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I was wondering whether the concepts of Fermi Level or chemical potential make any sense in the context of many-body problems. I understand that when one is working with a one-electron hamiltonian, the Fermi level is just the energy of the last occupied state, or, if there's a finite temperature, some energy between the HOMO and LUMO. But if we are dealing with a hamiltonian with interactions (briefly put, we have terms that look like $\hat{c}^\dagger \hat{c}^\dagger \hat{c} \hat{c}$ apart from the one-electron terms $\hat{c}^\dagger \hat{c}$), the one-particle picture is no longer valid: There are no one-electron states. Therefore my question: is it correct to say that the concept of Fermi level only makes sense in the context of one-electron hamiltonians? Or am I missing something? Is there perhaps a more general definition of the notion of Fermi energy, which doesn't require assuming that we are in a one-body picture?

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The Fermi energy $\xi_F$ and the chemical potential $\mu(T)$ are two distinct but related quantities. In principle the particle number is conserved. However it is hard to deal with a system with fixed particle number $\mathcal{N}$. In order to release this constraint we introduce a Lagrange multiplier $\mu(T)$ and allow the particle number to vary. The chemical potential is finally fixed from the particle number conservation equation. From this definition, you see that the chemical potential $\mu(T)$ is a well defined quantity even for interacting systems and is temperature dependent (this is important).

The Fermi energy $\xi_F$, is the chemical potential $\mu(T)$ at $T=0$. This again is a well defined quantity for interacting systems. Note that the chemical potential is temperature dependent but the Fermi energy is not.

In principle according to the definition i just introduced, your are right. The Fermi energy is the energy of the last occupied state, is a valid definition only for non-interacting systems. The reason why it is also used for interacting systems, is due to a the Fermi-liquid-theory introduced by Landau. Landau argued that at zero temperature, if some conditions are satisfied an system of interacting fermions behaves as non-interacting system but with renormalized parameters. Such systems are called Fermi liquids.

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