The mean-field approximation is at the heart of our understanding of complex systems, despite its fundamental limitation of completely neglecting correlations between the elementary constituents. In a recent work [Phys. Rev. Lett. 117, 130401 (2016)], 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 quantum correlations are generically found to 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, which we dub 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 sublinear in the boundary-to-bulk ratio of the clusters as $T\ensuremath{\rightarrow}0$, while it becomes faster than linear as
T grows. These results pave the way towards the development of semiclassical numerical approaches based on an approximate, yet systematically improved account of quantum correlations.
The time-dependent mean-field approximation for two dynamically coupled subsystems is extended to include correlation effects between the subsystems, allowing for decorrelation processes to develop in the reduced density matrices. The extended scheme is formulated in terms of the truncation to M terms of the Schmidt decomposition of the full density matrix. This M natural orbitals truncation scheme is compared to the exact numerical solution for a system of two coupled anharmonic oscillators in a factorized initial state. It is found that the approximation M=3 gives a good approximation to the exact results over several characteristic times of the system. \textcopyright{} 1996 The American Physical Society.
I present here some results on the mean field theory approach to the statistical mechanics of a D-dimensional array of Josephson junctions in the presence of a magnetic field. The mean field theory equations are obtained by computing the thermodynamical properties. In the high temperature region in the limit D→∞, where the problem is simplified, this limit defines the mean field approximation. Close to the transition point the system behaves very similar to a particular form of spin glasses, i.e., to gauge glasses. We have noticed that in this limit the evaluation of the coefficients of the high temperature expansion may be mapped onto the computation of some matrix elements for the q-deformed harmonic oscillator. The same arguments can be used to predict the thermodynamical properties in the mean field limit. These results can be extended to the low temperature phase using a conjecture on the equivalence of some system without disorder with appropriate random systems.