Sorry if this is a dumb question as I'm just starting out, but in this paper on Twisted Bilayer Graphene, the authors claim to use "standard perturbation theory" to derive equation (2)

$$ H_J = J \sum_{\langle ij \rangle} \sum_{a} \hat T_{i}^a \hat T_{j}^a= J \sum_{\langle ij \rangle} \sum_{a} c^\dagger_{i\alpha} T_{\alpha\beta}^a c_{i\beta }c^\dagger_{j\gamma} T_{\gamma \delta}^a c_{j \delta} $$

from equation (1)

$$ H= - t \sum_{\langle ij \rangle} \sum_{\alpha} c^\dagger_{i\alpha} c_{j\alpha}+ U\sum_{i}\left( \sum_{\alpha} n_{i\alpha} \right)^2 $$

in the large U limit, treating hopping as a perturbation.

Typical book references, e.g. Sachdev's book, don't really talk about doing perturbation theory to get effective Hamiltonians, just states and energy levels. I am hoping someone has some insight on this procedure or a link to a reference.

My best guess so far was trying to tediously use Brillouin-Wigner Perturbation theory to calculate a bunch of states to second order and then write $H_{eff} = \sum_n E_n |n\rangle \langle n|$ but I feel like this can't be the way to do it..

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