This is a follow up on a previous post which discussed the strange fact that in quantum many-body theory certain limits (e.g., zero frequency, and the long wavelength) do not necessarily commute.
Today I learnt that in 1960 Kohn and Luttinger pointed out that the following definitions of the ground state energy are not necessarily the same:
1. The lowest eigenvalue of the Hamiltonian in the thermodynamic limit.
2. The zero temperature limit of the free energy of an infinite system.
This observation was based on an examination of Goldstone's perturbation theory, which is based on 1.
Ward and Luttinger later showed that for spatially isotropic band structures and interactions the two are equivalent.
Metzner and Vollhardt [where I learned all this] claim that for the Hubbard model, to second order in U, the two are equivalent.
Aside: Kohn was very productive in the early 1960's! Another recent post referred to his seminal paper on insulators.