As noted by Sachev, and in a previous question, https://www.physicsoverflow.org/41602/, there cannot be quantum phase transitions for finite systems (with bounded local Hilbert space dimension). The reason for this is that for any finite system the Hamiltonian be described by a matrix. As a result, the spectral gap is finite.
However, in reality there are no infinite systems, so is there a good/rigorous definition of a quantum phase transition for finite systems? Experimentalists must look for other quantities to signal a phase transition -- I suspect that they typically look for the correlation functions diverging (or for a finite system I suppose they produce a spike) but is there anything to suggest this is "correct"? There are examples of phase transitions where the this does not happen (e.g. https://arxiv.org/abs/1512.05687). However, does using correlation functions work provided we promise our system satisfies certain conditions/properties? Are there other measures which are guaranteed to tell us something about phase transitions?