  Bipartite charge fluctuations (BCF) have been introduced to provide an
experimental indication of many-body entanglement. They have proved themselves
to be a very efficient and useful tool to characterize quantum phase
transitions in a variety of quantum models conserving the total number of
particles (or magnetization for spin systems). In this Letter, we study the BCF
in generic one-dimensional $\mathbb{Z}_2$ (topological) models including the
Kitaev superconducting wire model, the Ising chain or various topological
insulators such as the SSH model. The considered charge (either the fermionic
number or the relative density) is no longer conserved, leading to macroscopic
fluctuations of the number of particles. We demonstrate that at phase
transitions characterized by a linear dispersion, the BCF probe the change in a
winding number that allows one to pinpoint the transition and corresponds to
the topological invariant for standard models. Additionally, we prove that a
sub-dominant logarithmic contribution is still present at the exact critical
point. Its quantized coefficient is universal and characterizes the critical
model. Results are extended to the Rashba topological nanowires and to the XYZ
model.
