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Sachdev begins his book on Quantum Phase Transitions by asserting that, for a system on a finite lattice, the ground state energy of a Hamiltonian H(g) (where g is some coupling) is a smooth, analytic function of g. This paper: https://arxiv.org/pdf/1603.03943.pdf seems to suggest an exception to this rule in quantum optics, and mentions a similar paper dealing with a QPT in the finite Rabi model.

Why must a finite lattice model have a ground state energy which is an analytic function of the coupling? Is Sachdev making some implicit assumptions that are violated in these studies? How does one show in general that a system possessing a level crossing in the thermodynamic limit opens up a spectral gap when taken to a finite lattice?

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