  In a recent work [D. Malpetti and T. Roscilde, arXiv:1605.04223] we have
shown that in quantum many-body systems at finite temperature, two-point
correlations can be formally separated into a thermal part, and a quantum part
-- and that generically quantum correlations decay exponentially at finite
temperature, with a characteristic, temperature-dependent quantum coherence
length. The existence of these two different forms of correlation in quantum
many-body systems suggests the possibility of formulating an approximation
which affects quantum correlations only, without preventing the correct
description of classical fluctuations at all length scales. Focusing on lattice
boson and quantum Ising models, we make use of the path-integral formulation of
quantum statistical mechanics to introduce such an approximation -- that we dub
\emph{quantum mean-field} (QMF) approach, and which can be readily generalized
to a cluster form (cluster QMF or cQMF). The cQMF approximation reduces to
cluster mean-field theory at $T=0$, while at any finite temperature it produces
a family of systematically improved, semi-classical approximations to the
quantum statistical mechanics of the lattice theory at hand. Contrary to
standard MF approximations, the correct nature of thermal critical phenomena is
captured by any cluster size. In the two exemplary cases of the two-dimensional
quantum Ising model and of two-dimensional quantum rotors, we study
systematically the convergence of the cQMF approximation towards the exact
result, and show that the convergence is typically linear or sub-linear in the
{ boundary-to-bulk ratio} of the clusters as $T\to 0$, while it becomes faster
than linear as $T$ grows. These results pave the way towards the development of
semi-classical numerical approaches based on an approximate, { yet}
systematically improved account of quantum correlations.
