Does Ising model belong to the field of strongly correlated systems? How to make a judgement that whether a problem is within the field of strongly correlated systems? Do classical problems (not quantum mechanical) belong to this field?
 A: Here is a theoretical viewpoint that is strongly biased by my own research interest. Roughly speaking, I describe a system as strongly correlated if mean-field theory fails to predict the correct result. By mean-field theory, I mean any approximation in which one assumes that some correlation function between two randomly fluctuating quantities can be factorised as
$$ \langle A B \rangle \approx \langle A \rangle \langle B \rangle, $$
where $\langle\cdot\rangle$ denotes a statistical average with respect to a probability distribution or quantum state. If this approximation fails, then statistical or quantum correlations between the observables $A$ and $B$ are too strong to neglect. Obviously, depending on your system and the quantity you are interested in calculating, mean-field theory may or may not predict the correct result. Therefore there is a degree of subjectivity here. 
This idea can apply to quantum many-body systems such as spin systems (e.g. Heisenberg model) and classical stochastic many-body systems (e.g. the simple exclusion process).
