Is there a standard way for me to isolate 2 of N bands of a general $N\times N$ Hamiltonian? That is, I want to make a $2\times 2$ 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 $4\times 4$ 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.


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