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Recently the possibility of exciton condensates have shown up in some lattice TQFT models we've been studying, and I've been trying to learn more about them.

Suppose that we have a band insulator with band gap $\Delta$. If the binding energy between an electron and hole, $E_b$ is greater than the band gap, i.e. $E_b> \Delta$, then exciton pairs may form spontaneously, depleting the lower parts of the bands.

Is this a phase transition? It would seem to take insulator to insulator unless $\Delta = 0$.

If it is a phase transition, then what symmetry breaks? (Or is there some other, symmetry-protected entanglement developing?). If it's an approximate symmetry (say approximate exciton condensation), then what would the thermodynamic signature be?

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