Tight binding on a square lattice with three orbitals symmetries

Question

I came across a tricky problem while studying tight binding within the second quantization frame:

Consider a square lattice with one atom per unit cell, where each atom has three active hydrogen atom type orbitals with symmetries s, px, and py. Noting $\alpha = (s, p_x, p_y)$ the orbital index, k the lattice momentum and ($c_{k, \alpha}^\dagger c_{k, \alpha}$) the creation and anihilation operators, we consider the the intra-orbital tight binding Hamiltonian $H_{intra} = \sum_{k,\alpha} \epsilon_\alpha(k_x, k_y)c_{k, \alpha}^\dagger c_{k, \alpha}$. What is the expression of $\epsilon_\alpha(k_x, k_y)$?

In this scenario I only consider on-site and nearest neighbor hopping of spin-less electrons. I am not used to work with different orbitals so I'm not really aware of the different symmetries they present. Moreover, I don't really know if using wavefunction is useful in this case since we are already in the Fourier space? I can set different hopping parameters under a global name $t_\alpha$.

Lastly, should I re-write in the Hamiltonian in the first place to then identify the $\epsilon_\alpha(k_x, k_y)$? I would have one term for on-site hopping and one for the different transition possible with the hopping parameters for $x$ and $y$ directions.

Answer 1

I realized I forgot to answer this question, it could be useful to others.

First start with s-orbitals They have a full spherical symmetry so hopping in the x-direction is equivalent to a hopping in the y-direction. Thus the hopping parameter is given by: $$t_{x,s}=t_{y,s}=t$$

For p-orbitals, you cannot use the same symmetry. Instead you need to introduce hopping parameters $t_1$ and $t_2$ such that:

$$t_1 = t_{p_y, p_y}(\hat{y}) = t_{p_x, p_x}(\hat{x})$$

$$-t_2 = t_{p_y, p_y}(\hat{x}) = t_{p_x, p_x}(\hat{y})$$

$t_1$ connects orbitals pointing toward each other (positive amplitude), whereas $t_2$ connects orbitals oriented parallel to one another.

Then you can decompose the Hamiltonian for each possible hopping with the according parameters, first in the real space and then apply a Fourier transform on the creation/anihilation operators to obtain the expression of the energy. So you should have in the end $\epsilon_{k,s}$, $\epsilon_{k,p_x}$ and $\epsilon_{k,p_y}$.