How to make a $2times 2$ Hamiltonian using any $2$ levels of an $N$-level Hamiltonian?

Question

Is there a standard way for me to isolate 2 of N bands of a general $Ntimes N$ Hamiltonian? That is, I want to make a $2times 2$ Hamiltonian given a larger one. I was told that there is a general method called downfolding for effective Hamiltonians in condensed matter physics, but to my understanding, these project out parts of the Hamiltonian using various approximations to prioritize more significant physical effects (based off energy, some bias potential, symmetry, etc).
But, I am curious as to what tools I need to study any 2 levels of a larger Hamiltonian. For example, if I have a $4times 4$ Hamiltonian, is there a framework I can use to look only at various pairs of energy levels? That is, any of only levels {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, or {3,4}? I apologize if this is a simple question, but I was looking for a starting point (perhaps a good reference, key words I can use, and introductory steps?) for me to learn such techniques.

Jan Cillié Louw · Accepted Answer

Actually, downfolding (see section 5.2) on its own does not require any approximations. It also does what you are looking for. One defines two projectors $P, Q$ onto sub-Hilbert spaces, where we are interested in the subspace corresponding to $P$. Together $P+Q = 1$, i.e., they together project onto the full Hilbert space. One then gets an effective Hamiltonian, which can be quite complicated if the states in the two subspaces are coupled, which describes the dynamics of the bands you are interested in. Focusing on a particular eigenvalue $omega$ of $H$, the effective Hamiltonian can be expressed as
$$H_{text{eff}}(omega) = P H P + P H Q frac{1}{omega-QHQ}QHP = P H P + omega^{-1} P H Q HP + omega^{-1} O(omega /QHQ),$$
where we have expanded the Geometric series. Where we have used big Big O notation. This holds well when focusing on low energy region.

How to make a $2times 2$ Hamiltonian using any $2$ levels of an $N$-level Hamiltonian?

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