Detecting winding and Chern numbers in topological matter using the spectral function

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 different models of topological systems, such as the Kitaev model of topological superconductors, the long-range Su-Schrieffer-Heeger model, the Qi-Wu-Zhang model, the Haldane model on a hexagonal lattice, and a four-band model that is a physical realization of the Kitaev chain, and we 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. While the detection of bulk topological quantum numbers using the SPSF works well for one-dimensional systems, it has certain limitations for higher-dimensional systems, where it must be complementary with another measurement to ensure the matter under investigation is topological, i.e., having a nonzero Chern number.