Hamiltonian of tight binding model in second quantization is given as [tex]H = -t \sum_{<i,j>} a_i^{\dagger} a_j[/tex](adsbygoogle = window.adsbygoogle || []).push({});

After changing basis it is [tex]H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}}[/tex]

where [tex]E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}}[/tex]

where [itex]\vec{b}[/itex] is a nearest neighbour shift vector.

I've used this relation to calculate [itex]E_{\vec{k}} [/itex] for a square lattice, where nearest neighbours are points [itex](a,0)[/itex], [itex](-a,0)[/itex], [itex](0,a)[/itex], [itex](0,-a)[/itex]. The result is [tex] 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) [/tex]

For a traingular lattice I set point (0,0) in the middle so neighbours are [itex](a/2,a \sqrt{3}/2)[/itex], [itex](-a/2,-a \sqrt{3}/2)[/itex], [itex](a/2,-a \sqrt{3}/2)[/itex], [itex](-a/2,a \sqrt{3}/2)[/itex], [itex](a,0)[/itex], [itex](-a,0)[/itex]. The result is [tex] -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))[/tex]

Are these calculations correct?

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# I Dispersion relation in tight binding model

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