  The DC photocurrent can detect the topology and geometry of quantum materials
without inversion symmetry. Herein, we propose that the DC shot noise (DSN), as
the fluctuation of photocurrent operator, can also be a diagnostic of quantum
materials. Particularly, we develop the quantum theory for DSNs in gapped
systems and identify the shift and injection DSNs by dividing the second-order
photocurrent operator into off-diagonal and diagonal contributions,
respectively. Remarkably, we find that the DSNs can not be forbidden by
inversion symmetry, while the constraint from time-reversal symmetry depends on
the polarization of light. Furthermore, we show that the DSNs also encode the
geometrical information of Bloch electrons, such as the Berry curvature and the
quantum metric. Finally, guided by symmetry, we apply our theory to evaluate
the DSNs in monolayer GeS and bilayer MoS$_2$ with and without inversion
symmetry and find that the DSNs can be larger in centrosymmetric phase.
