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We can define a gapped quantum system theoretically by placing some conditions on the energy eigenvalues of (the elements of) a sequence of lattice hamiltonians in the thermodynamic limit, cf. this question. This is a theoretical definition, and it's not obvious to me how to convert it into an empirical definition. By an empirical definition, I mean an experiment or class of experiments that we can perform on the system to distinguish a gapped system from a system that is not gapped.

I believe, for example, that in quantum Hall systems, the fact that these systems are gapped corresponds to the incompressibility of the quantum Hall droplet. If we squeeze a quantum Hall droplet a little bit, the density of the droplet will not change. (In the case of a droplet with an edge, this is not really a gapped system, since arbitrarily small squeezes will produce chiral edge waves, but I think we can still think of the bulk as being gapped.) However, if we squeeze hard enough to produce quasiparticles in the droplet, then the density may change. I also think the incompressibility / gap places some constraints on absorption of light by QH systems, although I am less clear on this.

These kinds of experiments, like squeezing or shining light on a physical sample of some condensed matter, are what I have in mind when I ask for an empirical definition. My question is then:

  • Is there a general, empirical definition of a gapped quantum system (or gapped phase of matter) of the form "When we perform experiment X (or class of experiments $\{X_i\}$) on a gapped quantum system, we observe outcome Y (or outcomes $\{Y_i\}$."
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