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There is, of course, an interaction between fermions on nearest-neighbor sites. Since the van-der-Waals interaction between free atoms is short range and the lattice Wannier wave functions are localized this interaction is neglected with respect to all local interactions. In addition, in the DMFT approach employed here such an interation reduces to static, Hartree-type potentials which merely renormalize the chemical potential [47]
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The notation (Tτ b(τ )b†(τ ′))S, where b(τ ), b†(τ ) are bosonic operators in the Heisenberg representation, is meant to imply that the quantum mechanical and statistical averages are calculated within a path integral formalism with the action S. Namely, (Tτ b(τ )b†(τ ′))S = ∫ D [b]b(τ )b*(τ ′) exp(-S [b])/Z, where the functional integral is performed in the space of complex functions b(τ ) which obey periodic boundary conditions b(τ +β) = b(τ ), and Z = ∫ D [b] exp(-S [b]) [53]. In the case of fermions the corresponding functional integrals are performed with Grassmann (anticommuting) variables satisfying anti-periodic boundary conditions
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The notation (Tτ b(τ )b†(τ ′))S, where b(τ ), b†(τ ) are bosonic operators in the Heisenberg representation, is meant to imply that the quantum mechanical and statistical averages are calculated within a path integral formalism with the action S. Namely, (Tτ b(τ )b†(τ ′))S = ∫ D [b]b(τ )b*(τ ′) exp(-S [b])/Z, where the functional integral is performed in the space of complex functions b(τ ) which obey periodic boundary conditions b(τ +β) = b(τ ), and Z = ∫ D [b] exp(-S [b]) [53]. In the case of fermions the corresponding functional integrals are performed with Grassmann (anticommuting) variables satisfying anti-periodic boundary conditions.
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