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In Sachdev's Quantum Phase Transitions he defines a quantum phase transition in an infinite lattice as the point in which the ground energy as a function of the parameters of the system becomes non-analytic. In the article "Fidelity and quantum phase transitions" by Zhou (doi:10.1088/1751-8113/41/41/412001) he says that a pinch point in the fidelity per site as a function of the parameters is indicative of a quantum phase transition. I can't see the connection between the two. What is the relation between ground state energy and fidelity per site?

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