  Measuring physical observables requires averaging experimental outcomes over
numerous identical measurements. The complete distribution function of possible
outcomes or its Fourier transform, known as the full counting statistics,
provides a more detailed description. This method captures the fundamental
quantum fluctuations in many-body systems and has gained significant attention
in quantum transport research. In this letter, we propose that cusp
singularities in the full counting statistics are a novel tool for
distinguishing between ordered and disordered phases. As a specific example, we
focus on the superfluid-to-Mott transition in the Bose-Hubbard model and
introduce $Z_A(\alpha)=\langle \exp({i\alpha \sum_{i\in
A}(\hat{n}_i}-\overline{n}))\rangle $ with $\overline{n}=\langle n_i \rangle$.
Through both analytical analysis and numerical simulations, we demonstrate that
$\partial_\alpha \log Z_A(\alpha)$ exhibits a discontinuity near $\alpha=\pi$
in the superfluid phase when the subsystem size is sufficiently large, while it
remains smooth in the Mott phase. This discontinuity can be interpreted as a
first-order transition between different semi-classical configurations of
vortices. We anticipate that our discoveries can be readily tested using
state-of-the-art ultracold atom and superconducting qubit platforms.
