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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?

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    $\begingroup$ Typically proximity to a critical point (that is associated with a phase transition) is available in the scaling behavior of observables. The scaling law is controlled by the critical point, and it doesn't care about the finiteness of the system (the finiteness would appear as a scale, say $L$, in these scaling laws). $\endgroup$ – vik Aug 14 at 2:26
  • $\begingroup$ @vik , thanks! Is there a theoretical reason why such scaling laws don't care about the finiteness of the system undergoing the transition (that is, has it been rigorously shown that scaling behaviour still indicates a phase transition in finite systems)? Or is it just a heuristic that experimentalists use? $\endgroup$ – user138901 Aug 15 at 22:11
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    $\begingroup$ Yes, the size of the system may be thought of as a infra-red (low energy) cutoff, just what inverse-temperature does for time. Now it should be possible to continue using the continuum theory's scaling laws with the system-size featuring as a new infra-red scale. $\endgroup$ – vik Aug 17 at 17:40
  • $\begingroup$ Thanks again! I've found a bunch of references. If there's any references in particular that you'd recommend (particularly any that provide theoretical justification), let me know. Otherwise feel free to submit an answer and I'll accept it! $\endgroup$ – user138901 Aug 18 at 14:08
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    $\begingroup$ These issues are usually discussed under "finite size scaling", which is important for numerically accessing the thermodynamic limit above 1D. This [wwwhome.lorentz.leidenuniv.nl/~zuiden/CP_Website/2015/c8.pdf] appears to convey the basic idea. What I was noting earlier is entirely informed by these considerations. Hope it helps. $\endgroup$ – vik Aug 20 at 4:37

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