# Critical points of the three-dimensional Bose-Hubbard model from on-site atom number fluctuations. (arXiv:1903.05936v1 [cond-mat.quant-gas])

We discuss how positions of critical points of the three-dimensional

Bose-Hubbard model can be accurately obtained from variance of the on-site atom

number operator, which can be measured through atom-number-projection

spectroscopy. The idea that we explore is that the derivative of the variance,

with respect to the parameter driving the transition, has a pronounced maximum

close to critical points. We show that Quantum Monte Carlo studies of this

maximum lead to precise determination of critical points for the

superfluid-Mott insulator transition in systems with mean number of atoms per

lattice site equal to one, two, and three. We also extract from such data the

correlation-length critical exponent through the finite-size scaling analysis

and discuss how the derivative of the variance can be reliably computed from

numerical data for the variance. The very same conclusions apply to the

derivative of the nearest-neighbor correlation function, which can be obtained

from routinely measured time-of-flight images.