  Angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling
microscopy (STM) have become indispensable tools in the study of correlated
quantum materials. Both probe complementary aspects of the single-particle
excitation spectrum. Taken together, ARPES and STM have the potential to
explore properties of the electronic Green's function, a central object of
many-body theory. This review explicates this potential with a focus on
heavy-electron quantum criticality, especially the role of Kondo destruction. A
discussion on how to probe the Kondo destruction effect across the
quantum-critical point using ARPES and STM measurements is presented.
Particular emphasis is placed on the question of how to distinguish between the
signatures of the initial onset of hybridization-gap formation, which is the
"high-energy" physics to be expected in all heavy-electron systems, and those
of Kondo destruction, which characterizes the low-energy physics and, hence,
the nature of quantum criticality. Recent progress and possible challenges in
the experimental investigations are surveyed, the STM and ARPES spectra for
several quantum-critical heavy-electron compounds are compared, and the
prospects for further advances are outlined.
