Dispersion relation in tight binding model

[tobix10]

Hamiltonian of tight binding model in second quantization is given as $$H = -t \sum_{<i,j>} a_i^{\dagger} a_j$$
After changing basis it is $$H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}}$$
where $$E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}}$$
where $\vec{b}$ is a nearest neighbour shift vector.

I've used this relation to calculate $E_{\vec{k}}$ for a square lattice, where nearest neighbours are points $(a,0)$, $(-a,0)$, $(0,a)$, $(0,-a)$. The result is $$E_{\vec{k}} = -t[e^{i(k_x a + 0)} + e^{i(- k_x a + 0)} + e^{i(0 + k_y a)} + e^{i(0 - k_y a)}] = -2t (cos k_x a + cos k_y a)$$

For a traingular lattice I set point (0,0) in the middle so neighbours are $(a/2,a \sqrt{3}/2)$, $(-a/2,-a \sqrt{3}/2)$, $(a/2,-a \sqrt{3}/2)$, $(-a/2,a \sqrt{3}/2)$, $(a,0)$, $(-a,0)$. The result is $$-2t (cos k_x a + cos (k_x \frac{a}{2} +k_y \frac{\sqrt{3}}{2}a) + cos (k_x \frac{a}{2} -k_y \frac{\sqrt{3}}{2}a))$$

Are these calculations correct?

 

[eljose79]

What do you think about the results?

I would say that your calculations appear to be correct based on the given information. The results for the square lattice and triangular lattice both follow the same general form of -2t (cos k_x a + cos k_y a), with slight variations due to the different lattice structures. This makes sense because the Hamiltonian is still describing the same physical system, just with different lattice arrangements.

In terms of the results, they show that the energy levels for the square and triangular lattices have a similar dependence on the wavevector k, but with some differences due to the lattice symmetry. For example, the triangular lattice has additional terms that involve both k_x and k_y, whereas the square lattice only has terms involving either k_x or k_y.

Overall, your calculations and results seem reasonable and in line with what is expected for these types of lattice systems. However, it would be helpful to have more information about the specific system and the context for these calculations in order to fully assess their accuracy and significance.