ABSTRACT
In two-dimensional time-reversal symmetric topological insulators described by Dirac models, theZ2topological invariant can be described by the spin Chern number. We present a linear response theory for the spin Berry curvature that integrates to the spin Chern number, and introduce its spectral function that can be measured at finite temperature by momentum- and spin-resolved circular dichroism, which may be achieved by pump-probe type of experiments using spin- and time-resolved ARPES. As a result, the sign of the Pfaffian of theZ2invariant can be directly measured. A spin Chern number spectral function is further introduced from the optical spin current response, and is shown to be measurable from the spin-resolved opacity of two-dimensional materials under circularly polarized light, pointing to an optical measurement of theZ2invariant and a frequency sum rule. The spin Chern number expressed in real space is known to yield a spin Chern marker, and we propose that it may be measurable from spin-resolved local heating rate caused by circularly polarized light. A nonlocal spin Chern marker is further proposed to characterize the quantum criticality near topological phase transitions, and is shown to be equivalent to an overlap between spin-selected Wannier states.