The function BlockBandDiagonalize() can be used to reduce the number of basis (spin-)orbitals by making linear combinations of (spin-)orbitals, according to the tight-binding structure (hopping matrix elements) within the (spin-)orbitals. As a simple example to make the idea clear, consider the following 3-by-3 matrix: $$ M = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} $$ Now, assuming that $M$ corresponds to a tight-binding Hamiltonian defined on some basis, we can linearly combine the second and third basis orbitals, such that we get a single orbital which mix with the first orbital via the matrix M. Consider the following unitary rotation matrix: $$ U = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix} $$ Now, transforming the matrix $ M $ using the unitary matrix $ U $ results in: $$ M' = U M U^{T} = \begin{pmatrix} 0 & \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$ In the new representation, the first basis orbital only mixes with the second The basis orbital and not with the third one. The function BlockBandDiagonalize() can accept 3 types of objects as an arguments: Tight-binding object, Operator, or Matrix.
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