I have a very naive question : given a tight-binding model (with nearest-neighbor hoping) on a lattice defined by a Bravais lattice with a number of sublattices (for instance the honeycomb lattice is a triangular lattice with two sublattices), why is there a band associated with each sublattice ? For instance when a lattice is defined by a Bravais lattice with a 2 sublattice basis, the tight-binding model will have 2 bands.. But why is that ? There must be something very simple that I am not getting here... is it just because that to define a band structure, you need some translation invariance ?


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Think of a tight binding model without hopping. This situation would even be accurate for a bulk where the distance between atoms is very large. What do we have in this situation when we look at the energys of the atoms?

For simplicity lets only look at the s-orbitals and 2 sublattices A and B (= 2 Atoms in the unit cell) - you would find 2 different discrete energies: The energy of a s-orbital of the type A and the energy of a s-orbital of the type B. You only have to look into the unit cell to get that. And since the unit cell has 2 atoms in it there are two energys. If you would include p-orbitals, you would add 3 p-energies of the A type and 3 of the B type and so on...

Now "turn on" the hopping (doesn't matter how much neighbors you include). The interaction between the atoms now forms the bands, which means that the energies we have talked about previousely smear to the bands. So the actual number of bands equals the number of discrete energys you would have without interaction.

Another way to look at it would be the Hamiltonmatrix, of which the eigenenvalues are the energies we look for: When you have one atom in the unit cell and 4 orbitals (e.g. s,p$_x$,p$_y$,p$_z$) you have a 4x4 Matrix. A 4x4 matrix has 4 eigenvalues, which means 4 bands. So if you had 2 atoms in the unit cell with the same amount of orbitals you'd have an 8x8 matrix, and therefore 8 eigenvalues (= 8 bands).

The number of bands you get in a tight binding model is given by: $$ n_{\text{bands}} = n_{\text{orbitals}}*n_{\text{atoms in unit cell}} $$


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