I have a question regarding to how to construct an operator from k.p Hamiltonian. Maybe there are some problems in my understanding, I hope you can point me out and correct my description if I made something wrong below.
We know that the k.p Hamiltonian can be obtained by taking the inner product of the basis function that belongs to the same irreducible representation for a given point group. Another way is to write down the tight binding model explicitly and do the expansion. For example, the Dirac Hamiltonian obtained by expanding the tight binding element in the high symmetry point in graphene. But for the former, I don't how to construct an operator explicitly. As I understand, for a tight binding Hamiltonian, we can monitor how the basis was transformed so that we can write down the explicit matrix representation of a certain symmetry operation. I want to split this into 2 sub-questions.
Like I said above, if the effective k.p Hamiltonian can be obtained by doing an expansion around the high symmetry point in the tight binding element, does it mean that the basis of the k.p Hamiltonian matrix obtained from the point group approach must correspond to some basis in my tight binding matrix? If yes, that means I can use the symmetry operator in my tight binding form to act on the effective k.p form. If no, how can we construct it explicitly?