With increasing U/t there is a first-order transition from a metal to an insulator.

This leads to a discontinuity in the double occupancy at the transition, illustrated in the sketch above.

The double occupancy D is shown below versus U/t for t'=0.8t. For reference D=0.25 for a half-filled system at U=0.

The figure is taken from a 2008 PRL by Ohashi et al.

D is calculated with Cluster Dynamical Mean-Field Theory (DMFT)

1. A rough estimate of the magnitude of D can be found from the Hellman-Feynman theorem D= dE_0/dU where E_0 is the ground state energy.

In the Mott phase this is dominated by the antiferromagnetic Heisenberg exchange J ~ 4t^2/U. Hence, D ~ (t/U)^2

2. The discontinuity in D at the metal-insulator transition is relatively small, being about a 15 per cent change for T=0.1t, and less at higher temperatures. To me this suggests that in some sense the character of the metallic and insulating phases near the transition are not that different, just like a liquid and gas are hard to distinguish near the critical point.

3. These results are in contrast to Brinkmann-Rice theory [which ignores J] which gives D=0 in the Mott phase.

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