# Tight-binding model for decorated square lattice

How do I go about determining the tight-binding Hamiltonian for the crystal structure below? I have identified the primitive lattice vectors $$\mathbf{a}_1=(a,0)$$ and $$\mathbf{a}_2=(0,a)$$ for lattice constant $$a$$ and labelled the three basis atoms. All the atoms are identical and have one s-orbital each. All nearest-neighbour distances are equal.

It is assumed that the diagonal elements of the Hamiltonian are zero. Also, the parameter for nearest-neighbour hopping is given as $$t=\langle \phi|H|\phi\rangle$$ where $$|\phi\rangle$$ is an s-orbital. I understand the derivation for the square lattice, but the addition of a basis has confused me.

• What's your question? Do you want to write out a Hamiltonian or what? Nov 22, 2021 at 3:17
• @RoderickLee Exactly yes Nov 22, 2021 at 3:20
• isn't that just hoping terms with every pair of connected sites? And since there are three basis atoms, you can write the hopping term individually. Nov 22, 2021 at 3:26
• see my answer below Nov 22, 2021 at 3:28

$$H=-t\sum_{i,j}c_{(i,j),1}^\dagger c_{(i,j),2}+c_{(i,j),1}^\dagger c_{(i,j),3}+c_{(i,j),2}^\dagger c_{(i+1,j),1}+c_{(i,j),3}^\dagger c_{(i,j+1),1}+\mathrm{h.c.}$$ $$i$$ for $$x$$-index and $$j$$ for $$y$$-index.