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I have a tight binding Hamiltonian of a BCC lattice which is a 4x4 matrix in k space (the 4 elements correspond to 4 atoms that are in a unit cell)

I want to expand it for small k's around the symmetry points P or Gamma or H.

I'm looking at a paper by J. L. Ma˜nes, PHYSICAL REVIEW B 85, 155118 (2012), where he is doing this by some kind of Unitary transformation to the Hamiltonian that changes the basis from the four Bloch functions basis to the "symmetry-adapted basis". I didn't find an explanation in the paper to the operator he uses to go from one basis to another, and i don't understand what is this symmetry adapted basis anyway.

If anyone understand this i would be very happy to know :)

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.155118

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# Expanding the Hamiltonian around a point of symmetry

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