancement of ARPES and scanning
tunneling spectroscopy techniques.
