In this case, ρ ρ α α β β ( ω ω ) = 1 Z ∑ ∑ m , n 2 π π δ δ ( E n − − E m − − ω ω ) ⟨ ⟨ m ∣ ∣ ψ ψ α α ∣ ∣ n ⟩ ⟩ ⟨ ⟨ n ∣ ∣ ψ ψ β β † † ∣ ∣ m ⟩ ⟩ ( e − − β β E m − − ζ ζ e − − β β E n ) , where m and n are many-body states.