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 we can use a variational method made by Gutzwiller 1 , 2. He introduces this wave function, in the case of a single band system |ΨG⟩=∏i[1+(g−1)ˆDi]|Φ0⟩ where g is a real number between 0 and 1 that plays the role of a variational parameter, |Φ0⟩ is a Slater determinat on which we can apply Wick's theorem and ˆDi=ˆni,↑ˆni,↓ is the double occupation operator.
Here Bunemann et all. extend (2) to a multiband system, and it becomes |ΨG⟩=∏i[1+∑Γ(λi,Γ−1)ˆmi,Γ]|Φ0⟩ where ˆmi,Γ=|Γ⟩i⟨Γ|i=∏σ∈Γˆni,σ∏σ∈¯Γ(1−ˆni,σ) while λi,Γ plays now the role of variational parameters. Γ is the atomic eignestate of the atomic Hamiltonian.
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 assuming that Ψ0 is normalized, we have that the sum of all Wick's contractions of ˆc†i,αˆci,β with ˆP†iˆPi vanishes
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