In these notes here the tight binding model for graphene is worked out.
The tight Binding Hamiltonian is the usual:
$$H=-t\sum_{\langle i,j\rangle}(a_{i}^{\dagger}b_{j}+h.c.)$$
where two different sets of creation/annihilation operators are used because there are 2 different sub lattices in graphene (indicated with A and B).
Then it says at the bottom of page 3 that
It is convenient to write the TB eigenfunctions in the form of a spinor, whose components correspond to the amplitudes on the A and B atoms respectively
So if I understand correctly the wave function is a two component spinor with the first component corresponding to sub lattice A and the second to sub lattice B:
$$\psi=\begin{bmatrix}\psi_{A}(x) \\ \psi_{B}(x)\end{bmatrix} \quad .$$
The problem is that I don't understand the meaning of a wave function like that. In the case of spin for me it makes sense to have a two component wave function:
$$\psi=\begin{bmatrix}\psi_{up}(x) \\ \psi_{down}(x)\end{bmatrix} \quad ,$$ where for example $|\psi_{up}(x)|^2$ gives the probability density of finding the electron at position x with spin up.
But in the previous case what does it mean? Is $|\psi_{A}(x)|^2$ the probability density of finding the electron at x on sub lattice A? I can't make sense of this statement.
Any suggestions on how to interpret that?
Best Answer
The basis here consist of all the lattice sites, which can be label by the position of a unit cell, $x_i$ and the atom in this cell (A or B): $$ \phi_{i,A}(x), \phi_{i,B}(x)\leftrightarrow \phi_{i,\alpha} (i=A,B). $$ The arbitrary wave function than can be written es an expansion in this basis: $$ \psi(x)=\sum_{i,\alpha}c_{i,\alpha}\phi_{i,\alpha} = \sum_{i}c_{i,A}\phi_{i,A}(x) + \sum_{i}c_{i,B}\phi_{i,B}(x)=\psi_A(x) + \psi_B(x) $$ Since the orbitals $\phi_{i,\alpha}$ are localized on the sites, some matrix elements will be non-zero only between $A$ and $B$, while others only between $A$ and $A$ or $B$ and $B$, which makes it convenient to use the matrix notation.
As an instructive example, one could solve a problem of a one-dimensional chain with two identical masses in a unit cell.