I have a doubt on the constraints for the expecation values obtained by Bünemann et all.
First i want to introduce my notation
To analytically solve a tight-binding model, ˆH=∑i,j,σ,σ′tσ,σ′i,jˆc†i;σˆcj;σ′+∑iˆHi,at
Here Bunemann et all. extend (2) to a multiband system, and it becomes |ΨG⟩=∏i[1+∑Γ(λi,Γ−1)ˆmi,Γ]|Φ0⟩
They realize that in the limit of infinite coordination number, average values on the Gutzwiller wavefunction can be computed exactly if the following constraints are satised ⟨Ψ0|ˆP†iˆPi|Ψ0⟩=1⟨Ψ0|ˆP†iˆPiˆc†i,αˆci,β|Ψ0⟩=⟨Ψ0|ˆc†i,αˆci,β|Ψ0⟩
It is not clear the physical meaning of these constraints, given also Here by Metzner.
Mathematically, applying Wick's theorem to (4)
⟨Ψ0|ˆP†iˆPiˆc†i,αˆci,β|Ψ0⟩=⟨Ψ0|ˆP†iˆPi|Ψ0⟩⟨Ψ0|ˆc†i,αˆci,β|Ψ0⟩+⟨Ψ0|ˆP†iˆPiˆc†i,αˆci,β|Ψ0⟩contractions
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