Is the Rayleigh–Schrödinger perturbation theory ever useful for a many-body system? The Rayleigh-Schrodinger perturbation theory is introduced in every textbook on quantum mechanics. It seems that it can yield accurate results for many single-particle systems. Actually, in most  examples in the textbooks, the model considered is a single-particle one.
I am wondering whether the perturbation theory is useful for many-body systems. In the theory, it is assumed that the perturbed eigenstate is somehow close to the unperturbed one, i.e., their overlap should be close to 1. However, two many-body wave function generally are very close to being orthogonal.
 A: Textbook perturbation theory can be used for many-body systems just as well as for one-particles systems, except that one needs to calculated the matrix elements of the Hamiltonian and perturbation in respect to many-body states. This can be done both in real space (e.g., calculating the matrix elements between electronic Slater determinants) or in the Fock space (where the operators and the wave functions are those in second quantized form). This however remains cumbersome, and since about 1950s quantum field theory language and diagrammatic expansions became commonplace. Importantly, RSPT (and related Brillouin-Wigner PT) remains the language that is frequently used to understand and interpret the results obtained by more sophisticated methods.
Examples:

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*Kondo's original paper on the Kondo effect is a good example of using related Brillouin-Wigner perturbation theory (essentially the version of RSPT for scattering problems) to calculate the scattering from magnetic impurities up to the third order (nobody bothered to look at the third order before him - perhaps because working with the Slater determinants is hard).

*Fetter&Walecka in their classical book provide very instructive calculation of the electron gas ground state energy using the Reyleigh-Schrödinger perturbation theory (and, if I am not mistaken, reproducing it later with the Goldstone expansion and/or Feynman-Dyson one.)

A: As mentioned in the previous answer, we can still use perturbation theory for many-body systems. I would like to give a concrete example. In general, the Hamiltonian of an interacting Fermion system can be given by $$H=H_0+H'=\sum_{\mu\nu}\langle\mu|\hat{h}|\nu\rangle c^{\dagger}_\mu c_\nu+\frac{1}{2}\sum_{\mu \nu \alpha \beta}\langle\mu \nu|\hat{V}|\alpha \beta \rangle c^{\dagger}_\mu c^{\dagger}_{\nu} c_\beta c_\alpha $$ If the interaction term is weak enough, we can treat it as a perturbation and apply first-order perturbation theory to the interaction term, i.e. the first-order energy correction is given by $\langle H'\rangle=\langle \Omega_0|H'|\Omega_0\rangle$ where $|\Omega_0\rangle$ is the non-interacting many-body ground state(which is just a filled Fermi sea up to the Fermi surface). Now the problem is to compute $\langle \Omega_0|c^{\dagger}_\mu c^{\dagger}_{\nu} c_\beta c_\alpha|\Omega_0\rangle$. Note that for this matrix element to be non-zero, $\alpha \neq \beta$. Also, we should have $\alpha=\nu$ and $\beta=\mu$ or $\alpha=\mu$ and $\beta=\nu$. Then we get $$\langle \Omega_0|c^{\dagger}_\mu c^{\dagger}_{\nu} c_\beta c_\alpha|\Omega_0\rangle=(\delta_{\alpha \mu}\delta_{\beta \nu}-\delta_{\alpha \nu}\delta_{\beta \mu})n_\alpha n_\beta$$. The minus sign comes from the anti-commutation rule for fermions. By substituting this into $\langle H' \rangle$ we get $$\langle H' \rangle = \frac{1}{2}\sum_{\mu \nu \alpha \beta}\langle\mu \nu|\hat{V}|\alpha \beta \rangle (\delta_{\alpha \mu}\delta_{\beta \nu}-\delta_{\alpha \nu}\delta_{\beta \mu})n_\alpha n_\beta = \frac{1}{2}\sum_{\mu \nu}(\langle\mu \nu|\hat{V}|\mu \nu \rangle -\langle\mu \nu|\hat{V}|\nu \mu \rangle)n_\mu n_\nu$$
Note that the first-order perturbation here corresponds to the Hartree-Fock approximation, with the first term being the Hartree term and second term the Fock term.
