Excitation spectrum in BCS theory and mean field theory I've recently been learning about the BCS theory of superconductivity. An extremely rough idea is as follows: given the interacting BCS Hamiltonian
$$
H = \sum_{\vec{k}\sigma} \xi_{\vec{k}} c^{\dagger}_{\vec{k}\sigma} c_{\vec{k}\sigma} + \sum_{\vec{k}, \vec{k}'} V_{\vec{k} \vec{k}'} c^{\dagger}_{\vec{k} \uparrow} c^{\dagger}_{-\vec{k} \downarrow} c_{-\vec{k}' \downarrow} c_{\vec{k}' \uparrow} ,
$$
we apply a mean-field approximation by replacing in the interaction term (schematically)
$$
c^{\dagger} c^{\dagger} c c \rightarrow \langle c^{\dagger} c^{\dagger} \rangle c c + c^{\dagger} c^{\dagger} \langle c c \rangle - \langle c^{\dagger} c^{\dagger} \rangle \langle c c \rangle
$$
The resulting Hamiltonian is  quadratic and therefore solvable. We find, in particular, that the mean-field Hamiltonian obtains a gap and that the excitations above the gap (known as Bogoliubov quasiparticles) are superpositions of particle and hole. A complete mean-field solution requires solving the self-consistency condition, known as the gap equation in BCS theory.
Onto the question: to verify whether the above mean field theory gives a reasonable picture of the ground state is easy: you can simply calculate the energy expectation value of the mean-field ground state and find that it is indeed lower in energy than the Fermi sea, so that the mean field theory is useful as a variational approach. But what about the Bogoliubov quasiparticles? These quasiparticles are commonly discussed as a central qualitative feature of BCS theory. But how can we say that the quasiparticle solutions of the mean field Hamiltonian have anything to do qualitatively with the excitations of the interacting theory? In other words:

*

*How can we quantitatively see that the gap found in the mean field theory implies a gap in the interacting theory?

*How can we see that the Bogoliubov particle-hole superposition picture carries over from the mean field theory to the interacting theory?

Clearly, the question here is more  general than just for BCS theory. The more general question is: what information, quantitative or qualitative, can we reliably infer from an interacting theory from its mean field theory? Perhaps this question might be easier to answer from the path integral perspective, where we have a systematic way of going beyond mean field theory by computing fluctuations about the saddle point. But from the Hamiltonian formalism, it seems very hard to say anything with confidence beyond possibly the structure of the ground state.
I asked my professor about this, and his argument was in terms of the renormalization group. Roughly, the claim was that mean field theory (when it works) identifies for you possible fixed points of the RG for the original Hamiltonian. If you choose the "right" mean field theory, then the Hamiltonian and all its eigenstates will be adiabatically connected to those of the mean-field Hamiltonian, and you can indeed infer the structure of the excitation spectrum from mean field theory. Hopefully I'm not misquoting his answer, but in either case it feels like killing a fly with a sledgehammer. Surely there ought to be a more direct way to see that mean field theory can faithfully describe the original Hamiltonian before appealing to deep RG arguments.
 A: Thanks to the OP for this great question.
Applications of the mean field (MF) theory and the theory itself, seem to be an overkill, but on deeper thought they are simply techniques to construct tractable dynamics for various systems. Although I do not possess enough background on the BCS theory or superconductivity, applying MF theory is a great idea.
A specific answer to this question is in the realm of latest research, so this answer will provide some references to other MF systems, particularly mean field games (MFGs), which have explored the eigen spectrum of the underlying system and applied the path integral (Feynman-Kac lemma) to solve the systems. Overall, these works are using control theory (an extension of Hamiltonian variational formulations of interacting coupled systems of large scale populations with quadratic cost functions) for modeling emergent behavior, such as in superconductivity. Solving the problem in the OP will indeed require delving into various structures of coupling functions (in the cost function in the case of control theory).
In the case of homogeneous flocking, this paper explores the eigenstructure to show it's relationship to the stability of the quadratic MFG model for flocking of birds. This related paper shows the spectral relationship to the quadratic MFG model for agents (micro-states of the populations) with nonlinear Langevin dynamics, but for local but nonlinear interaction coupling functions. This paper shows the direct correspondence between the Schrodinger theory and MFGs by applying the path integral to compute solutions to quadratic MFGs of agents with nonlinear Langevin dynamics and exploring the eigen spectrum of the underlying PDEs via a novel variable transformation or change of variables. Finally, this paper establishes the relationship analytically in the case that the agents in the MFG with nonlinear dynamics are perturbed by state dependent noise.
